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Comportement en grand temps et intégrabilité decertaines équations dispersives sur l’espace de Hardy
Ruoci Sun
To cite this version:Ruoci Sun. Comportement en grand temps et intégrabilité de certaines équations dispersives surl’espace de Hardy. Equations aux dérivées partielles [math.AP]. Université Paris-Saclay, 2020.Français. NNT : 2020UPASS111. tel-03092314
Thès
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tNNT:2020UPA
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Comportement en grand temps etintégrabilité de certaines équationsdispersives sur l’espace de Hardy
Thèse de doctorat de l’université Paris-Saclay
Ecole Doctorale de Mathématiques Hadamard (EDMH) n 574Spécialité de doctorat : Mathématiques fondamentales
Unité de recherche : Université Paris-Saclay, CNRS, Laboratoire demathématiques d’Orsay, 91405, Orsay, France
Référent : Faculté des sciences d’Orsay
Thèse présentée et soutenue à Orsay, le 26 juin 2020, par
Ruoci SUN
Composition du jury :
Sandrine GRELLIER PrésidenteProfesseure, Université d’OrléansErwan FAOU Rapporteur & ExaminateurDirecteur de recherche, (HDR), Université de Rennes 1,École Normale Supérieure de RennesDidier PILOD Rapporteur & ExaminateurProfesseur, University of BergenOana POCOVNICU ExaminatriceProfesseure, Heriot-Watt University, Maxwell Institute forMathematical SciencesFrédéric ROUSSET ExaminateurProfesseur, Université Paris-Saclay
Patrick GÉRARD Directeur de thèseProfesseur, Université Paris-Saclay
1
Remerciements
Je voudrais tout d’abord exprimer ma plus grande reconnaissance envers mon directeur de these, PatrickGerard. Son enthousiasme communicatif, sa rigueur de pensee, sa generosite intellectuelle m’ont pro-fondement marque. Il m’a fait decouvrir des domaines de recherche passionnants. Il m’a soutenu toujourslors de difficultes de recherche et de preparation des articles. Ses explications sont toujours hyper claires etcoherentes. J’ai pu en profiter pendant ces quatre annees de these et je suis conscient que c’est une grandechance. J’ai ete aussi particulierement impressionne par sa capacite a s’interesser a tous les domaines desmathematiques et donc a pouvoir toujours repondre a mes questions quel que soit le sujet. C’est pour moiun modele a suivre. Il m’a aide aussi enormement dans la langue francaise, je lui en suis tres reconnaissant.
Je remercie vivement Erwan Faou et Didier Pilod d’avoir accepte avec obligeance de relire ma these endetail, et d’ecrire sur elle leur rapport. L’attention de mathematiciens de leur envergure, la minutie deleur travail (dont j’ose a peine imaginer le temps qu’il leur a coute), ainsi que la pertinence de leursremarques, sont autant d’honneurs dont je mesure la portee.
Sandrine Grellier, Oana Pocovnicu et Frederic Rousset me font une vraie faveur en donnant de leur tempspour sieger a mon jury : puissent ces mots leur temoigner ma gratitude. La sollicitude des mathematiciensaccomplis pour leurs cadets est pour moi un constant motif d’emerveillement.
Je remercie toute l’equipe analyse numerique et equations aux derivees partielles (ANEDP) pour son ac-cueil et son dynamique. Merci a Frederic Paulin et Stephane Nonnenmacher pour m’accepter d’entrer al’Ecole dotorale de mathematiques Hadamard (EDMH). Je voudrais remercier aussi l’institut de Fields aToronto pour trois semaines extraordinaires que j’y ai passes, et ou pointerent les germes du chapitres 2 a4 du present manuscrit. Je voudrais exprimer mes remerciements a Catherine Sulem et Jean-Claude Sautpour leurs acceuil et pour leurs aides a obtenir mon visa de Canada. Je voudrais remercier Claude Zuilyet Laurent Thomann pour ses invitations aux colloques ’analyse asympototique’ en Italie et a Marseille.Merci egalement a David Dos Santos Ferreira, a Frederic Rousset et aux tous les organisateurs(trices)des conferences Journees EDPs qui portent toujours des exposes magnefiques chaque annee. Merci aHerbert Koch pour son accueil a Bonn en janvier 2020. Je voudrais remercier egalement Sijue Wu pourson invitation a Michigan, qui me permet de presenter mes resultats par poster.
Les voyages effectues au cours de ma these ont beaucoup stimule ma creativite. Merci a tous ceux quiles ont finances, en particulier, a l’EDMH, a l’ANR ANAE et a l’equipe ANEDP. Je voudrais exprimermes remerciements a Marie-Christine Myoupo, Clotilde d’Epenoux, Laurie Vincent, Severine Simon,Florence Rey, Estelle Savinien et Ophelie Molle pour leurs tres grande efficacite pour des affaires ad-ministratives. Merci a Mathilde Rousseau, Laurent Dang et le service informatique a LMO pour leursaides a la visio-conference de ma soutenance et a resoudre des problemes de l’ordinateur dans mon bureau.
Je profite aussi de l’occasion pour remercier tous les professeurs que j’ai pu avoir dans ma scolarite etqui m’ont donne envie de faire des mathematiques, specialement a l’Ecole Normale Superieure (ENS),a l’Universite de Paris-Sud 11, a l’Universite de Paris-Dauphine et a l’Universite Paris VI . Je voudraisd’abord remercier mes trois tuteurs a l’ENS: Bastien Mallein, Bertrand Maury et Cyril Imbert pourleurs guides et soutiens sur ma scolarite a l’ENS. Je voudrais exprimer mes grands remerciements a mesprofesseurs Thomas Alazard, Scott Armstrong, Nalini Anantharaman, Patrick Bernard, Olivier Biquard,Yann Brenier, Nicolas Burq, Raphael Cerf, Jean-Yves Chemin, Olivier Debarre, Thomas Duquesne,Jacques Fejoz, Patrick Gerard, Ilia Itenberg, Christophe Lacave, Nicolas Lerner, Mathieu Lewin, Yvan
Martel, Stephane Mischler, Stephane Nonnenmacher, Frederic Rousset, Eric Sere, Sylvia Serfaty, Em-manuel Trelat, Claude Viterbo, etc pour leur cours excellents que j’ai suivis pendant mes scolarite deLicence 3 au Master 2, qui m’ont incite a choisir EDPs hamiltoniennes comme domaine de recherche.
Je remercie egalement a Guy David, Maria Paula Gomez Aparicio, David Harari et Hans Henrik Rughpour me donner la chance d’enseigner les TDs a l’Universite Paris-Sud. L’experience de 350 heures deTDs sur mathematiques me permet de toucher d’autre domaines mathematiques dehors analyse des EDPs.
La recherche etant bien plus interessante lorsqu’elle est collective, je suis tres heureux d’avoir discutemon sujet avec Piotr Bizon, Walter Craig, Jean-Marc Delort, Alessio Figalli, Benoıt Grebert, ZaherHani, Slim Ibrahim, Thomas Kappeler, Joachim Krieger, Tai-Ping Liu, Fabricio Macia, Nader Mas-moudi, Clement Mouhot, Sohrab Shahshahani, Avraham Soffer, Gigliola Staffilani, John Francis Toland,Nikolay Tzvetkov, Michael Weinstein, Ping Zhang etc. Les discussions avec eux m’ont aide beaucoupdans ma recherche. Je les suis tres reconnaissant.
J’ai eu la chance de rencontrer des jeunes edpistes du monde: Siddhant Agrawal, Leo Bigorgne, CharlesCollot, Elek Csobo, Stefan Czimek, Yu Deng, Thibault de Poyferre, Anne-Sophie de Suzzoni, ChenjieFan, Francesco Fanelli, Olivier Graf, Chenlin Gu, Jiao He, Jiaxi Huang, Cecile Huneau, Maxime Ingre-meau, Jacek Jendrej, Yang Lan, Camille Laurent, Hugo Lavenant, Zongyuan Li, Xian Liao, BaopingLiu, Jiaqi Liu, Shuang Miao, Alexis Michelat, Quang Huy Nguyen, Chenmin Sun, Pierre Roux, JulienSabin, Annalaura Stingo, Qingtang Su, Joseph Thirouin, Victor Vilaca da Rocha, Yuexun Wang, AllenYilun Wu, Shengquan Xiang, Haiyan Xu, Pin Yu, Xu Yuan, Haitian Yue, Katherine Zhiyuan Zhang,Lei Zhao, Zhiyan Zhao, Zehua Zhao, Jiqiang Zheng, Hui Zhu etc. Merci a eux pour leur conversation,mathematique ou non, qui m’a eleve l’esprit et l’ame.
Je suis tres reconnaissant a mes collegues a l’Universite Paris-Sud, qui ont cre une atmosphere amicaletres agreable, tout particulierement Yang Cao, Zhangchi Chen, Linxiao Chen, Clementine Courtes, Lu-cile Devin, Wei-Guo Foo, Agnes Gadbled, Ning Guo, Weikun He, Zhizhong Huang, Magda Khalile, YasirAmmar Kilic, Camille Labourie, Thomas & Luc Lehericy, Bingxiao Liu, Sasha Minet, Claire Brecheteau,Jeanne Nguyen, Jingrui Niu, Davi Obeta, Gabriel Pallier, Anthony Preux, Yi Pan, Zicheng Qian, CagriSert, Julien Sedro, Changzhen Sun, Salim Tayou, Yisheng Tian, Simeng Wang, Xiaozong Wang, Bo Xia,Songyan Xie, Cong Xue, Daxin Xu, Yeping Zhang.
Je suis tres heureux de rencontrer mes camarades a l’ENS: Elie Casbi, Kaitong Hu, Jialun Li, ShengyuanZhao, Linyuan Liu, Shinan Liu, Disheng Xu, Lizao Ye, Ruxi Shi, Zhihao Duan, Hugo Federico, Censi Li,Tristan Ozuch-Meersseman, Eliot Pacherie, Jiaxin Qiao, Yichen Qin, Julien Sazadaly, Aurelien Velleret,Hua Wang, Yilin Wang, Mingchen Xia, Junyi Zhang, Yi Zhang, Tunan Zhu, Peng Zheng, etc. Leursbrillances exceptionnelles m’ont impressione beaucoup. Je les suis aussi reconnaissant.
Finalement, je voudrais exprimer mes plus grands remerciements a ma famille qui me soutient en tout letemps.
1
Resume On s’interesse dans cette these a trois modeles d’equations hamiltoniennes dispersives nonlineaires : l’equation de Schrodinger cubique defocalisante filtree par le projecteur de Szego ΠT, qui enlevetous les modes de Fourier strictement negatifs, sur le tore T := R/2πZ (NLS–Szego cubique), l’equationde Schrodinger quintique focalisante filtree par le projecteur de Szego ΠR sur la droite R (NLS–Szegoquintique) et l’equation de Benjamin–Ono (BO) sur la droite. Comme pour les deux modeles precedents,l’equation de BO peut encore s’ecrire sous la forme d’une equation de Schrodinger quadratique filtree parle projecteur de Szego ΠR. Ces trois modeles nous donnent l’occasion d’etudier les proprietes qualitativesde certaines ondes progressives, le phenomene de croissance de normes de Sobolev, le phenomene de dif-fusion non lineaire et certaines proprietes d’integrabilite de systemes dynamiques hamiltoniens. Le but decette these est de comprendre l’influence des operateurs non locaux ΠT et ΠR sur des equations de typede Schrodinger et d’adapter les outils lies a l’espace de Hardy sur le cercle et sur la droite. On appliqueaussi la methode de forme normale de Birkhoff, l’argument de concentration–compacite, qui est precisea travers le theoreme de decomposition en profils, et la transformee spectrale inverse pour resoudre cesproblemes. Dans le troisieme modele, la theorie de l’integrabilite permet de faire le lien avec certainsaspects algebriques et geometriques.
Mots− clefs : Equation de Schrodinger non lineaire, Projecteur de Szego, Equation de Benjamin–Ono,Espace de Hardy, Stabilite orbitale, Onde progressive, Turbulence d’onde, Forme normale de Birkhoff,Concentration–compacite, Seuil de diffusion, Multi-soliton, Paire de Lax, Coordonnees d’action–angle,Operateur de Toeplitz
Abstract We are interested in three non linear dispersive Hamiltonian equations : the defocusing cu-bic Schrodinger equation filtered by the Szego projector ΠT that cancels every negative Fourier modes,leading to the cubic NLS–Szego equation on the torus T := R/2πZ ; the focusing quintic Schrodingerequation, which is filtered by the Szego projector ΠR, leading to the quintic NLS–Szego equation on theline R ; and the Benjamin–Ono (BO) equation on the line. Similarly to the other two models, the BOequation on the line can be written as a quadratic Schrodinger-type equation that is filtered by the Szegoprojector ΠR. These three models allow us to study their qualitative properties of some traveling waves,the phenomenon of the growth of Sobolev norms, the phenomenon of non linear scattering and some pro-perties about the complete integrability of Hamiltonian dynamical systems. The goal of this thesis is toinvestigate the influence of the non local operators ΠT and ΠR on some one-dimensional Schrodinger-typeequations and to adapt the tools of the Hardy space on the torus and on the line. We also use the Birkhoffnormal form transform, the concentration–compactness argument, refined as the profile decompositiontheorem, and the inverse spectral transform in order to solve these problems. In the third model, theintegrability theory allows to establish the connection with some algebraic and geometric aspects.
Keywords : Non linear Schrodinger equation, Szego projector, Benjamin–Ono equation, Hardy space,Orbital stability, Traveling wave, Wave turbulence, Birkhoff normal form, Concentration–compactness,Scattering mass threshold, Multi-soliton, Lax pair, Action–angle coordinates, Toeplitz operator
2
Cette these est dediee a ma famille.
Table des matieres
1 Introduction generale 5
1.1 Presentation du contexte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.1 L’equation de Szego sur le tore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.2 L’equation de Szego sur la droite . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2 Enonces des resultats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.1 L’equation de NLS–Szego cubique sur le tore . . . . . . . . . . . . . . . . . . . . . 14
1.2.2 L’equation de NLS–Szego quintique sur la droite . . . . . . . . . . . . . . . . . . . 17
1.2.3 L’equation de Benjamin–Ono sur la droite . . . . . . . . . . . . . . . . . . . . . . . 21
1.3 Perspectives de recherche . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.3.1 L’equation de NLS–Szego amortie . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.3.2 Unicite des etats fondamentaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.3.3 Transformation de diffusion inverse (IST) de l’equation de BO . . . . . . . . . . . 28
1.4 Preliminaires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.4.1 Decomposition en profils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.4.2 Probleme de Cauchy pour l’equation de NLS–Szego cubique sur la tore . . . . . . 29
2 Long time behavior of the NLS–Szego equation on the torus 37
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.1.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2 The cubic Szego equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.2.1 The Lax pair structure and L∞-estimate . . . . . . . . . . . . . . . . . . . . . . . . 42
2.2.2 Wave turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.2.3 A special case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.3 Long time behavior for small data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.3.1 The case α ≥ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.3.2 The case 0 ≤ α < 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.4 Orbital stability of the plane wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.4.1 The proof of Theorem 2.1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.4.2 Long time Hs-stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.4.3 Long time Hs-stability in the case α = 0 . . . . . . . . . . . . . . . . . . . . . . . . 58
2.5 Comparison to the NLS equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3
4 TABLE DES MATIERES
3 Traveling waves of NLS–Szego equation on the line 673.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.2 Profile decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.3 Orbital stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.3.1 Proof of theorem 3.1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763.3.2 The special case γ = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.4 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.4.1 Persistence of regularity for scattering . . . . . . . . . . . . . . . . . . . . . . . . . 803.4.2 The open problem of uniqueness of ground states . . . . . . . . . . . . . . . . . . . 81
4 Integrability of the BO equation on the line 854.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.1.2 Organization of this paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.1.3 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.2 The Lax operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.2.1 Unitary equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.2.2 Spectral analysis I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.2.3 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.2.4 Lax pair formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.3 The action of the shift semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.4 The manifold of multi-solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.4.1 Differential structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.4.2 Spectral analysis II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104.4.3 Characterization theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1144.4.4 The stability under the Benjamin–Ono flow . . . . . . . . . . . . . . . . . . . . . . 115
4.5 The generalized action–angle coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 1184.5.1 The associated matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1194.5.2 Inverse spectral formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1214.5.3 Poisson brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234.5.4 The diffeomorphism property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1264.5.5 A Lagrangian submanifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284.5.6 The symplectomorphism property . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.6 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1334.6.1 The simple connectedness of UN . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1334.6.2 Covering manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Chapitre 1
Introduction generale
1.1 Presentation du contexte
Dans cette these, on s’interesse au comportement en grand temps et a l’integrabilite de certaines equationsdispersives. L’etude des equations aux derivees d’evolution non lineaires (EDPs) est motivee par lamodelisation d’ evolutions temporelles de certaines quantites definies sur un milieu continu pour desprobemes issus des sciences physiques, naturelles, sociales et d’autres domaines relies directement auxmathematiques. Pendant un demi-siecle, l’analyse des EDPs s’est beaucoup consacree a l’etude de l’exis-tence et de l’unicite de solutions locales et globales dans des espaces fonctionnels bien choisis. Par contrasteavec les equations differentielles ordinaires dont les solutions sont a valeurs dans des espaces vectorielsou des varietes lisses de dimension finie, les equations aux derivees partielles peuvent etre vues commedes systemes dynamiques dont les orbites sont incluses dans des espaces vectoriels ou dans des varietesde dimension infinie. Grace aux outils de l’analyse fonctionnelle, des notions de solution faible et forteont pu voir le jour, permettant de definir des trajectoires dans des espaces de fonctions avec differentsniveaux de regularite.
Une equation aux derivees partielles est dite dispersive si ses solutions se decrivent sous la forme d’ondesqui se propagent a des vitesses qui varient en fonction de la longueur d’onde. Une equation dispersivesemi lineaire s’ecrit le plus souvent sous la forme
∂tw(t) + Lw(t) = N (w(t)), (1.1.1)
ou L est un operateur anti-auto-adjoint non borne et densement defini sur certain espace d’Hilbert H, etN est une fonctionnelle non lineaire definie sur le domaine de definition de L. L’equation (1.1.1) peut sereformuler a l’aide de la formule de Duhamel,
w(t) = e−(t−t0)Lu(t0) +
∫ t
t0
e−(s−t0)LN (w(s))ds. (1.1.2)
Par exemple, pour resoudre le probleme de Cauchy en basse regularite de l’equation de Schrodingersemi-lineaire
i∂tw + ∆w = N (w), (t, x) ∈ R×M (1.1.3)
ou M est Rd ou Td, avec T = R/2πZ, on utilise souvent la theorie de Strichartz–Ginibre–Velo qui permet
5
6 CHAPITRE 1. INTRODUCTION GENERALE
de convertir l’estimation dispersive
‖eit∆ϕ‖Lp(Rd) .d,p |t|d( 1p−
12 )‖ϕ‖
Lpp−1 (Rd)
, ϕ ∈ Lpp−1 (Rd), 2 ≤ p ≤ +∞ (1.1.4)
en des inegalites controlant la norme en espace–temps de Lebesgue de la solution. On renvoie a Strichartz[130], Ginibre–Velo [58, 59, 60], Tao [138] et Bahouri–Chemin–Danchin [5] etc. pour le cas M = Rd, Bour-gain [13] pour le cas M = Td, Burq–Gerard–Tzvetkov [18, 19, 20] pour le cas M est une variete compactequelconque, etc.
Apres avoir resolu le probleme de Cauchy, on s’interesse au comportement en grand temps d’une solutionde l’equation (1.1.1), en particulier la stabilite d’ondes progressives, le phenomene de turbulence faibledu point de vue de la croissance de normes de Sobolev, le phenomene de la diffusion non lineaire, etc.Par exemple, on considere l’equation de Schrodinger cubique defocalisante sur le tore Td,
i∂tw + ∆w = |w|2w, (t, x) ∈ R× Td (1.1.5)
et on pose la question : la solution t ∈ R 7→ u(t) ∈ Hs(T) est-elle uniformement bornee sur R, pour touts ≥ 0 ?
La reponse est positive lorsque d = 1 a l’aide de la suite de lois de conservation engendree par une pairede Lax decouverte par Zakharov–Shabat [152]. Ces lois de conservation controlent toutes les Hs-normesde Sobolev de la solution (dans le cas s ≥ 0 entier, voir Faddeev–Takhtajan [33], Grebert–Kappeler [65],Gerard [46], Sun [131] ; dans le cas s > 1 non entier, voir Kappeler–Schaad–Topalov [82] ; dans le cas− 1
2 < s < 0, voir Koch–Tataru [91] et Killip–Visan–Zhang [90]).
Proposition 1.1.1. Si w0 ∈ C∞(T) et w ∈ C∞(R × T) resout l’equation (1.1.5) en dimension d = 1avec la donnee initiale w(0) = w0, on a supt∈R ‖w(t)‖Hs(T) < +∞, pour tout s ≥ 0.
Lorsque d ≥ 2, Colliander–Keel–Staffilani–Takaoka–Tao [29] et Guardia–Haus–Procesi [67] ont montreque pour tout d ≥ 2, K 1 et s > 1, il existe une solution w : t 7→ w(t) ∈ Hs(Td) globale de l’equation(1.1.5) telle que ‖w(0)‖Hs < K−1 et ‖w(T )‖Hs > K pour un certain T > 0. Par consequent, il n’existepas de loi de conservation controlant la norme Hs, si s > 1. Dans le cas ou Td est remplacee par l’espacede produit R × Td, lorsque d ≥ 2, Hani–Pausader–Tzvetkov–Visciglia [69] ont montre, a l’aide d’uneprocedure de diffusion modifiee, qu’il existe des solutions dont l’orbite n’est pas bornee dans Hs pour sassez grand.
Dans le cas M = T2, l’equation (1.1.3) cubique admet le terme nonlineaire completement resonant,N = NR,
NR(w) =∑n∈Z2
∑(n1,n2,n3)∈ΓR(n)
wn1wn2
wn3
ein·x, w =∑n∈Z2
wnein·x, R ∈ Z
⋂[0,+∞), (1.1.6)
ou ΓR(n) = (n1, n2, n3) ∈ (Z2)3 : n1− n2 + n3 = n et ‖n1‖2R2 −‖n2‖2R2 + ‖n3‖2R2 −‖n‖2R2 ∈ [−R,R],et Hani [68] a trouve qu’il existe une solution globale de l’equation (1.1.3) dont l’orbite n’est pas bornee.
On s’interesse a la relation entre l’existence d’une infinite de lois de conservations lineairement independantes,et le phenomene de croissance de normes de Sobolev de la solution d’une equation du type (1.1.1) arbi-traire. On pose une autre question : existe-il une equation du type (1.1.1) qui admet simultanement ces
1.1. PRESENTATION DU CONTEXTE 7
deux phenomenes ?
La reponse est positive. Considerons l’equation de Szego cubique sur le tore T
i∂tw = ΠT(|w|2w), (t, x) ∈ R× T, (1.1.7)
ou ΠT : L2(T)→ L2(T) est projecteur de Szego qui supprime tous les modes de Fourier negatifs,
ΠTf(x) =∑n≥0
fneinx, si f ∈ L2(T), f(x) =
∑n∈Z
fneinx, (1.1.8)
avec fn = 12π
∫ 2π
0f(x)e−inxdx. On definit L2
+(T) = ΠT(L2(T)) et Hs+(T) := L2
+(T)⋂Hs(T). Introduite
par Gerard–Grellier [47, 49, 51, 52], cette equation admet simultanement une paire de Lax qui permet deconstruire des variables d’action–angle, de trouver une formule explicite pour le flot a l’aide d’une transfor-mation spectrale inverse, et des solutions turbulentes engendrees par des donnees initiales generiquementdefinies dans le sous-espace C∞+ (T) := C∞(T)
⋂L2
+(T). La structure de paire de Lax engendre unesuite de lois de conservation lineairement independantes de l’equation (1.1.7), mais aucune parmi ellesne controle la norme Hs
+(T), si s > 12 . Inspiree par ces resultats, O. Pocovnicu a trouve des resultats
similaires sur l’equation de Szego cubique sur la droite dans [119, 120]. H. Xu [148, 149] a etudie uneperturbation lineaire de l’equation de Szego cubique sur le tore et elle a trouve une solution qui admetune croissance de norme de Sobolev exactement exponentielle. Elle a aussi implemente la strategie deHani–Pausader–Tzvetkov–Visciglia [69] dans le cadre de l’equation de ’guide d’onde’ dans Xu [150]. J.Thirouin a etabli une estimation de norme de Sobolev en grand temps de l’equation de Schrodinger frac-tionnelle dans [140]. Il a mene l’etude d’une equation de Szego quadratique. Dans Thirouin [141], il atrouve l’estimation optimale de la croissance de norme de Sobolev et dans Thirouin [142] il a classifiecompletement les ondes progressives de cette equation. On renvoie aussi a Biasi–Bizon–Evnin [10] pourla croissance de norme de Sobolev dans le cas d’une autre variante de l’equation de Szego cubique.
Rappelons maintenant certains resultats sur l’equation de Szego cubique, d’abord sur la tore, ensuite surla droite, qui sont relies aux resultats principaux de cette these.
1.1.1 L’equation de Szego sur le tore
Definition 1.1.2. Sur L2(T), on introduit le produit scalaire 〈f, g〉L2(T) = 12π
∫ 2π
0f(x)g(x)dx et la forme
symplectique ωL2(T)(f, g) := Im〈f, g〉L2(T). L’energie ETSzego(w) = 1
4‖w‖4L4(T) est densement definie sur
L2+.
Alors l’equation de Szego cubique sur le tore (1.1.7) est de type hamiltonien, ∂tw = XETSzego
(w), ou le
champ hamiltonien de ETSzego par rapport a ωL2(T) est donne par
dETSzego(w)(h) = ωL2(T)(h,XET
Szego(w)), ∀w, h ∈ H
12+(T).
Notons que, malgre la forme (1.1.1), cette equation est totalement non dispersive, puisque l’operateur Lest nul !
Proposition 1.1.3. (Gerard–Grellier [47]). Soit w0 ∈ H12+(T), il existe une unique fonction w ∈
Cb(R, H12+(T)) qui resout l’equation (1.1.7) avec la donnee initiale w(0) = w0 et l’application du flot
w0 ∈ H12+(T) 7→ w ∈ L∞(R, H
12+(T)) est continue. Si s > 1
2 et w0 ∈ Hs(T), alors w ∈ C∞(R, Hs+(T)).
8 CHAPITRE 1. INTRODUCTION GENERALE
Soient w ∈ L2+(T) et b ∈ L∞(T), l’operateur de Hankel de symbole w et l’operateur de Toeplitz de
symbole b sont deux operateurs bornes sur L2+(T), qui sont definis par
HTw(h) = ΠT(wh), TT
b (h) = ΠT(bh), ∀h ∈ L2+(T). (1.1.9)
L’operateur de Hankel HTw est anti-lineaire et l’operateur de Toeplitz TT
b est lineaire. L’operateur dedecalage ST : L2
+(T)→ L2+(T) est defini par
STh(x) = eixh(x), ∀x ∈ T et h ∈ L2+(T). (1.1.10)
Ensuite l’operateur de Hankel decale KTw : L2
+(T)→ L2+(T) est defini par
KTw = (ST)∗Hw = HwS
T = H(ST)∗w. (1.1.11)
Alors on a le theoreme suivant.
Theoreme 1.1.4. (Gerard–Grellier [47]). Soient s > 12 et w ∈ C∞(R, Hs
+(T)) resout l’equation de Szego
cubique sur la tore i∂tw = ΠT(|w|2w), on definit ATw = −iTT
|w|2 + i2 (HT
w)2 et CTw = −iTT
|w|2 + i2 (KT
w)2.Alors on a
∂tHTw = [AT
w, HTw], ∂tK
Tw = [CT
w,KTw]. (1.1.12)
On identifie les elements w ∈ L2+(T) aux traces des fonctions holomorphes sur le disque unite w ∈
Hol(D(0, 1)), D(0, 1) = z ∈ C : |z| < 1, telles que sup0≤r<1
∫ 2π
0|w(reix)|2dx < +∞ par la correspon-
dance suivante
w ∈ L2+(T) 7→
w : z ∈ D(0, 1) 7→∑n≥0
wnzn
, w(x) = limr→1
w(reix) (1.1.13)
qui etablit un isomorphisme d’espace de Hilbert de l’espace L2+(T) vers l’espace de Hardy sur le disque
unite (voir Chapter 17 of Rudin [124]).
Theoreme 1.1.5. (Gerard–Grellier [51]). Soient w0 ∈ H12+(T) et w ∈ C(R, H
12+(T)) la solution de
l’equation (1.1.7) avec la donnee initiale w(0) = w0, alors on a la formule suivante,
w(t, z) =
⟨(IdL2
+(T) − ze−it(HTw0
)2
eit(Kw0)2
(ST)∗)−1
e−it(HTw0
)2
w0,1
⟩L2(T)
, (1.1.14)
ou la fonction 1(x) = 1 est constante.
Definition 1.1.6. L’ensemble M(N)T consiste en toutes les fractions rationnelles w ∈ L2+(T) de la
forme w(x) = A(eix)B(eix) , ou B(0) = 1, les polynomes A ∈ C≤N−1[X], B ∈ C≤N [X] sont premiers entre eux
et tels que deg(A) = N − 1 ou deg(B) = N , les racines de B sont dehors le disque unite ferme D(0, 1).Ici C≤N [X] designe l’ensemble des polynomes P ∈ C[X] de degre deg(P ) ≤ N .
Muni du produit interieur de L2+(T), l’ensemble M(N)T est une sous-variete kahlerienne de L2
+(T) dedimension complexe dimC(M(N)T) = 2N . Un theoreme qui remonte a Kronecker [94] dit qu’une fonctionw ∈M(N)T si et seulement si le rang de l’operateur de Hankel HT
w est egal a N .
Definition 1.1.7. L’ensemble M(N)Tgen consiste en toutes les fonctions w ∈ M(N)T telles que 1 /∈Im(HT
w) et les N vecteurs (HTw)2k(1)k=1,2,··· ,N sont lineairement independants.
1.1. PRESENTATION DU CONTEXTE 9
L’ensembleM(N)Tgen est une partie ouverte deM(N)T dont le complementaire est de mesure de Lebesgue
nulle. Munie de la forme symplectique ωL2(T), M(N)Tgen est une variete symplectique de dimension
dimR(M(N)Tgen) = 4N . Les actions sont donnees par
ΩTN = (IT1 , IT2 , · · · , ITN ; JT
1 , JT2 , · · · , JT
N ) ∈ R2N : ITk > JTk > ITk+1 > 0.
La forme symplectique canonique sur la variete ΩTN × T2N est νT =
∑Nk=1(dITk ∧ dϕT
k + dLTk ∧ dθTk ).
Theoreme 1.1.8. (Gerard–Grellier [49]). Il existe un symplectomorphisme reellement analytique
χTN : (M(N)Tgen, ωL2(T))→ (ΩT
N × T2N ,νT) (1.1.15)
telle que ETSzego
(χTN
)−1(IT1 , · · · , ITN ;LT
1 , · · · , LTN ;ϕT
1 , · · · , ϕTN ; θT1 , · · · , θTN ) = 1
4
∑Nk=1(|ITk |2 − |LT
k |2).
L’application χN introduit les coordonnees action–angle pour l’equation de Szego cubique sur le tore. Si
χTN (w) = (IT1 (w), · · · , ITN (w);LT
1 (w), · · · , LTN (w);ϕT
1 (w), · · · , ϕTN (w); θT1 (w), · · · , θTN (w)) ∈ ΩT
N × T2N ,
alors l’equation (1.1.7) est linearisee sous ces coordonnees, si w : t ∈ R 7→ w(t) ∈M(N)T resout (1.1.7),
∂t(ITk w)(t) = ∂t(L
Tk w)(t) = 0, ∂t(ϕ
Tk w)(t) =
(ITk w)(t)
2, ∂t(θ
Tk w)(t) = − (LT
k w)(t)
2.
Plus precisement, la famille 12I
Tk (w)k≥1 consiste en l’ensemble des valeurs propres de l’operateur (HT
w)2
et la famille 12L
Tk(w)k≥1 consiste en l’ensemble des valeurs propres de l’operateur (KT
w)2. De plus,l’application χT
N admet une generalisation en dimension infinie. (voir Gerard–Grellier [49]). La solutionw : t 7→ w(t) ∈ M(N)T est quasi-periodique (voir Gerard–Grellier [51]). Le resultat suivant met enevidence un phenomene de turbulence faible pour l’equation (1.1.7).
Theoreme 1.1.9. (Gerard–Grellier [50]). Il existe une partie Gδ-dense V ⊂ C∞+ (T) telle que si w0 ∈ V,alors la solution w : t ∈ R 7→ w(t) ∈ C∞+ telle que la donnee initiale est w(0) = w0 verifie les proprietessuivantes :
• il existe une suite temporelle tn tendant vers +∞, telle que ∀s > 12 , ∀M ∈ N, on a
limn→+∞
‖w(tn)‖Hs(T)
tMn
= +∞;
• il existe une suite temporelle tn tendant vers +∞, telle que
limn→+∞
w(tn) = w0 dans C∞+ (T).
Par ailleurs, pour tout w0 ∈ C∞+ (T), il existe une constante Cs > 0 telle que
‖w(t)‖Hs(T) ≤ Cs‖w0‖Hs(T)eCs‖w0‖2Hs(T)|t|, ∀t ∈ R. (1.1.16)
Comme l’union⋃N≥1M(N)T est dense dans C∞+ (T) et V est Gδ-dense dans C∞+ (T), on voit que le
comportement en grand temps de l’equation de Szego cubique (1.1.7) depend de facon sensible de sadonnee initiale.
10 CHAPITRE 1. INTRODUCTION GENERALE
Remarque 1.1.10. On considere l’equation hamiltonienne avec la meme energie sans le projecteur deSzego ΠT : L2
+(T)→ L2+(T) sur la tore :
i∂tV = |V |2V, (t, x) ∈ R× T. (1.1.17)
Alors on a V (t, x) = eit|V0|2V0(x) et ‖V (t)‖Hs ' |t|s, pour tout s ≥ 0, si |V0| n’est pas une fonctionconstante. Donc le projecteur de Szego ΠT : L2(T)→ L2
+(T) accelere le transfert d’energie vers les hautesfrequences.
Xu [148, 149] a mene l’etude d’une equation de Szego cubique perturbee par un terme lineaire
i∂tw = ΠT(|w|2w) + α
∫Tw, α ∈ R. (1.1.18)
Dans ce cas, (HTw, A
Tw) n’est plus une paire de Lax, mais (KT
w, CTw) est encore une paire de Lax. Lorsque
α > 0 et w0(x) = eix +√α, on note w : t ∈ R 7→ w(t) ∈ Hs
+(T) la solution de l’equation (1.1.18), alorson a la croissance exponentielle exacte de la norme de Sobolev, lorsque t→ ±∞,
‖w(t)‖Hs(T) ' e(2s−1)√α|t|, ∀s > 1
2.
Thirouin [141, 142] a trouve l’estimation optimale de la croissance de norme de Sobolev et a classifiecompletement les ondes progressives de l’equation de Szego quadratique
i∂tw = 2JΠT(|w|2) + Jw2, ∀(t, x) ∈ R× T, J := J(u) =
∫T|w|2w ∈ C. (1.1.19)
1.1.2 L’equation de Szego sur la droite
Considerons l’equation cubique sur la droite (voir Pocovnicu [119, 120])
i∂tw = ΠR(|w|2w), (t, x) ∈ R× R, (1.1.20)
ou ΠR : L2(R)→ L2(R) est projecteur de Szego sur la droite,
ΠRf(x) =1
2π
∫ +∞
0
eixξ f(ξ)dx, ∀f ∈ L2(R). (1.1.21)
On definit les espaces de Sobolev filtres par L2+(R) := ΠR(L2(R)) et Hs
+(R) = L2+(R)
⋂Hs(R), pour tout
s ≥ 0. Le theoreme de Paley–Wiener (Theorem 19.2 of Rudin [124]) donne l’identification entre l’espaceL2
+ et l’espace de Hardy sur le demi-plan de Poincare C+ = z ∈ C : Imz > 0, on a l’isomorphismed’Hilbert suivant w ∈ L2
+(R) 7→ w ∈ H2(C+) = g ∈ Hol(C+) : ‖g‖H2(C+) < +∞, ou
‖g‖Hp(C+) = supy>0
(∫R|g(x+ iy)|pdx
) 1p
, si p ∈ (0,+∞), (1.1.22)
Alors l’equation (1.1.20) est aussi de type hamiltonien par rapport a la forme symplectique ωL2(R)(f, g) =
Im〈f, g〉L2(R) avec le produit scalaire 〈f, g〉L2(R) =∫R fg. L’energie de l’equation (1.1.20) est ER
Szego(w) =14
∫R |w|
4, pour tout w ∈ L4(R)⋂L2
+(R). L’equation de Szego sur la droite est globalement bien posee
sur Hs+(R), pour tout s ≥ 1
2 .
1.1. PRESENTATION DU CONTEXTE 11
Definition 1.1.11. Soient w ∈ L2+(R) et b ∈ L∞(R), l’operateur de Hankel HR
w : L2+(R) → L2
+(R)
et l’operateur de Toeplitz TRb : L2
+(R) → L2+(R) sont definis par HR
w(h) = ΠR(wh) et TRb (h) = ΠR(bh)
respectivement.
Theoreme 1.1.12. (Pocovnicu [119]). Soient s > 12 et w ∈ C∞(R, Hs
+(R)) resout l’equation (1.1.20),
on definit un operateur ARw = −iTR
|w|2 + i2 (HR
w)2. Alors ∂tHRw = [AR
w, HRw].
Ainsi l’equation (1.1.20) admet aussi une paire de Lax. En revanche l’operation de decalage n’existe plusmais devient un semi-groupe (S(η))η≥0 avec S(η) : L2
+(R)→ L2+(R) defini S(η)f = eηf , ou eη(x) = eiηx,
x ∈ R, dont l’adjoint est donne par S(η)∗ = Te−η . Inspire par la definition 1.1.6, on introduit la variete
M(N)R, qui consiste en toutes les fractions rationnelles sous la forme w(x) = A(x)B(x) , ou B(0) = 1, les
polynomes A ∈ C≤N−1[X], B ∈ C≤N [X] etant premiers entre eux et tels que deg(A) = N − 1 oudeg(B) = N , les racines de B etant contenues dans le demi-plan inferieur de Poincare C−. Muni duproduit interieur de L2
+(R), l’ensemble M(N)R est une sous-variete kahlerienne de L2+(R) de dimension
dimC(M(N)R) = 2N . De plus M(N)R est le revetement universel de la variete M(N)T donnee dans ladefinition 1.1.6. (voir Sun [135]) Le theoreme de Kronecker [94] donne la caracterisation par le rang desoperateurs de Hankel
M(N)R = w ∈ L2+(R) : rang(HR
w) := dimC(ImHRw) = N. (1.1.23)
Definition 1.1.13. Soit w ∈M(N)R, il existe une unique fonction g ∈ Im(HRw) = Im((HR
w)2) telle queHRwg = w. L’ensemble des valeurs propres de (HR
w)2 est σpp((HRw)2) = λ2
1, λ22, · · · , λ2
N, ou λk = λk(w),tel que 0 < λk ≤ λk+1. Il existe une base orthonormee du sous-espace Im(HR
w) notee par ekNk=1 formeepar des vecteurs propres qui satisfont HR
wek = λkek, pour tout k = 1, 2, · · · , N .
Theoreme 1.1.14. (Pocovnicu [120]). Soit w0 ∈ M(N)R et notons W (t) := eit2 (HR
w0)2
, l’operateur dedecalage infinitesimal compresse T : Im(HR
w0)→ Im(HR
w0) est defini par
Tf(x) := xf(x)−(
limx→+∞
xf(x)
)(1− g0). (1.1.24)
ou la fonction g0 ∈ Im(HRw0
) verifie que HRw0g0 = w0, donnee dans la definition 1.1.13. Soit j = 1, 2, · · · , N
fixe et on note Mj ⊂ N qui consiste en l’ensemble de tous les indices k tels que HRw0ek = λjek. On definit
βj := 〈g0, ej〉L2(R) et l’operateur lineaire S(t) : Im(HRw0
)→ Im(HRw0
) par
〈S(t)ek, ej〉L2(R) =
λj
2πi(λ2k−λ
2j )eit2 (λ2
k−λ2j )(λjβjβk − λkβkβj
), si k ∈ 1, 2, · · · , N\Mj
λ2j
2πβjβkt+ 〈Tej , ek〉L2(R), si k ∈Mj .
Si on note w : t ∈ R 7→ w(t) ∈ Im(HRw0
) la solution de l’equation de Szego associee a la donnee initialew(0) = w0, alors on a la formule explicite suivante
w(t, x) =i
2π〈w0,W (t) (S(t)− x)
−1W (t)g0〉L2(R), ∀(t, x) ∈ R× R. (1.1.25)
Remarque 1.1.15. Soit s > 12 , comme l’union
⋃N≥1M(N)R est dense dans Hs
+(R), la formule (1.1.25)se generalise au cas d’une solution a donnee initiale arbitraire dans Hs
+(R) telle que x 7→ xw0(x) ∈ L∞(R)(voir Theorem 1.4 de Pocovnicu [120]).
12 CHAPITRE 1. INTRODUCTION GENERALE
Definition 1.1.16. Une fonction w ∈ M(N)R est dite generique si l’operateur (HRw)2 n’admet que des
valeurs propres non nulles simples 0 < λ21 < λ2
2 < · · · < λ2N . On note M(N)Rgen l’ensemble de telles
fonctions generiques, qui est une partie ouverte dense de M(N)R.
Remarque 1.1.17. Une fonction generique w ∈ M(N)Rgen verifie que 〈w, ek〉L2(R) 6= 0, pour tout k =
1, 2, · · · , N , ou les fonctions propres ekNk=1 sont donnees dans la definition 1.1.13, d’apres Gerard–Pushnitski [57].
Alors (M(N)Rgen, ωL2(R)) est une variete symplectique de dimension dimR(M(N)Rgen) = 4N . On definit
YN := (−ΩN ) × (R∗+)N × RN × TN , ou (−ΩN ) = 0 < IR1 < IR2 < · · · < IRN. Muni de la formesymplectique canonique,
νR =
N∑k=1
(dIRk ∧ dϕR
k + dLRk ∧ dθRk
), ϕR
k ∈ R, θRk ∈ T.
YN est une variete symplectique reellement analytique de dimension dimR(Y) = 4N .
Theoreme 1.1.18. (Pocovnicu [120]). Il existe un symplectomorphisme reellement analytique
χRN : (M(N)Rgen, ωL2(R))→ (YN ,ν
R),
tel que ERSzego(χR
N )−1 ne depend que des variables d’actions (IR1 , IR2 , · · · , IRN ;LR
1 , LR2 , · · · , LR
N ) ∈ (−ΩN )×(R∗+)N .
Remarque 1.1.19. L’application χRN introduit les coordonnees d’action–angle generalisees. La moitie
des actions et des angles IRk , ϕRk , k = 1, 2, · · · , N , coıncide dans le cas de l’equation de Szego cubique
sur la tore. La famille 14π I
Rk (w)k≥1 consiste en l’ensemble des valeurs propres de l’operateur de Hankel
(HRw)2. Cependant, l’autre moitie est completement differente. Dans le cas T, le deuxieme operateur de
Lax est trouve, c’est l’operateur de Hankel decale Kw = HTwS
T dont ses valeurs propres et ses fonctionspropres donnent le reste des coordonnees. Dans le cas R, l’action de decalage devient le semi-groupe(S(η)∗)η≥0. On definit LR
k (w) = 4|〈w, ek〉L2(R)|2, ou ek est la fonction propre associee a la valeur propre
λ2j = 1
4π IRk (w) de l’operateur de Hankel (HR
w)2, donnee par la definition 1.1.13. On a LRk (w) > 0, pour
tout k = 1, 2, · · · , N , d’apres la remarque 1.1.17.
Definition 1.1.20. Une onde solitaire de l’equation de Szego cubique sur la droite (1.1.20) est unesolution lisse w : t ∈ R 7→ w(t) ∈ C∞+ (R) qui s’ecrit sous la forme w(t, x) = e−iωtw0(x − ct), pourcertaines constantes c, ω ∈ R, ou w0 = w(0).
Inspiree de la classification des ondes progressives pour l’equation de Szego sur la tore (voir Gerard–Grellier [47]), Pocovnicu a classifie l’ensemble des ondes solitaires de l’equation (1.1.20) a l’aide dutheoreme de Beurling–Lax qui permet de caracteriser tous les sous-espaces translation–invariants del’espace de Hardy L2
+(R).
Theoreme 1.1.21. (Pocovnicu [119]). Une fonction w ∈ C(R, H12+(R)) est une onde solitaire si et
seulement s’il existe C, p ∈ C telles que Imp < 0 telle que
w(t, x) =Ce−iωt
x− ct− p(1.1.26)
ou les constantes c et ω dependent de C et p.
1.2. ENONCES DES RESULTATS 13
De plus, Pocovnicu a montre la stabilite orbitale de l’onde solitaire (1.1.26) par la methode de concentration–compacite (voir Lions [101, 102]), qui est amelioree comme le theoreme de decomposition en profils parGerard [45] (voir aussi Bahouri–Gerard [6]). Ce theoreme est recapitule dans la sous-section 1.4.1.
Theoreme 1.1.22. (Pocovnicu [119]). Soient a, r > 0, considere le cylindre
C(a, r) = x ∈ R 7→ α
x− p∈ C : |α| = a, Imp = −r ⊂ H∞+ (R). (1.1.27)
Pour tout ε > 0, il existe δ = δ(ε, a, r) > 0 telle que ∀w0 ∈ H12+(R) verifiant infφ∈C(a,r) ‖w0−φ‖
H12 (R)
< δ,
on note w : t ∈ R 7→ w(t) ∈ H12+(R) la solution de l’equation de Szego sur la droite (1.1.20), alors
supt∈R
infφ∈C(a,r)
‖w(t)− φ‖H
12 (R)
< ε.
Definition 1.1.23. Une donnee initiale w0 ∈ M(N) est dite fortement generique si l’operateur H2w0
a ses valeurs propres non nulles simples 0 < λ21 < λ2
2 < · · · < λ2N avec 〈w0, ek〉L2(T), pour tout
k = 1, 2, · · · , N et |〈w0, ej〉L2(T)| 6= |〈w0, ek〉L2(T)| pour tout k 6= j. L’ensemble des fonctions fortementgeneriques est note par M(N)sgen.
Theoreme 1.1.24. (Pocovnicu [120]). Soit w0 ∈ M(N)sgen une donnee initiale fortement generiquepour l’equation de Szego cubique sur la droite (1.1.20), la solution correspondante s’ecrit comme unesomme de N -solitons et d’un reste. Plus precisement, on a
u(t, x) =
N∑j=1
Cje−iωjt
x− cjt− pj+ ε(t, x), lim
t→±∞‖ε(t, x)‖Hs+(R) = 0, ∀s ≥ 0,
ou ωj = λ2j , les constantes pj ∈ C− et Cj ∈ C dependent de λj , ej et w0.
Theoreme 1.1.25. (Pocovnicu [120]). Soient s > 12 , il existe des solutions w : t ∈ R 7→ w(t) ∈ Hs
+(R)de l’equation de Szego, telles que limt→±∞ ‖w(t)‖Hs(R) = +∞.
1.2 Enonces des resultats
L’equation de Schrodinger cubique sur le tore de dimension 1 n’a aucune solution turbulente d’apresla proposition 1.1.1. D’un autre cote, le projecteur de Szego ΠT : L2(T) → L2
+(T) accelere le transfertd’energie vers les frequences hautes pour l’equation (1.1.17). On s’interesse donc a l’influence du projecteurde Szego sur l’equation de NLS cubique (1.1.5) dans le cas d = 1, en introduisant l’equation de NLS–Szegocubique defocalisante sur le tore
i∂tu+ ∂2xu = ΠT(|u|2u), (t, x) ∈ R× T. (1.2.1)
Sur la droite, on s’interesse a l’influence du projecteur de Szego ΠR : L2(R)→ L2+(R) donne par (1.1.21)
sur l’equation de Schrodinger L2-critique focalisante sur la droite
i∂tw + ∂2xw = −|w|4w, (t, x) ∈ R× R. (1.2.2)
Cette equation admet l’invariance galileenne et un seuil de masse, au–dessus duquel il existe des solutionsqui explosent en temps fini, et sous lequel la solution diffuse toujours. (voir Cazenave–Weissler [25, 26],
14 CHAPITRE 1. INTRODUCTION GENERALE
Cazenave–Lions [24], Weinstein [145] et Dodson [31]) En ajoutant le projecteur de Szego ΠR devant leterme non lineaire, on obtient l’equation de NLS–Szego quintique focalisante sur la droite,
i∂tu+ ∂2xu = −ΠR(|u|4u), (t, x) ∈ R× R . (1.2.3)
On voit tout de suite des grandes differences entre l’equation (1.2.2) et l’equation (1.2.3). Les resultatssur les equations de NLS–Szego sfont l’objet des chapitres 2 et 3 de cette these, voir [133, 134]. Quant auchapitre 4, il est consacre a l’equation de Benjamin–Ono (BO) sur la droite,
∂tu = H∂2xu− ∂x(u2), (t, x) ∈ R× R, (1.2.4)
ou u est a valeurs reelles et H est la transformation d’Hilbert H = −isign(D), definie par
Hf(ξ) = −isign(ξ)f(ξ), ∀f ∈ L2(R). (1.2.5)
avec sign(±ξ) = ±1, pour tout ξ > 0. On remarque que le projecteur de Szego peut s’ecrire sous la formeΠR = 1
2 (IdL2(R) + iH). On utilise l’abreviation Lp(R) = Lp(R,C). L’espace Lp(R,R) consiste en toutesles Lp-fonctions a valeurs reelles. Comme dans les travaux precedents, on peut en fait ecrire l’equationde BO sous la forme d’une equation de Schrodinger non lineaire filtree par le projecteur de Szego ΠR. Sion pose v = ΠRu, ou u resout l’equation (1.2.4), alors
i∂tv − ∂2xv + ∂x[v2 + 2ΠR(|v|2)] = 0, (t, x) ∈ R× R. (1.2.6)
1.2.1 L’equation de NLS–Szego cubique sur le tore
Ce projet correspond au chapitre 2 et a l’article [133]. L’equation de NLS-Szego cubique defocalisante surle tore (1.2.1) peut s’obtenir a partir de l’equation de Szego cubique (1.1.7) en ajoutant le terme dispersif∂2x. Afin de mesurer la gradation de dispersivite, on rajoute un parametre devant le laplacien ∂2
x. L’objetprincipal de cette sous-section est l’equation suivante
i∂tu+ εα∂2xu = ΠT(|u|2u), 0 < ε < 1, α ≥ 0. (1.2.7)
On munit encore L2(T) de la forme symplectique ωL2(T)(f, g) = Im〈f, g〉L2(T), en sorte que l’equation(1.2.7) est hamiltonienne avec l’energie definie dans H1
+(T) :
Eα,ε(u) =εα
2‖∂xu‖2L2 +
1
4‖u‖4L4 , u ∈ H1
+. (1.2.8)
L’equation (1.2.7) admet encore deux autres lois de conservation,
Q(u) = ‖u‖2L2 , I(u) = Im
∫Tu∂xu = ‖|D| 12u‖2L2(T).
Rappelons l’estimation d’interpolation
supt∈R‖u(t)‖Hs ≤ ‖u0‖1−2s
L2 ‖u0‖2sH
12, ∀s ∈ [0,
1
2].
Afin de montrer l’existence et l’unicite de la solution globale de l’equation (1.2.7), on utilise l’inegalitede Brezis–Gallouet [17], le theoreme d’Aubin–Lions–Simon (voir Theorem II.5.16 de Boyer–Fabrie [15])et l’inegalite de Trudinger (voir Yudovich [151], Vladimirov [144], Ogawa [114] et Gerard–Grellier [47]).L’existence et l’unicite dans les espaces de Sobolev en basse regularite peuvent etre obtenues comme dansBourgain [13]. Dans ce qui suit, on ne considere que les estimations dans l’espace de Sobolev en hauteregularite.
1.2. ENONCES DES RESULTATS 15
Proposition 1.2.1. Soient s ≥ 12 et u0 ∈ Hs
+(T), il existe une unique fonction u ∈ C(R, Hs+(T)) qui
resout l’equation (1.2.7) avec la donnee initiale u(0) = u0. Pour tout T > 0, l’application flot u0 ∈Hs
+(T) 7→ u ∈ C([−T, T ], Hs+(T)) est continue.
Cette proposition est recapitulee au theoreme 1.4.2 plus loin. On obtient deux ensembles de resultatsconcernant le comportement en grand temps des solutions de l’equation (1.2.7). Le premier resultatconcerne l’estimation en grand temps de la norme de Sobolev de la solution. Si la donnee initiale u0 estbornee par ε, on cherche un intervalle dans lequel la solution u(t) reste encore bornee par O(ε).
Theorem 1.2.2. Soit s > 12 , il existe deux constantes as ∈]0, 1[ et Ks > 0 telles que si 0 < ε 1 et
u0 ∈ Hs+ avec ‖u0‖Hs = ε, u resout l’equation (1.2.7) avec u(0) = u0, alors on a
sup|t|≤ asε4−α
‖u(t)‖Hs ≤ Ksε, si α ∈ [0, 2];
sup|t|≤ asε2‖u(t)‖Hs ≤ Ksε, si α > 2.
(1.2.9)
De plus, dans le cas α > 2 et s ≥ 1, l’intervalle temporel Iαε = [−asε2 ,asε2 ] est maximal au sens ou
∀ 0 < ε 1, il existe une donnee uε0 ∈ C∞+ telle que ‖uε0‖Hs ' ε et pour tout β > 0, et on a
sup|t|≤ 1
ε2+β
‖u(t)‖Hs & ε| ln ε| 12 ε, u(0) = uε0.
Remark 1.2.3. Dans le cas α ∈ [0, 2), on utilise la methode de forme normale de Birkhoff, qui estsimilaire a Bambusi [7], Grebert [64], Gerard–Grellier [48] et Faou–Gauckler–Lubich [34] etc. En revanche,on ne sait pas si l’intervalle temporel [− as
ε4−α ,asε4−α ] est optimal. Les termes de resonances a 6 indices
dans l’equation homologique ne peut pas etre supprimes par notre transformation de forme normale deBirkhoff.(voir le chapitre 2)
Le deuxieme ensemble de resultats concerne la stabilite en grand temps d’une onde plane. On definitl’onde plane em : x 7→ eimx, pour tout m ∈ N et s ≥ 1. Soit u = u(t, x) une solution de l’equation (1.2.7)telle que ‖u(0)− em‖Hs = ε. Par conservation de energie (2.1.6), on deduit l’estimation suivante :
supt∈R‖u(t)‖H1 .‖u0‖H1
ε−α2 , ∀0 < ε < 1, α ≥ 0. (1.2.10)
En revanche, aucune information sur la stabilite de l’onde plane em n’est obtenue a partir de (1.2.10)lorsque ε→ 0+. Compte tenu du phenomene de croissance de norme de Sobolev pour l’equation de Szegocubique (1.1.7) sur le tore (voir theoreme 1.1.9), le phenomene de turbulence d’onde pour (1.2.7) vadependre de la taille de sa dispersion. On commence par trois resultats sur la stabilite de l’onde planedans le cas ou la dispersion est polynomiale en ε, εα∂2
x avec 0 ≤ α ≤ 2. Le prochain theoreme indique lastabilite orbitale de l’onde plane em par rapport a la norme H1 pour l’equation (1.2.7).
Theoreme 1.2.4. Soient ε ∈]0, 1[, α ∈ [0, 2] et m ∈ N, il existe Cm > 0 telle que si ‖u(0)− em‖H1 = ε,alors on a
supt∈R
infθ∈R‖u(t)− eiθem‖H1 ≤ Cmε1−
α2 .
Pour tout t ∈ R, la borne inferieure est atteint lorsque θ = arg um(t). Des resultats similaires sont obtenuspar Zhidkov [153, Sect. 3.3] et Gallay–Haragus [43, 44] pour l’equation de Schrodinger cubique (1.1.5)avec d = 1. Dans le cas de basse dispersion, le theoreme 1.2.4 ameliore l’estimation (1.2.10). Pour tout
φ ∈ H12+ , on note u : t ∈ R 7→ Sα,ε(t)φ ∈ H
12+ la solution de l’equation (1.2.7) telle que u(0) = φ.
16 CHAPITRE 1. INTRODUCTION GENERALE
Corollary 1.2.5. Soit m ∈ N, on a sup 0<ε<10≤α≤2
sup‖φ−em‖H1≤ε supt∈R ‖Sα,ε(t)φ‖H1 <∞.
En comparant avec le phenomene de marguerite dans Gerard–Grellier [47, 48, 51], le terme dispersif εα∂2x
empeche la croissance de la norme H1, pour l’equation (1.2.7), si 0 ≤ α ≤ 2. En utilisant le changementde variable u(t) = ei arg um(t)(em + ε1−
α2 v(t)), un argument de bootstrap conduit a la stabilite orbitale en
grand temps de l’onde plane em par rapport aux normes de Sobolev en hautes frequences.
Proposition 1.2.6. Soient s ≥ 1 et m ∈ N, il existe deux constantes bm,s ∈]0, 1[ et Lm,s > 0 telles quesi 0 ≤ α < 2 et ‖u(0)− em‖Hs = ε ∈]0, 1[, alors on a
sup|t|≤ bm,s
ε1−α
2
infθ∈R‖u(t)− eiθem‖Hs ≤ Lm,sε1−
α2 . (1.2.11)
On aimerait trouver un intervalle temporel plus grand dans lequel l’estimation (1.2.11) soit encore vraiepar la transformation de forme normale de Birkhoff. Mais les coefficients devant les modes de Fourier dehautes frequences dans l’equation homologique peuvent etre arbitrairement grands, si α ∈]0, 2[. Donc onrevient au cas α = 0 et on considere l’equation suivante.
i∂tu+ ∂2xu = ΠT(|u|2u), (t, x) ∈ R× T.
Dans ce cas, on a le theoreme suivant.
Theorem 1.2.7. Pour tout s ≥ 1, pour tout m ∈ N, il existe trois constantes dm,s, εm,s ∈ (0, 1) etKm,s > 0 telles que si ‖u(0)− em‖Hs = ε ∈ (0, εm,s), alors
sup|t|≤ dm,s
ε2
infθ∈R‖u(t)− eiθem‖Hs ≤ Km,sε.
Des resultats similaires ont ete etablis par Faou–Gauckler–Lubich [34] pour l’equation de Schrodingercubique (1.1.5) pour tous les d ∈ N. (voir le chapitre 2 pour la comparaison entre l’equation (1.2.7) et(1.1.5)) Apres avoir etabli ces resultats de stabilite, on va construire des solutions pour l’equation (1.2.7)qui sont grandes par rapport a leur donnees initiales.
Theorem 1.2.8. Il existe une constante K > 0 telle que pour tout 0 < δ 1, on note U : t ∈ R 7→U(t) ∈ C∞+ (T) la solution de l’equation de NLS-Szego suivante
i∂tU + ν2∂2xU = ΠT(|U |2U), U(0, x) = eix + δ, (1.2.12)
ou ν = e−πK2δ2 , alors on a ‖U(tδ)‖H1 ' 1
δ avec tδ := πδ√
4+δ2.
En d’autres termes, le support de l’energie de la solution de l’equation (1.2.12) est transfere vers lesgrandes modes de Fourier. Ce resultat est similaire au cas de l’equation de Szego cubique (1.1.7) obtenupar Gerard–Grellier [50, 51, 52] et au cas l’equation de Schrodinger cubique (1.1.5) avec d = 2 obtenupar Colliander–Keel–Staffilani–Takaoka–Tao [29]. En comparant avec le theoreme 1.2.4, on montre quele phenomene de turbulence faible demeure si la dispersion de l’equation de NLS–Szego est exponentiel-lement basse par rapport a la perturbation de l’onde plane e1 : x 7→ eix.
Remarque 1.2.9. La deuxieme partie du theoreme 1.2.2 est une consequence du theoreme 1.2.8. En
fait, a chaque α > 2 fixe, on fait la dilatation u(t, x) = εU(ε2t, x) avec e−πK2δ2 = ν = ε
α−22 . Alors u resout
(1.2.7) avec la donnee initiale u(0, x) = ε(eix + δ) et on a
‖u( tδ
ε2 )‖H1 = ε‖U(tδ)‖H1 ' ε
δ' ε√
(α− 2)| ln ε| ε,
1.2. ENONCES DES RESULTATS 17
et tδ
ε2 '√
(α−2)| ln ε|ε2 1
ε2+β , pour tout β > 0. En revanche, cette methode ne marche pas dans le cascritique α = 2. Si u resout
i∂tu+ ε2∂2xu = ΠT(|u|2u), u(0, x) = ε(eix + δ),
on fait le changement de variable U(t, x) = ε−1u(ε−2t, x) et on obtient l’equation (1.2.12) avec ν = 1.Grace aux theoremes 1.2.4 et 1.2.7, on a des estimations suivantes
supt∈R‖u(t)‖H1 = O(ε), sup
|t|≤ d1,s
ε2δ2
‖u(t)‖Hs = O(ε), ∀0 < δ 1, ∀0 < ε < 1,
pour tout s > 12 . Le probleme de trouver l’intervalle temporel optimal dans le cas α = 2 du theoreme 1.2.2
reste ouvert.
1.2.2 L’equation de NLS–Szego quintique sur la droite
Ce projet correspond au chapitre 3 de la these et a l’article Sun [134]. On rappelle d’abord la definitionde la diffusion pour une solution de (1.2.3).
Definition 1.2.10. Soit s ≥ 0 fixe, une solution globale u ∈ C(R;Hs+(R)) de l’equation (1.2.3) est dite
Hs-diffuse vers l’avant en temps s’il existe une fonction u+ ∈ Hs+(R) telle que
limt→+∞
‖eit∂2xu+ − u(t)‖Hs(R) = 0.
Une solution globale u ∈ C(R;Hs+(R)) de l’equation (1.2.3) est dite Hs-diffuse vers l’arriere en temps s’il
existe une fonction u− ∈ Hs+(R) telle que
limt→−∞
‖eit∂2xu− − u(t)‖Hs(R) = 0.
Dans le cas d’une donnee petite dans L2, l’equation (1.2.3) et (1.2.2) sont globalement bien posees dansl’espace de Hardy L2
+(R) et leurs solutions L2-diffusent simultanement a l’avant et a l’arriere en temps.La preuve est basee sur l’inegalite de Strichartz, et est similaire a celle de Cazenave–Weissler [25, 26].
Proposition 1.2.11. Il existe ε0 > 0 telle que si ‖u0‖L2 ≤ ε0, il existe une unique fonction u ∈ C(R;L2+)
qui resout l’equation (1.2.3) et u L2-diffuse simultanement a l’avant et a l’arriere en temps.
Il y a trois lois de conservation communes pour (1.2.2) et (1.2.3)
M(u) = ‖u‖2L2 , P (u) = 〈Du, u〉L2 , ENLS(u) =‖∂xu‖2L2
2−‖u‖6L6
6, (1.2.13)
ou D = −i∂x et on choisit u ∈ H1+(R) pour (1.2.3), w ∈ H1(R) pour (1.2.2). Si u ∈ C(R;H1
+(R)) resout
l’equation (1.2.3), alors la quantite de mouvement P (u) = ‖|D| 12u‖2L2(R) et la masse M(u) controlent la
norme H12 de la solution, ce qui permet de resoudre le probleme de Cauchy de l’equation de NLS–Szego
L2-(sur)critique sur la droite, pour tous les donnees initiales u0 ∈ H1+(R).
Theoreme 1.2.12. Soient m ≥ 0, λ = ±1 et u0 ∈ H1+(R), il existe une unique fonction u ∈ C(R;H1
+(R))qui resout l’equation de NLS–Szego,
i∂tu+ ∂2xu = λΠR(|u|2mu), u(0, x) = u0(x), (t, x) ∈ R× R. (1.2.14)
18 CHAPITRE 1. INTRODUCTION GENERALE
Demonstration. L’existence et l’unicite sont obtenues par l’injection de Sobolev. Dans le cas λ = −1,pour montrer que la solution soit globale, il suffit d’utiliser l’inegalite de Gagliardo–Nirenberg suivante,
‖u‖L2m+2(R) .m ‖|D|12u‖
mm+1
L2(R)‖u‖1
m+1
L2(R), ∀m ≥ 0, ∀u ∈ H12+(R). (1.2.15)
et les trois lois de conservation (1.2.13) pour obtenir que
supt∈R‖∂xu(t)‖2L2(R) .m ‖∂xu0‖2L2(R) + ‖|D| 12u0‖2mL2(R)‖u0‖2L2(R).
En revanche, lorsque la donnee initiale w0 ∈ H1(R) telle que ENLS(w0) < 0 et w0 ∈ L2(R, x2dx), alors ilest bien connu que la solution w : t ∈ R 7→ w(t) ∈ H1(R) de l’equation (1.2.2) associee a la donnee initialew0 explose en temps fini, grace a l’identite du viriel (voir Glassey [61], Cazenave [22, 23], Ogawa–Tsutsumi[115] etc.). On fait reference a Perelman [118] et Merle–Raphael [106] pour une description detaillee dela dynamique de l’explosion pour l’equation (1.2.2).
On va montrer que l’equation (1.2.3), admet une onde progressive qui s’ecrit sous la forme u(t, x) =eiωtQ(x + ct) pour certaines constantes ω, c ∈ R. La fonction u resout l’equation (1.2.3) si et seulementsi le profil Q resout l’equation elliptique non locale suivante
∂2xQ+ ΠR(|Q|4Q) = ωQ+ cDQ. (1.2.16)
Pour m ≥ 2 et γ ≥ 0, on definit une fonctionnelle qui est invariante par la translation spatiale, par larotation de phase et par la dilatation interieure et exterieure,
I(γ)m (f) :=
‖∂xf‖mL2(R)‖f‖m+2L2(R) + γ‖|D| 12 f‖2mL2(R)‖f‖
2L2(R)
‖f‖2m+2L2m+2(R)
, ∀f ∈ H1(R)\0. (1.2.17)
on a I(γ)m (f) = I
(γ)m (fλ,µ,y,θ), ou fλ,µ,y,θ(x) = λeiθf(µx+ y), ∀x, y, θ ∈ R et ∀λ, µ > 0.
Definition 1.2.13. La borne inferieure de I(γ)m est notee par J
(γ)m = inff∈H1
+(R)\0 I(γ)m (f). L’ensemble
de ses minimiseurs est note par
G(γ)m = f ∈ H1
+(R)\0 : I(γ)m (f) = J (γ)
m =⋃a,b>0
G(γ)m (a, b), (1.2.18)
ou G(γ)m (a, b) = f ∈ G(γ)
m : ‖f‖L2(R) = a, ‖f‖L2m+2(R) = b.
Alors on a
G(γ)m (a, b) = λf(µ·) ∈ H1
+(R)\0 : f ∈ G(γ)m (1, 1), λ = a−
1m b
m+1m et µ = b
2m+2m a−
2m+2m .
Une consequence du theoreme de decomposition en profil (theoreme 1.4.1) est le resultat suivant, qui
indique l’existence de miniseurs de I(γ)m .
Theoreme 1.2.14. Soient m ≥ 2 et γ ≥ 0, si (fn)n∈N ⊂ H1+(R) est une suite minimisante de I
(γ)m telle
que ‖fn‖L2(R) = ‖fn‖L2m+2(R) = 1 et limn→+∞ I(γ)m (fn) = J
(γ)m , il existe un profil U ∈ G
(γ)m (1, 1), une
fonction strictement croissante ψ : N→ N et une suite a valeurs reelles (xn)n∈N tels que
limn→+∞
‖fψ(n) − τxnU‖H1(R) = 0, avec τyU(x) = U(x− y), ∀x, y ∈ R. (1.2.19)
1.2. ENONCES DES RESULTATS 19
Remarque 1.2.15. Si H1+(R) est remplace par H1(R), alors il existe w ∈ H1(R)\0 telle que ‖w‖L2(R) =
‖w‖L2m+2(R) = 1, I(γ)m (w) = minf∈H1(R)\0 I
(γ)m (f) et la limite (1.2.19) marche encore.
Soient m ≥ 2, γ ≥ 0 et f ∈ H1+\0 est un minimiseur de I
(γ)m , alors d
dε
∣∣∣ε=0
log I(γ)m (f + εh) = 0, pour
tout h ∈ H1+. f resout l’equation d’Euler–Lagrange
m‖f‖m+2L2 ‖∂xf‖m−2
L2 ∂2xf + 2(m+ 1)J (γ)
m ΠR(|f |2mf)
=((m+ 2)‖f‖mL2‖∂xf‖mL2 + 2γ‖|D| 12 f‖2mL2 )f + 2γm‖f‖2L2‖|D|12 f‖2m−2
L2 Df.(1.2.20)
On fixe m = 2. Ensuite l’equation (1.2.16) est identifiee comme l’equation suivante
‖Q‖4L2
3J(γ)2
∂2xQ+ ΠR(|Q|4Q) =
2‖Q‖2L2‖∂xQ‖2L2 + γ‖|D| 12Q‖4L2
3J(γ)2
Q+2γ‖Q‖2L2‖|D|
12Q‖2L2
3J(γ)2
DQ.
Un minimiseur Q(γ) ∈ G(γ)2 sera appele un etat fondamental de la fonctionnelle I
(γ)2 , si ‖Q(γ)‖4L2 = 3J
(γ)2 .
Lorsque u(t, x) = eiωtQ(γ)(x + ct) resout (1.2.3) et Q(γ) ∈ G(γ)2 (
4
√3J
(γ)2 , b) pour certaines constantes
b, ω > 0 et c, γ ≥ 0, alors c = 0 si et seulement si γ = 0.
Si γ = 0, alors on a ‖∂xQ(γ)‖2L2 = ω2
√3J
(0)2 and ‖Q(γ)‖6L6 = 3ω
2
√3J
(0)2 .
Si γ > 0, alors on a
‖∂xQ(γ)‖2L2 =
√3J
(γ)2
8γ(4γω − c2), ‖|D| 12Q(γ)‖2L2 =
√3J
(γ)2 c
2γ, ‖Q(γ)‖6L6 = 3
√3J
(γ)2 (
ω
2+c2
8γ).
De plus, l’inegalite ‖|D| 12Q(γ)‖2L2 ≤ ‖Q(γ)‖L2‖∂xQ(γ)‖L2 implique que c2 ≤ 4γ2ωγ+2 .
Meme si les etats fondamentaux ne sont pas encore tous classifies, on a une version faible de H1-stabiliteorbitale par le theoreme 1.2.14 et par la conservation de la H
12 -norme.
Theoreme 1.2.16. Soient ε, b > 0 et γ ≥ 0, il existe δ = δ(b, ε, γ) > 0 telle que si
inff∈G(γ)
2 (4√
3J(γ)2 ,b)
‖u0 − f‖H1+(R) < δ,
alors on a supt∈R infΨ∈
⋃C(γ)−1≤θ≤C(γ) G
(γ)2 (
4√
3J(γ)2 ,θb)
‖u(t)−Ψ‖H1+(R) < ε, ou u est la solution de l’equation
(1.2.3) avec la donnee initiale u(0) = u0 et C(γ) :=
(inff∈H1
+(R)\0‖|D|
13 f‖L2(R)
‖f‖L6(R)
)−16
√J
(γ)2
1+γ .
Remarque 1.2.17. Le probleme d’unicite de l’etat fondamental d’une equation elliptique non locale estdifficile a resoudre. (Voir Frank–Lenzmann [40] et Frank–Lenzmann–Silvestre [41] pour les Laplaciensfractionnaires dans R et aussi Lenzmann–Sok [99] pour un principe de rearrangement de Fourier) Pourun γ ≥ 0 general , on a seulement la stabilite orbitale avec dilatation parce que l’unicite des etats fon-
damentaux de I(γ)2 est inconnue, la norme de L6 de l’etat fondamental Ψ qui s’approche u(t) est aussi
inconnue. On a seulement un domaine‖f‖L6
C(γ) ≤ ‖Ψ‖L6 ≤ C(γ)‖f‖L6 , ou f est l’etat fondamental qui
s’approche de la donnee initiale u0.
20 CHAPITRE 1. INTRODUCTION GENERALE
Toutefois, dans un cas particulier, on peut montrer l’unicite des etats fondamentaux a dilatation, arotation de phase et a translation spatiale pres, en utilisant l’inegalite de Cauchy–Schwarz.
Proposition 1.2.18. Dans le cas γ = m = 2, on a
J(2)2 = min
f∈H1+\0
I(2)2 (f) =
8π2
3, G
(2)2 = x 7→ λeiθ
µx+ y + i∈ H1
+ : ∀λ, µ > 0 ∀θ, y ∈ R. (1.2.21)
Si u(t, x) = eiωtQ(x+ ct) est l’onde progressive de l’equation (1.2.3) et Q ∈ G(2)2 , alors on a
3c2 = 8ω, ‖Q‖4L2 = 8π2, ‖Q‖6L6 =3πc2√
2, ‖|D| 12Q‖2L2 =
πc√2, ‖∂xQ‖2L2 =
πc2
2√
2.
On en deduit que l’onde progressive uc(t, x) = e3c2it
8 Qc(x + ct) est H1-stable orbitalement, pour tout
c > 0, ou Qc ∈ G(2)2 (
4√
8π2, 6
√3πc2√
2).
Theoreme 1.2.19. Pour tout ε, c > 0, il existe une constante δε,c > 0 telle que
inff∈G(2)
2 (4√
8π2, 6√
3πc2√2
)
‖u0 − f‖H1 < δ,
alors on a supt∈R inff∈G(2)
2 (4√
8π2, 6√
3πc2√2
)‖u(t) − f‖H1 < ε, ou u resout l’equation (1.2.3) avec la donnee
initiale u(0) = u0.
On trouvera des resultats similaires de classification des etats fondamentaux au moyen de l’inegalite deCauchy–Schwarz dans Foschi [39], Gerard–Grellier [47, 52], Pocovnicu [119] etc.
Dans le cas γ = 0, on a
I(0)2 (f) :=
‖∂xf‖2L2‖f‖4L2
‖f‖6L6
, ∀f ∈ H1(R)\0.
Tous les etats fondamentaux dans H1(R) de I(0)2 sont completement classifies dans Weinstein [145].
On sait que minf∈H1(R)\0 I(0)2 (f) = π2
4 . Il existe une unique fonction radiale a valeurs positives qui
vaut R(x) =4√3√
cosh(2x)telle que (I
(0)2 )−1(π
2
4 ) = λeiθR(µ · −y) : λ, µ > 0 θ, y ∈ R. L’onde progrs-
sive w(t, x) = eiωtR(x) est une solution instable de l’equation de Schrodinger L2-critique focalisante
(1.2.2) au sens suivant : il existe une suite w(n)0 = (1 + 1
n )R ⊂ H1(R) telle que w(n)0 → R, lorsque
n → +∞, mais la solution maximale correspondante w(n) s’explose en temps fini. On note un etat fon-
damental par Q(0) ∈ G(0)2 (
4
√3J
(0)2 , ‖Q(0)‖L6) de I
(0)2 dans l’espace de Hardy H1
+(R). Comme R /∈ H1+ et
Q+ : x 7→ 1x+i ∈ H
1+, on a π2
4 = I(0)2 (R) < J
(0)2 = I
(0)2 (Q(0)) ≤ I(0)
2 (Q+) = 4π2
3 .
La proposition 1.2.11 implique que les solutions de l’equation (1.2.3) et de l’equation (1.2.2) L2-diffusentsimultanement a l’avant et a l’arriere en temps, lorsque leurs donnees initiales sont suffisament petites.De plus, Dodson [31] a montre que la solution globale de l’equation de Schrodinger L2-critique focalisante(1.2.2) L2-diffusent simultanement a l’avant et a l’arriere en temps, si sa donnee initiale admet la masse‖w0‖ < ‖R‖L2 . Vu le resultat d’instabilite de Weinstein [145], le seui”l de masse de l’equation (1.2.2)
pour L2-diffusion egale la masse de l’etat fondamental R ∈ H1(R) de I(0)2 .
1.2. ENONCES DES RESULTATS 21
Par contre, le theoreme de stabilite orbitale (theoreme 1.2.16) implique que le seuil de masse de l’equation
(1.2.3) pour L2-diffusion est strictement plus petit que la masse de l’etat fondamental Q(0) ∈ H1+ de I
(0)2 .
On definit E ⊂ R∗+ l’ensemble des ε > 0 telles que si ‖u0‖L2 < ε, la solution correspondante de l’equation(1.2.3) L2-diffuse simultanement a l’avant et a l’arriere en temps. Si une H1-solution de l’equation (1.2.3)L2-diffuse, alors elle H1-diffuse aussi et son Lr-norme decroıt, pour tout 2 < r ≤ +∞. Pour cette raison,les ondes progressives ne peuvent pas L2-diffuser. Les solutions qui s’approchent des ondes progressivesne L2-diffusent non plus.
Corollary 1.2.20. On a sup E < ‖Q(0)‖L2 =4
√3J
(0)2 . Plus precisement, il existe une fonction u ∈
C(R;H1+(R)) qui resout l’equation (1.2.3), telle que ‖u(0)‖L2 < ‖Q(0)‖L2 , et telle que u ne L2-diffuse ni
a l’avant ni a l’arriere en temps.
Remark 1.2.21. La masse de l’etat fondamental de I(2)2 est strictement plus grande que la masse de
l’etat fondamental de I(0)2 . ‖Q(2)‖4L2 = 8π2 > 4π2 ≥ ‖Q(0)‖4L2 .
E(Q(2)) = − πc2
4√
2< 0 = E(Q(0)).
L’identification du seuil de masse pour la diffusion des solutions de l’equation (1.2.3) demeure un problemeouvert. L’equation (1.2.3) n’a pas d’invariance galileenne, qui est necessaire pour utiliser des estimationsde type Morawetz. On fait reference a Dodson [31], Killip–Visan [88], Killip–Visan–Zhang [89] pour plusde detail dans le cas non filtre.
1.2.3 L’equation de Benjamin–Ono sur la droite
Ce paragraphe correspond au chapitre 4 et a l’article Sun [135]. L’equation de Benjamin–Ono (BO) surla droite (1.2.4)
∂tu = H∂2xu− ∂x(u2), (t, x) ∈ R× R,
a ete d’abord introduite par Benjamin [8] puis par Ono [116] pour modeliser la propagation d’ondesinternes longues non lineaires dans un fluide stratifie par deux couches de differentes densites, jointes parune region mince ou la densite varie de facon continue, la couche inferieure est infinie. L’equation (1.2.4)admet une solution globale et unique dans l’espace C(R, Hs(R,R)), pour tout s ≥ 0. (Voir Tao [137] pours ≥ 1, Burq–Planchon [21] pour s > 1
4 , Ionescu–Kenig [80], Molinet–Pilod [108] et Ifrim–Tataru [79] pours ≥ 0, etc.) Si u = u(t, x) est solution de l’equation de BO (1.2.4), alors uc,y : (t, x) 7→ cu(c2t, c(x−y)) l’estaussi. Une solution u = u(t, x) s’appelle une onde solitaire s’il existe R ∈ C∞(R) qui resout l’equationelliptique non locale suivante
HR′ +R−R2 = 0, R(x) > 0 (1.2.22)
et u(t, x) = Rc(x − y − ct), ou Rc(x) = cR(cx), pour certaines constantes c > 0 et y ∈ R. L’equation(1.2.22) admet une unique solution a translation pres,
R(x) =2
1 + x2, ∀x ∈ R, (1.2.23)
trouvee par Benjamin [8] et dont l’unicite est montree dans Amick–Toland [4]. On s’interesse aux solutionsmulti-soliton de l’equation (1.2.4).
22 CHAPITRE 1. INTRODUCTION GENERALE
Definition 1.2.22. Soit N un entier strictement positif, une fonction u : R → R s’appelle un N -soliton s’il existe c1, c2, · · · , cN > 0 et x1, x2, · · · , xN ∈ R tels que u(x) =
∑Nj=1Rcj (x − xj). On note
UN ⊂ L2(R,R) l’ensemble des N -solitons.
L’ensemble UN est une sous-variete reellement analytique simplement connexe de l’espace de HilbertL2(R,R). On a dimR UN = 2N . Tous les espaces tangents de UN sont inclus dans un sous-espace auxiliaire
T := h ∈ L2(R, (1 + x2)dx) : h(R) ⊂ R,∫Rh = 0, (1.2.24)
sur lequel on definit un 2-tenseur alterne ω ∈ Λ2(T ∗) par
ω(h1, h2) =i
2π
∫R
h1(ξ)h2(ξ)
ξdξ, ∀h1, h2 ∈ T . (1.2.25)
Grace a l’inegalite de Hardy, ω ∈ Λ2(T ∗) est bien defini. Muni de la 2-forme ω : u ∈ UN 7→ ω ∈Λ2(T ∗), (UN , ω) est une variete symplectique. Soit une fonction lisse f : UN → R, son champ de vecteurshamiltonien Xf ∈ X(UN ) s’ecrit sous la forme
Xf : u ∈ UN 7→ ∂x∇f(u) ∈ Tu(UN ), (1.2.26)
ou ∇f(u) est la derivee de Frechet : df(u)(h) = 〈h,∇uf(u)〉L2 , pour tout h ∈ Tu(UN ). Le crochet dePoisson entre deux fonctions lisses f, g : UN → R est defini par
f, g : u ∈ UN 7→ ωu(Xf (u), Xg(u)) = 〈∂x∇uf(u),∇ug(u)〉L2 ∈ R.
Alors l’equation de BO (1.2.4) dans la variete UN est un systeme hamiltonien
∂tu = XE(u), E(u) =1
2〈|D|u, u〉
H−12 ,H
12− 1
3
∫Ru3. (1.2.27)
Le systeme (1.2.27) admet une unique solution globale t ∈ R 7→ u(t) ∈ UN . On s’inspire de la constructiondes coordonnees de Birkhoff pour l’equation de BO sur la tore T = R/2πZ dans Gerard–Kappeler [54].On va construire les coordonnees d’action–angle generalisees pour le systeme (1.2.27).
Soit ΩN := (r1, r2, · · · , rN ) ∈ RN : rj < rj+1 < 0, ∀j = 1, 2, · · · , N − 1 l’ensemble des actions et
ν =∑Nj=1 drj ∧ dαj la forme symplectique canonique sur ΩN × RN . On a le resultat principal.
Theoreme 1.2.23. Il existe un symplectomorphisme reellement analytique ΦN : (UN , ω)→ (ΩN×RN , ν)tel que
E Φ−1N (r1, r2, · · · , rN ;α1, α2, · · · , αN ) = − 1
2π
N∑j=1
|rj |2. (1.2.28)
Remarque 1.2.24. Par consequent, la variete UN est simplement connexe. Elle est le revetement uni-versel de la variete des potentiels a N “gap” de l’equation de Benjamin–Ono sur le tore decrite parGerard–Kappeler [54]. Dans l’appendice 4.6, on fournit une preuve directe de ces resultats topologiques,qui est independante du theoreme 1.2.23.
1.2. ENONCES DES RESULTATS 23
Remarque 1.2.25. L’application ΦN : u ∈ UN 7→ (I1(u), I2(u), · · · , IN (u); γ1(u), γ2(u), · · · , γN (u)) ∈ΩN × RN introduit les coordonnees d’action–angle generalisees de l’equation de BO dans la variete deN -solitons,
Ik, E(u) = 0, γk, E(u) =Ik(u)
π, ∀u ∈ UN . (1.2.29)
Theoreme 1.2.23 donne une description complete de l’orbite du flot du systeme (1.2.27) a conjugaison reelanalytique pres. Soit u : t ∈ R 7→ u(t) ∈ UN la solution de l’equation (1.2.27), on note rk(t) = Ik u(t)les coordonnees d’actions et αk(t) = γk u(t) les coordonnees d’angles generalises αk(t) = γk u(t), ona donc
rk(t) = rk(0), αk(t) = αk(0)− rk(0)t
π, ∀k = 1, 2, · · · , N. (1.2.30)
On donnera le detail de l’application ΦN : UN → ΩN × RN dans la definition 4.5.1 et le theoreme 4.5.2.
Afin d’etablir le lien entre les coordonnees action–angle et les parametres de translation–dilatation d’unN -soliton, on introduit la matrice d’inversion spectrale associee a ΦN , definie par
M : u ∈ UN 7→ (Mkj(u))1≤j,k≤N ∈ CN×N , Mkj(u) =
2πiIk(u)−Ij(u)
√Ik(u)Ij(u) , if j 6= k,
γj(u) + πiIj(u) , if j = k,
(1.2.31)
ou Ik, γk : U → R sont donnees par remarque 4.1.3. On peut montrer la caracterisation polynomialesuivante de UN .
Proposition 1.2.26. Une fonction a valeurs reelles u appartient a UN si et seulement s’il existe unpolynome Qu ∈ C[X] de degre N dont les racines sont contenues dans le demi-plan inferieur de Poincare
C−, et u = Πu + Πu, avec Πu = iQ′uQu
. De plus, Qu est unique et est le polynome caracteristique de la
matrice M(u) ∈ CN×N donnee par (1.2.31).
Le probleme de trouver une formule explicite pour la solution d’equation (1.2.27) devient donc un
probleme algebrique. Plus precisement, un N -soliton s’ecrit sous la forme u(x) =∑Nj=1Rcj (x − xj)
si et seulement si ses parametres de translation–dilatation xj − c−1j i1≤j≤N ⊂ CN− sont des racines du
polynome caracteristique Qu(X) = det(X −M(u)) dont les coefficients sont exprimees en fonction descoordonnees action–angle (Ij(u), γj(u))1≤j≤N ∈ ΩN ×RN . La proposition 1.2.26 est recapitulee avec plusde detail dans la proposition 4.4.1 et dans le theoreme 4.4.8, qui caracterise la variete UN en termesde theorie spectrale pour un certain operateur de Lax. En particulier, la formule d’inversion spectrale
Πu = iQ′uQu
avec Qu(x) = det(x−M(u)) est recapitulee a la formule (4.5.11), dont on deduit l’injectivite
de l’application d’action–angle ΦN : UN → ΩN ×RN . Soit u : t ∈ R 7→ u(t) ∈ UN la solution de l’equationde BO (1.2.4), on a donc la formule explicite suivante
u(t, x) = 2Im〈(M(u0)− (x+ t
πV(u0)))−1
X(u0), Y (u0)〉CN , (t, x) ∈ R× R, (1.2.32)
ou la produit scalaire de CN est 〈X,Y 〉CN = XTY , et, pour tout u ∈ UN , la matrice V(u) ∈ CN×N etles vecteurs X(u), Y (u) ∈ CN sont definis par
√2πX(u)T = (
√|I1(u)|,
√|I2(u)|, · · · ,
√|IN (u)|),
√2π−1Y (u)T = (
√|I1(u)|−1,
√|I2(u)|−1, · · · ,
√|IN (u)|−1),
V(u) =
I1(u)I2(u)
. . .IN (u)
.
24 CHAPITRE 1. INTRODUCTION GENERALE
Resume de la preuve du theoreme 1.2.23
La construction des coordonnees action–angle est fondee sur la structure de paire de Lax pour l’equationde BO (1.2.4), decouverte par Nakamura [110], voir aussi Bock–Kruskal [12]. La sous-section 4.2 estconsacree a l’etude du spectre de l’operateur de Lax Lu : h ∈ H1
+(R) 7→ −i∂xh − ΠR(uh) ∈ L2+(R) a
symbole u ∈ L2(R,R), ou ΠR est le projecteur de Szego donnee par (1.1.21) et L2+(R) est l’espace de
Hardy defini par (4.1.13). Lu est un operateur auto-adjoint sur L2+(R) borne par le bas et son spectre
essentiel est σess(Lu) = [0,+∞). Si de plus x 7→ xu(x) ∈ L2(R), on montre que toutes les valeurs propresde Lu sont simples et negatives, grace a une identite auxiliaire, decouverte par Wu [146]. Ensuite onintroduit une fonction generatrice
Hλ(u) = 〈(Lu + λ)−1Πu,Πu〉L2 , if λ ∈ C\σ(−Lu), (1.2.33)
qui engendre l’ensemble des hierarchies BO, et fournit une suite de lois de conservation controlant toutesles normes de Sobolev.
Dans la sous-section 4.3, on etudie le semi-groupe de decalage (S(η)∗)η≥0 qui opere sur l’espace de HardyL2
+(R), ou S(η)f = eηf et eη(x) = eiηx. On obtient une version faible du theoreme de Beurling–Lax,qui se ramene a la resolution d’ une equation differentielle a coefficients constants, et etablit qu’un sous-espace invariant sous l’action du generateur infinitesimal G = i d
dη
∣∣η=0+S(η)∗ s’ecrit toujours sous la
formeC≤N−1[X]
Q , pour un certain polynome unitaire Q dont les racines sont contenues dans le demi-planinferieur de Poincare C−.
La sous-section 4.4 est consacree a la description geometrique de la variete de N -solitons. On etablit sastructure reel analytique et sa structure symplectique d’abord. Ensuite, on continue d’etudier le spectre del’operateur de Lax Lu a symbole u ∈ UN . L’operateur Lu admet exactement N valeurs propres distinctesλu1 < λu2 < · · · < λuN < 0 et l’espace de Hardy se decompose sous la forme
L2+ = Hcont(Lu)
⊕Hpp(Lu), Hcont(Lu) = Hac(Lu) = ΘuL
2+, Hpp(Lu) =
C≤N−1[X]
Qu. (1.2.34)
ou Qu est le polynome caracteristique de u donnee par la proposition 1.2.26 et Θu = QuQu
est une fonctioninterieure sur le demi-plan superieur de Poincare C+. On montre la proposition 1.2.26 et on identifiela matrice M(u) dans (1.2.31) comme la matrice de l’operateur G|Hpp(Lu) associee a la base spectraleϕu1 , ϕu2 , · · · , ϕuN, ou ϕuj ∈ Ker(λuj −Lu) est telle que ‖ϕuj ‖L2 = 1 et
∫R uϕ
uj > 0. La fonction generatrice
Hλ dans (1.2.33) s’identifie comme la transformee de Borel–Cauchy de la mesure spectrale de Lu associeeau vecteur Πu, ce qui entraıne la stabilite de UN sous le flot de l’equation de BO (1.2.4) dans H∞(R,R).Par consequent le systeme (1.2.27) admet une unique solution globale u : t ∈ R 7→ u(t) ∈ UN .
Le theoreme 1.2.23 est montre dans la sous-section 4.5. Les variables d’angles (generalisees) sont desparties reelles des elements de la matrice M(u), c’est-a-dire que γj : u ∈ UN 7→ Re〈Gϕuj , ϕuj 〉L2 ∈ R et lesvariables d’action sont Ij : u ∈ UN 7→ 2πλuj ∈ R. Grace a la structure de paire de Lax dL(u)(XHλ(u)) =
[Bλu , Lu], ou L : u ∈ UN 7→ Lu ∈ B(H1+, L
2+) est R-affine et Bλu est un operateur anti-auto-adjoint sur
L2+(R), on a les formules suivantes concernant le crochet de Poisson,
2πλj , γk = 1j=k, γj , γk = 0 on UN , 1 ≤ j, k ≤ N. (1.2.35)
ce qui implique que ΦN : u ∈ UN 7→ (I1(u), I2(u), · · · , IN (u); γ1(u), γ2(u), · · · , γN (u)) ∈ ΩN × RN estune immersion reel analytique. ΦN est alors un diffeomorphisme grace au theoreme d’inversion globale
1.2. ENONCES DES RESULTATS 25
de Hadamard. L’injectivite de ΦN peut aussi etre montree par la formule d’inversion spectrale Πu =Q′uQu
avec Qu(X) = det(X−G|Hpp(Lu)). Enfin, on montre que ΦN : (UN , ω)→ (ΩN ×RN , ν) preserve la forme
symplectique en etudiant la sous-variete lagrangienne ΛN :=⋂Nj=1 γ
−1j (0) ⊂ UN .
Dans l’appendice 4.6, on etablit la simple connexite de la variete UN et un revetement de UN vers lavariete des N -gap potentiels sans utiliser la structure d’integrabilite.
Travaux connexes
L’equation de Benjamin–Ono sur la droite et sur le tore a ete largement etudiee dans le domaine de EDPsdepuis les annees 1970. On renvoie a Saut [128] pour un excellent resume de ces resultats. Matsuno [104]a trouve l’expression explicite de multi-solitons en suivant la methode de transformation bilineaire deHirota [75] dans le cas de l’equation de KdV. Les solutions multiphase sont construites par Satsuma–Ishimori [126]. Les ondes solitaires sont classifiees par Amick–Toland [4], ce resultat est revisite par letheoreme 1.2.23 et la proposition 1.2.26. Dans le travail de Dobrokhotov–Krichever [30], les solutionsde multi-phase sont construites par une methode dite d’integration en zones finies, et ces auteurs ontegalement obtenu une formule d’inversion pour des solutions de multi-phase. Par rapport a ce dernierresultat, nous etablissons une description geometrique de la transformation d’inversion spectrale, puisquenotre application action–angle ΦN : UN → ΩN × RN est un symplectomorphisme reel analytique. Deplus, la formule d’inversion spectrale
Πu(x) = iQ′u(x)
Qu(x), Qu(x) = det(x−G|Hpp(Lu)) = det(x−M(u)), ∀x ∈ R. (1.2.36)
fournit une connexion spectrale entre l’operateur de Lax Lu et le generateur infinitesimal G. Dans le casde l’equation de BO quantique, Nazarov–Sklyanin [111] ont introduit aussi une fonction generatrice Hλ.Leur methode est developpee dans Moll [109] pour l’equation de BO classique. La stabilite asymptotiquede l’onde solitaire et de solutions ayant pour donnee initiale la somme de profiles de solitons largementsepares a ete obtenue par Kenig–Martel [85].
Concernant la theorie de l’integrabilite pour l’equation de BO, apres que la paire de Lax a ete trouvee,Ablowitz–Fokas [1], Coifman–Wickerhauser [28], Kaup–Matsuno [84] et Wu [146, 147] ont obtenu desresultats remarquables sur la transformation spectrale directe et inverse pour l’equation de BO sur ladroite. Gerard–Kappeler ont construit les coordonnees de Birkhoff pour l’equation de BO sur le toredans [54]. Notons que l’equation de KdV (voir Kappeler–Poschel [81]) et NLS (voir Grebert–Kappeler[65]) integrables sur la tore T admettent aussi de telles coordonnees de Birkhoff. Elles sont associees al’introduction de sous-varietes de potentiels “finite gap” permettant de transferer la theorie des systemeshamiltoniens en dimension finie vers le cas d’un systeme hamiltonien en dimension infinie. On renvoie aMatveev [105] pour un resume anterieur sur les travaux dans le domaine de ’finite gap potentials’.
Comme on l’a vu, l’equation de Szego cubique sur la tore T (voir Gerard–Grellier [47, 49, 51, 52]) et surla droite R (voir Pocovnicu [119, 120]) admettent des coordonnees d’action–angle globales (generalisees)sur les varietes generiques des fonctions rationnelles de rang fini. De plus la formule d’inversion spectrale,qui est similaire a la formule (1.2.32) de l’equation de Szego cubique permet d’exprimer explicitement leflot de Szego en fonction de la variable temporelle et de la variable de donnee initiale. Le semi-groupe dedecalage (S(η)∗)η≥0 et son generateur infinitesimal G = i d
dη
∣∣η=0+S(η)∗ sont aussi utilises dans Pocovnicu
[119, 120] pour etablir les coordonnees d’action–angle et la formule explicite de la solution.
26 CHAPITRE 1. INTRODUCTION GENERALE
Remarque 1.2.27. En comparant avec l’equation de Szego cubique sur la droite (1.1.20) (voir theoreme1.1.14 et Pocovnicu [120]), qui admet la paire de Lax (HR
w, ARw), on observe que l’operateur de Hankel HR
w
est compact. Il n’admet donc que du spectre discret. En revanche, dans le cas de l’equation de BO surla droite avec les solutions de multi-solitons (Sun [135]), l’operateur de Lax admet a la fois du spectrecontinu [0,+∞[ et du spectre purement ponctuel qui est une partie finie de ]−∞, 0[.
Remarque 1.2.28. On peut aussi comparer les theoremes 1.1.8, 1.1.18 et 1.2.23. Dans le cas de l’equationde Szego cubique (voir Gerard–Grellier [49] et Pocovnicu [120]), l’operateur de Hankel HR
w peut avoirune valeur propre degeneree. Donc les coordonnees d’action–angle (generalisees) de l’equation de Szegocubique sur la tore (resp. sur la droite) ne sont que generiquement definies dans M(N)T (resp. M(N)R).En revanche, pour l’equation de BO sur la tore (voir Gerard–Kappeler [54]) ou sur la droite (voir Wu[146], Sun [135]), l’operateur de Lax n’admet que des valeurs propres simples dans les regimes appropries.Ainsi les coordonnees de Birkhoff (resp. d’action–angle) sont globalement bien definies dans la varietedes N -gap potentiels (resp. des N -solitons).
Enfin, contrairement aux equations de type Szego, l’equation de BO admet une suite de lois de conser-vation qui controlent toutes les normes de Sobolev Hs (Voir Ablowitz–Fokas [1], Coifman–Wickerhauser[28] dans le cas 2s ∈ N et Talbut [136] dans le cas − 1
2 < s < 0 et les lois de conservation controlant lesnormes de Besov etc).
1.3 Perspectives de recherche
Dans ce paragraphe, on decrit trois perspectives de recherche ulterieure.
1.3.1 L’equation de NLS–Szego amortie
On s’interesse au phenomene de turbulence faible de l’equation de NLS–Szego amortie sur la tore
i∂tu+ ∂2xu+ iα〈u,1〉L2(T) = ΠT(|u|2u), (t, x) ∈ R× T, α > 0. (1.3.1)
La motivation pour etudier ce modele est basee sur les deux resultats suivant. D’une part, l’equation deSzego cubique amortie
i∂tV + iα〈V,1〉L2(T) = ΠT(|V |2V ), (t, x) ∈ R× S1, α > 0, (1.3.2)
introduite dans Gerard–Grellier [53] admet un phenomene de turbulence faible different du comportementen grand temps de l’equation de Szego cubique sur la tore (1.1.7). Il existe une partie ouverte non vide
U ⊂ H12+(T) telle que pour tout s > 1
2 , si la donnee initiale V (0) ∈ U⋂Hs
+(T), alors on a
limt→+∞
‖V (t)‖Hs = +∞.
D’autre part, l’equation de NLS–Szego cubique defocalisante sur la tore (1.2.7), qui est rappelee ici
i∂tu+ ∂2xu = ΠT(|u|2u), (t, x) ∈ R× T. (1.3.3)
se comporte comme l’equation de Schrodinger cubique (1.1.5) (d=1) lorsque la dispersion est grande,se comporte comme l’equation de Szego cubique lorsque la dispersion est petite. On s’interesse donc al’influence des resultats de stabilites par le terme amorti iα〈u,1〉L2(T). Il faut d’abord etudier l’existencede la solution turbulente de l’equation (1.3.2), c’est-a-dire que limt→+∞ ‖u(t)‖Hs(T) = +∞, pour tout
1.3. PERSPECTIVES DE RECHERCHE 27
s > 12 . On note t ∈ R 7→ Sα(t)φ (respectivement t 7→ S(t)φ) la solution de l’equation de NLS–Szego
amortie (1.3.2) (respectivement l’equation de NLS–Szego (1.3.3)) avec la donnee initiale u(0) = φ. Alorsla masse de la solution est une fonctionnelle de Lyapunov de l’equation (1.3.2). Plus precisement, pour
tout φ ∈ H12+(S1), on a
∂t‖Sα(t)φ‖2L2(T) + 2α|〈Sα(t)φ,1〉L2(T)|2 = 0. (1.3.4)
Alors ‖Sα(t)φ‖L2(T) decroıt et converge, lorsque t → +∞. En outre, la quantite de mouvement P (u) =
〈Du, u〉L2 = ‖|D| 12u‖2L2(T) est conservee le long du flot de l’equation (1.3.2). Par consequent, lorsque
limt→+∞ ‖Sα(t)φ‖L2(T) = 0, alors on a
limt→+∞
‖Sα(t)φ‖Hs(T) = +∞, ∀s > 1
2
compte tenu de l’inegalite d’interpolation
‖|D| 12φ‖L2(T) = ‖|D| 12Sα(t)φ‖L2(T) ≤ ‖Sα(t)φ‖1−12s
L2(T)‖|D|sSα(t)φ‖
12s
L2(T).
Pour mieux comprendre le comportement en temps long des solutions de (1.3.2), on utilise le principe deLaSalle, qui conduit a la proposition suivante.
Proposition 1.3.1. Soient u0 ∈ H12+(S1), u∞ est une limite H
12−faible de (Sα(t)u0) lorsque t→ +∞,
alors on a (Sα(t)u∞|1) = 0. C’est-a-dire S(t)u∞ = Sα(t)u∞ est solution commune de l’equation (1.3.2)et de l’equation (1.3.3).
1.3.2 Unicite des etats fondamentaux
La deuxieme perspective de recherche concerne l’unicite a rotation de phase, a translation spatiale et a
dilatation pres, des etats fondamentaux de la fonctionnelle I(γ)2 dans la sous-section 1.2.2, lorsque γ ≥ 0
est arbitraire. Rappelons que
I(γ)2 (f) :=
‖∂xf‖2L2(R)‖f‖4L2(R) + γ‖|D| 12 f‖4L2(R)‖f‖
2L2(R)
‖f‖6L6(R)
, D = −i∂x, ∀f ∈ H1(R)\0. (1.3.5)
Un premier resultat intermediaire est le suivant.
Proposition 1.3.2. Soient m ∈ N et m ≥ 2, si f ∈ G(γ)m , alors P (f) ∈ G
(γ)m , ou P (f)(x) =
12π
∫ +∞0|f(ξ)|eixξdξ. Plus precisement, il existe deux constantes a, b ∈ R telles que f(ξ) = |f(ξ)|ei(aξ+b),
for every ξ ≥ 0.
Par consequent, il suffit de montrer l’unicite a dilatation pres des etats fondamentaux qui ont une trans-
formee de Fourier strictement positive. Si h(ξ) = |Q(ξ)| = Q(ξ), pour un certain Q ∈ G(γ)2 , apres la
dilatation h 7→ Ah(B·), alors l’equation d’Euler–Lagrange (1.2.20) devient l’equation suivante
(EL(δ)) (1 + δ(γ)ξ + ξ2)h(ξ) = P (h, h, h, h, h)(ξ), (1.3.6)
ou
P (h1, h2, h3, h4, h5)(ξ) =
∫ξ1+ξ3+ξ5=ξ2+ξ4+ξ
ξj≥0
Π5j=1hj(ξj)dξ1dξ2dξ3dξ4
28 CHAPITRE 1. INTRODUCTION GENERALE
et δ : [0,+∞)→ [0,+∞) est une fonction continue. Lorsque γ = 2, on a l’unicite de la solution. Lorsqueδ s’approche de δ(2), si on suppose que hδ, gδ verifient l’equation (1.3.6), alors on a
hδ − gδ =3P (hδ − gδ, hδ, hδ, hδ, hδ) + 2P (hδ, hδ − gδ, hδ, hδ, hδ) +O(‖hδ − gδ‖2L1)
1 + δ(γ)ξ + ξ2
On introduit ensuite l’operateur Lδ : L1(R+)→ L1(R+) tel que
Lδ(ϕ)(ξ) := ϕ(ξ)− 3P (ϕ, hδ, hδ, hδ, hδ) + 2P (hδ, ϕ, hδ, hδ, hδ)
1 + δ(γ)ξ + ξ2, ∀ξ ≥ 0.
Alors on a ‖Lδ(hδ − gδ)‖L1 = O(‖hδ − gδ‖2L1). Si l’operateur Lδ(2) est inversible, alors on peut espererutiliser le theoreme d’inversion locale pour montrer l’unicite des etats fondamentaux a rotation de phase,a translation spatiale et a dilatation pres est valable, au moins lorsque γ est au voisinage de 2.
1.3.3 Transformation de diffusion inverse (IST) de l’equation de BO
Les transformations spectrales directes et inverses pour l’equation de Benjamin–Ono ont ete formellementdecrites dans Ablowitz–Fokas [1]. Le probleme de Cauchy par des methodes de transformation spectraleinverse (IST) a ete rigoureusement resolu d’abord par Coifman–Wickerhauser [28] pour des donneespetites et suffisamment decroissantes. Ces auteurs ont montre en particulier que, pour un tel potentiel,le spectre de l’operateur de Lax est absolument continu. Y. Wu a resolu completement le probleme dediffusion direct pour des donnees arbitraires mais suffisamment decroissantes dans [146, 147]. Wu a etudiele spectre de Lu et a etabli l’existence, l’unicite et les proprietes asymptotiques de solutions de Jostassociees au probleme de diffusion direct, ce qui fournit un cadre precis pour aborder le probleme dediffusion inverse. Une solution u ∈ L2(R,R) de l’equation de BO sur la droite s’ecrirait alors comme lasomme d’une partie L2-diffusive et d’une partie multi-soliton. On s’interesse a l’interaction de ces deuxparties. On rappelle que l’operateur de Lax s’ecrit sous la forme
Lu : h ∈ H1+(R) 7→ Dh−ΠR(uh) ∈ L2
+(R), u ∈ L2(R,R), D = −i∂x. (1.3.7)
Alors Lu est un operateur auto-adjoint non borne, qui est une perturbation relativement compacte deD : h ∈ H1
+(R) 7→ Dh ∈ L2+(R).
Proposition 1.3.3. (Wu [146] et Sun [135]). Si u ∈ L2(R, (1 + x2)dx) est a valeurs reelles, λ ∈ R etϕ ∈ Ker(λ− Lu), alors on a ∣∣∣ ∫
Ruϕ∣∣∣2 = −2πλ
∫R|ϕ|2. (1.3.8)
Si u ∈ L∞(R,R) en plus, le spectre purement ponctuel de Lu est une partie finie dans ]−∞, 0[.
Une question interessante est aussi de savoir si la proposition 1.3.3 reste vraie, lorsque la conditionu ∈ L∞(R,R) et la condition x 7→ xu(x) ∈ L2(R) sont enlevees.
1.4 Preliminaires
Ce paragraphe etablit quelques resultats preliminaires. On rappelle le theoreme de decomposition enprofils de Gerard [45] et on detaille la preuve de l’existence et l’unicite des solutions globales de l’equation(1.4.3).
1.4. PRELIMINAIRES 29
1.4.1 Decomposition en profils
L’injection de Sobolev H1(R) → L∞(R) n’est pas compacte. En revanche, toute suite bornee dans H1(R)s’ecrit, a une sous-suite pres, comme la somme presque orthogonale d’une suite tendant vers zero dansl’espace de Lebesgue Lp(R) pour tout 2 < p ≤ +∞, et d’une superposition de suites de translatees–dilateesde profiles fixes (voir Gerard [45]). On obtient ainsi des versions precisees du principe de concentration–compacite de Lions [101, 102]. L’enonce qui suit transfere le resultat de Hmidi–Keraani [74] au cadre del’espace de Hardy.
Theoreme 1.4.1. (Gerard [45], Hmidi–Keraani [74]). Pour toute suite (fn)n∈N+ bornee dans H1+, il
existe une sous-suite, notee par (fφ(n))n∈N+ , une suite des profils (U (j))j∈N+ ⊂ H1+ et une suite a indice
double (x(j)n )n,j∈N+ ⊂ R telle que si j 6= k, on a |x(j)
n − x(k)n | → +∞, telles que, pour tout l ∈ N+, on ait
fφ(n)(x) =
l∑j=1
U (j)(x− x(j)n ) + r(l)
n (x), ∀x ∈ R, (1.4.1)
avec liml→+∞ lim supn→+∞ ‖r(l)n ‖Lp = 0, ∀p ∈]2,+∞]. Pour tout l ∈ N+ et s ∈ [0, 1], alors on a la
propriete de presque orthogonalite
∣∣∣‖|D|sfφ(n)‖2L2 −l∑
j=1
‖|D|sU (j)‖2L2 − ‖|D|sr(l)n ‖2L2
∣∣∣→ 0, D = −i∂x, lorsque n→ +∞. (1.4.2)
Ce theoreme est souvent utilise pour etablir l’existence de minimiseurs de certaines fonctionnelles, pourla stabilite orbitale d’ondes progressives, pour la diffusion non lineaire etc. (Voir par exemple Bahouri–Gerard [6], Gallagher–Gerard [42], Hmidi–Keraani [73], Pocovnicu [119], Krieger–Luhrmann [92], Gerard–Lenzmann–Pocovnicu–Raphael [56], Sun [134] etc.)
1.4.2 Probleme de Cauchy pour l’equation de NLS–Szego cubique sur la tore
On va montrer que l’equation de NLS–Szego cubique est globalement bien posee dans les espaces Hs,pour tout s ≥ 1
2 . i∂tu(t, x) + ∂2
xu(t, x) = Π[|u|2u](t, x) t ∈ R, x ∈ Tu(0, ·) = u0 ∈ L2
+
(1.4.3)
avec Π = ΠT : u =∑n∈Z une
inx ∈ L2(T) 7→∑n≥0 une
inx 7→ L2+(T). On rappelle que Hs
+ :=
Hs(T)⋂L2
+(T).
Theoreme 1.4.2. Soient s ≥ 12 , u0 ∈ Hs
+, il existe une unique fonction u ∈ C(R;Hs+)⋂C1(R;Hs−2
+ )qui resout l’equation (1.4.3). L’application flot u0 7→ u est continue de Hs
+ vers C(R;Hs+). En outre, la
norme ‖ · ‖H
12+
est une loi de conservation de l’equation (1.4.3).
La preuve se divise en deux parties. La premiere partie est de montrer le cas s > 12 par un theoreme du
point fixe et la loi de conservation u 7→ ‖u‖H
12
. La deuxieme partie traite le cas s = 12 par un argument
de compacite.
30 CHAPITRE 1. INTRODUCTION GENERALE
Cas s > 12
Soit s > 12 , Hs(T) est une algebre de Banach et on a l’injection de Sobolev compacte Hs(T) → L∞(T).
Il faut d’abord prouver l’existence et l’unicite locale de l’equation (1.4.3) par le theoreme du point fixe.
Lemme 1.4.3. Soit s ≥ 0, on a l’inegalite suivante
‖fg‖Hs(T) .s ‖f‖L∞(T)‖g‖Hs(T) + ‖g‖L∞(T)‖f‖Hs(T),
pour tout f, g ∈ Hs(T)⋂L∞(T). En particulier, si s > 1
2 , Hs(T) → L∞(T) et on a
‖fg‖Hs(T) .s ‖f‖Hs(T)‖g‖Hs(T),
pour tout f, g ∈ Hs(T).
Demonstration. Voir Alinhac–Gerard [3] et Bahouri–Chemin–Danchin [5] dans le cas ou T est remplaceepar R.
Proposition 1.4.4. Soient s > 12 et ε > 0, il existe T := T (ε) > 0 telle que ∀u0 ∈ Hs
+, si ‖u0‖Hs(T) < ε,l’equation (1.4.3) admet une unique solution dans l’espace C([−T, T ];Hs
+). De plus l’application du flot
u0 7→ u est lipchitzienne continue Bε −→ C([−T, T ];Hs+), ou Bε est la boule fermee de l’espace Hs
+
centree en l’origine de rayon ε.
Demonstration. Il suffit de montrer que l’application K : u 7→ (t 7→ eit∆u0− i∫ t
0ei(t−s)∆Π[|u(s)|2u(s)]ds)
est contractante de BT,2ε dans lui-meme, ou BT,2ε est la boule fermee de rayon 2ε centree sur l’originedans l’espace C([−T, T ];Hs
+), pour un certain T > 0. Grace au lemme 1.4.3, un tel T existe toujours caron a les estimations suivantes
‖Ku(t)‖Hs ≤ ‖eit∆u0‖Hs +
∫ T
−T‖|u(τ)|2u(τ)‖Hsdτ ≤ ‖u0‖Hs + 2CsT‖u‖3L∞t Hsx ,
‖Ku(t)−Kv(t)‖Hs ≤∫ T
−T‖|u(τ)|2u(τ)− |v(τ)|2v(τ)‖Hsdτ
≤ 4Cs
∫ T
−T‖u(τ)− v(τ)‖Hs(‖|u(τ)‖2Hs + ‖v(τ)‖2Hs)dτ
≤ 8CsT‖u− v‖L∞(−T,T ;Hs)
[‖u‖2L∞(−T,T ;Hs) + ‖v‖2L∞(−T,T ;Hs)
],
pour tout t ∈ [−T, T ] et ∀u, v ∈ BT,2ε. L’existence de la solution locale est donc obtenue.
Soient deux solutions u, v ∈ C([−T, T ];Hs+) de l’ equation (1.4.3) telles que u(σ) = v(σ), pour un certain
σ ∈ [−T, T ], alors pour tout intervalle I ⊂ [−T, T ] qui contient σ tel que
8Cs|I|(‖u‖2C([−T,T ];Hs(T)) + ||v||2C([−T,T ];Hs(T))) < 1,
on a ‖u− v‖L∞(I;Hs) ≤ 4Cs∫I‖u(τ)− v(τ)‖Hs(‖u(τ)‖2Hs+‖v(τ)‖2Hs)dτ ≤
12‖u− v‖L∞(I;Hs). Alors u = v
sur l’intervalle I et l’unicite globale est obtenue.
1.4. PRELIMINAIRES 31
Soient u0, v0 ∈ Bε, on note u, v leur flots associes dans l’espace C([−T, T ];Hs+). Plus precisement,
u, v ∈ BT,2ε. Pour tout t ∈ [0, T ], on a
‖u(t)− v(t)‖Hs ≤‖u0 − v0‖Hs + 4Cs
∫ t
0
‖u(τ)− v(τ)‖Hs(‖u(τ)‖2Hs + ‖v(τ)‖2Hs)dτ
≤‖u0 − v0‖Hs exp[4Cs
(‖u‖2L∞(−T,T ;Hs) + ‖v‖2L∞(−T,T ;Hs)
)T],
grace a l’inegalite de Gronwall. La continuite lipchitzienne de l’application du flot est donc obtenue.
Soit s > 12 , on note Ismax,u0
l’intervalle maximal associe a la donnee initiale u0. On a donc le critere del’explosion suivante.
Proposition 1.4.5. En ecrivant Imax,u0=]− T−, T+[, on a T+ < +∞ =⇒ limt↑T+ ||u(t)||Hs = +∞ et
T− < +∞ =⇒ limt↓−T− ||u(t)||Hs = +∞.
Afin d’obtenir l’existence de la solution globale, il suffit de montrer que ‖u(t)‖Hs n’explose jamais en touttemps. On introduit deux lois de conservation de l’equation (1.4.3).
Proposition 1.4.6. Soient s > 12 , u ∈ C(I,Hs
+) une solution de l’equation (1.4.3) definie sur un
certain intervalle I ⊂ R. Alors ‖u(t)‖L2 = ‖u(0)‖L2 et ‖|D| 12u(t)‖L2 = ‖|D| 12u(0)‖L2 pour tout t ∈ I, ou
‖|D| 12ϕ‖L2 =∑k≥0 k|ϕk|2, pour tout ϕ ∈ H
12+ .
Demonstration. Si u0 ∈ C∞+ (T), alors son flot associe u ∈ C∞(I × T). Dans ce cas,
d
dt‖u(t)‖2L2 = 2Re〈u, ∂tu〉 = −2Im〈u, ∂2
xu−Π[|u|2u]〉 = 2Im[‖∂xu(t)‖2L2 + ‖u(t)‖2L4
]= 0
d
dt‖u(t)‖2
H12
= 2Re〈1i∂xu, ∂tu〉 = −2Im〈1
i∂xu, ∂
2xu−Π[|u|2u]〉 = 2Im‖u(t)‖2
H32− 1
2
∫T∂x|u(t, x)|4dx = 0
Si u0 ∈ Hs+ pour un certain s > 1
2 , soit t0 ∈ I fixe, alors l’intervalle
I ′ := t ∈ I : ‖u(t)‖L2 = ‖u(t0)‖L2 et ‖u(t)‖H
12
= ‖u(t0)‖H
12
est fermee dans I. Il suffit de montrer que I ′ est ouverte. Soit t1 ∈ I ′, il existe une suite un(t0) ∈ C∞+ (T)telle que un(t0) −→ u(t0) dans Hs
+. On prend ε := supn∈N ‖u0‖Hs , il existe donc une famille des flotsun ∈ C∞([−T, T ] × T) associes a un0 par la proposition 1.4.4 pour un certain T = T (ε). On a donc
‖un(t)‖L2 = ‖un(t0)‖L2 et ‖|D| 12un(t)‖L2 = ‖|D| 12un(t0)‖L2 , ∀t ∈ [−T, T ]. Grace a la continuite del’application du flot u(t0) ∈ Hs
+ 7→ u ∈ C(−T, T ;Hs+), on a donc [−T, T ] ⊂ I ′. Alors I ′ = I.
On utilise alors l’inegalite de Brezis–Gallouet [17] pour finir la preuve du cas s > 12 du theoreme 1.4.2.
Lemme 1.4.7. Soit s > 12 , il existe une constante Cs > 0 telle que
‖u‖L∞(T) ≤ Cs‖u‖H 12 (T)
√√√√log
(1 +‖u‖Hs(T)
‖u‖H
12 (T)
), (1.4.4)
pour tout u ∈ Hs(T).
32 CHAPITRE 1. INTRODUCTION GENERALE
Demonstration. Supposons que u =∑|k|≤N uke
ikx +∑|k|>N uke
ikx, pour tout N ∈ N, on a
‖u‖L∞(T) ≤∑|k|≤N
|uk|+∑|k|>N
|uk|
≤
∑|k|≤N
1
1 + |k|
12 ∑|k|≤N
(1 + |k|)|uk|2 1
2
+
∑|k|>N
1
k2s
12 ∑|k|>N
k2s|uk|2 1
2
.s‖u‖H
12
(ln(1 +N))12 + ‖u‖HsN
12−s.
Ensuite, on choisit Ns− 12 = b ‖u‖Hs‖u‖
H12
c+ 1, alors ‖u‖HsN12−s ≤ ‖u‖
H12
et
1 + (ln(1 +N))12 ≤ 1 +
(ln 2 +
2
2s− 1ln(1 +
‖u‖Hs‖u‖
H12
)
) 12
.s
(ln 2 + ln(1 +
‖u‖Hs‖u‖
H12
)
) 12
.
Alors ‖u‖L∞(T) .s ‖u‖H 12
(1 + (ln(1 +N))12 ) .s
(ln(1 + ‖u‖Hs
‖u‖H
12
)
) 12
.
Grace aux lemmes 1.4.3 et 1.4.7, on peut montrer que la solution construite dans la proposition 1.4.4 estglobale.
Preuve du theoreme 1.4.2 lorsque s > 12 . En utilisant la formule de Duhamel, on a ∀t ∈ Imax,u0
⋂R+
‖u(t)‖Hs ≤ ‖u0‖Hs +
∫ t
0
‖|u(τ)|2u(τ)‖Hsdτ.
Comme Hs⋂L∞(T) est une algebre de Banach, on a
‖|u(τ)|2u(τ)‖Hs .s ‖u(τ)‖2L∞‖u(τ)‖Hs .s ‖u(τ)‖2H
12‖u(τ)‖Hs ln
(1 +‖u(τ)‖Hs‖u0‖
H12
),
ou on applique l’inegalite de Brezis–Gallouet (1.4.4) dans la deuxieme estimation ci-dessus. Grace a la loide conservation ‖u‖
H12
, on a ‖|u(τ)|2u(τ)‖Hs .s,‖u0‖H
12
‖u(τ)‖Hs ln (2 + ‖u(τ)‖Hs). Alors il existe une
constante C = C(‖u0‖12
H , s) > 0 telle que
‖u(t)‖Hs ≤‖u0‖Hs +
∫ t
0
‖|u(τ)|2u(τ)‖Hsdτ.
≤‖u0‖Hs + C
∫ t
0
‖u(τ)‖Hs ln (2 + ‖u(τ)‖Hs) dτ
≤ exp[ln(2 + ‖u0‖H
12
) exp(Ct)]
ou on utilise une version logarithmique de l’inegalite de Gronwall dans la derniere etape. Donc
‖u‖L∞(0,T ;Hs) < +∞,
pour tout T ∈ Imax,u0
⋂R+. Alors Imax,u0
= R.
Pour tout T > 0, l’application flot est donc lipschitzienne de Hs+ vers C([R]−T, T ];Hs
+), grace a la loi deconservation ‖u‖
H12
. L’application flot est donc continue de Hs+ vers l’espace de Frechet C(R;Hs
+).
1.4. PRELIMINAIRES 33
Cas s = 12
La preuve se decompose en trois etapes. Il faut d’abord construire la solution faible de l’equation (1.4.3).Ensuite on montre l’unicite de la solution faible. En utilisant les lois de conservation construites dans laproposition 1.4.6, on verifie que la solution faible est fortement continue en temps a la fin. D’abord, onintroduit le theoreme d’Aubin–Lions–Simon et un corollaire du theoreme d’Ascoli.
Lemme 1.4.8 (Aubin–Lions–Simon). Soient X0, X1, X2 trois espaces de Banach tels que l’injectionX0 → X1 est compacte et l’injection X1 → X2 est continue. L’espace
W := u ∈ L∞(0, T ;X0) : ∂tu ∈ L∞(0, T ;X2)
muni de la norme ‖u‖L∞(0,T ;X0) + ‖∂tu‖L∞(0,T ;X2). Alors l’injection W → C([0, T ];X1) est compacte.
Demonstration. Cf le Theoreme II.5.16 de la reference [15].
On designe par Cw(R;Hs(T)) l’ensemble des fonctions x : t ∈ R 7→ x(t) ∈ Hs(T) telle que la fonctiont 7→ 〈x(t), ϕ〉Hs,H−s est continue sur R, pour tout v ∈ H−s(T).
Lemme 1.4.9. Soient s ∈ R, k > 0, une suite de fonctions (xn)n∈N ⊂ C(R;Hs+)⋂C1(R;H−k+ ) telles
que
supn∈N
[‖xn‖L∞(R;Hs+) + ‖∂txn‖L∞(R;H−k+ )
]< +∞.
Alors il existe une sous-suite nk → +∞ et x ∈ Cw(R;Hs+) telles que 〈xnk , ϕ〉L2 −→ 〈x, ϕ〉L2 uni-
formement sur tout intervalle compact de R, pour tout ϕ ∈ C∞(T).
Demonstration. On definit un sous-ensemble de C∞(T) :
D := f ∈ C∞(T) : ∃N ∈ N, λj ∈ Q, −N ≤ j ≤ N tels que f(x) =
N∑n=−N
λneinx
Alors ∀s ∈ R fixe, D est un sous-espace denombrable dense de Hs(T). Pour tout fonction v ∈ D, lafamille (〈xn(t), v〉Hs,H−s)n∈N est ponctuellement bornee
supn∈N|〈xn(t), v〉Hs,H−s | ≤ ‖v‖H−s · sup
ε>0‖xn(t)‖Hs < +∞, ∀t ∈ R.
De plus, la famille (〈xn(t), v〉Hs,H−s)n∈N est equi-continue,
|〈xn(t), v〉Hs,H−s − 〈xn(t′), v〉Hs,H−s | ≤ ‖v‖Hk supε>0
supt∈R‖∂txn(t)‖H−k · |t− t′|, ∀t, t′ ∈ R
D’apres theoreme d’Ascoli et le processus diagonal, on suppose que t 7→ 〈xn(t), v〉Hs,H−s converge uni-formement sur tout les intervalles compacts dans R, pour tout v ∈ D, quitte a extraire une sous-suite.Alors on definit
gv(t) := limn→+∞
〈xn(t), v〉Hs,H−s , ∀t ∈ R.
La fonction v ∈ D 7→ gv(t) ∈ C est lineaire continue par rapport a la topologie de Hs. Alors elle admetun unique prolongement dans H−s(T) qui est representee par la forme v 7→ 〈x(t), v〉, pour une certainex(t) ∈ Hs(T), grace au theoreme de Riesz. Donc ∀t ∈ R, xn(t) x(t) dans Hs(T), et
‖x(t)‖Hs ≤ supn∈N‖xn(t)‖L∞(R;Hs(T)).
34 CHAPITRE 1. INTRODUCTION GENERALE
Apres avoir construit x, il reste a verifier sa continuite faible. Pour tout ϕ ∈ H−s(T;C2) I ⊂ R intervallecompact et η > 0, ∃v ∈ D tel que ‖v − ϕ‖H−s < η
2 supε>0 supt∈R ‖xε(t)‖Hs. Alors
supt∈I|〈xn(t)− x(t), ϕ〉Hs,H−s | ≤ sup
t∈I|〈xn(t)− x(t), v〉Hs,H−s |+ ‖v − ϕ‖H−s · sup
t∈I[‖xn(t)‖Hs + ‖x(t)‖Hs ]
On a lim supn→+∞ supt∈I |〈xn(t) − x(t), ϕ〉Hs,H−s | ≤ η, ∀η > 0. Donc 〈xn, ϕ〉 −→ 〈x, ϕ〉 uniformementen I et x ∈ Cw(R;Hs(T)).
Proposition 1.4.10. Soit u0 ∈ H12+ , il existe une fonction u ∈ Cw(R;H
12+)⋂C1w(R;H
− 32
+ ) qui resoutl’equation (1.4.3) faiblement, i.e.
d
dt〈u(t), ϕ〉
H12 ,H−
12
+ 〈∂2xu(t), ϕ〉
H−32 ,H
32
= 〈Π[|u(t)|2u(t)], ϕ〉L2 , ∀t ∈ R. (1.4.5)
pour tout ϕ ∈ C∞(T). De plus, ‖u(t)‖H
12≤ ‖u0‖
H12
pour tout t ∈ R.
Demonstration. Soient un0 ∈ C∞(T) −→ u0 dans H12+ . Alors M := 1 + supn∈N ‖un0‖H 1
2 (T)< +∞. Il existe
donc un ∈ C∞(R×R) qui est la solution de l’equation (1.4.3) qui associe a un0 , pour tout n ∈ N. Comme‖ · ‖
H12 (T)
est une loi de conservation de l’equation (1.4.3), on a supn∈N ‖un‖L∞(R;H12 (T))
= M < +∞.
Par l’injection de Sobolev H12 (T) → Lp(T), pour tout p ∈ [1,∞[, on a
‖Π[|un(t)|2un(t)]‖L2(T) ≤ ‖un(t)‖3L6(T) ≤ C‖un(t)‖3
H12 (T)≤ CM3,
pour tout t ∈ R. Alors supn∈N ‖∂tu‖L∞(R;H
− 32
+ ).M3 < +∞.
Grace a l’injection compacte H12+ → Hs
+, pour tout s < 12 , quitte a extraire une suite, on suppose que
un → u dans l’espace C([0, T ];Hs+) pour tout T > 0 et s ∈] 1
3 ,12 [, d’apres le lemme de Aubin–Lions–Simon
(Lemme 1.4.8) et le processus diagonal. Comme s > 13 , on a l’injection Hs(T) → L
21−2s (T) → L6(T).
Alors
supt∈[0,T ]
‖Π[|un(t)|2un(t)− |u(t)|2u(t)]‖L2(T)
≤C‖un − u‖L∞([0,T ];L6(T))
[‖un‖2L∞([0,T ];L6(T)) + ‖u‖2L∞([0,T ];L6(T))
].M6‖un − u‖L∞([0,T ];Hs(T)) → 0,
(1.4.6)
lorsque n→ +∞, ∀T > 0.
D’autre part, en utilisant le lemme 1.4.9 , on a u ∈ Cw(R;H12+) et pour tout ϕ ∈ C∞(T), on suppose
〈un, ϕ〉L2 = 〈un, ϕ〉H
12 ,H−
12→ 〈u, ϕ〉
H12 ,H−
12
= 〈u, ϕ〉L2 (1.4.7)
converge uniformement sur tous les intervalles compacts de R, quitte a extraire une sous-suite. Grace auxformules (1.4.6) et (1.4.7), on a la convergence uniforme de 〈∂tun, ϕ〉
H−32 ,H
32−→ d
dt 〈u(t), ϕ〉H
12 ,H−
12
sur
tout compact de R. Alors u ∈ C1w(R;H
− 32
+ ) et la formule (1.4.5) est obtenue. Comme un(t) u(t) dans
H12+ , pour tout t ∈ R. On a
‖u(t)‖H
12≤ lim inf
n→+∞‖un(t)‖
H12
= lim infn→+∞
‖un0‖H 12
= ‖u0‖H
12.
1.4. PRELIMINAIRES 35
Apres avoir construit la solution faible de l’equation (1.4.3), on va montrer qu’elle est unique associee a
u0 ∈ H12+ en utilisant la formule Duhamel et l’inegalite de Trudinger [143].
Lemme 1.4.11. Il existe une constante universelle C > 0 telle que
‖u‖Lp(T) ≤ C√p‖u‖
H12 (T)
, (1.4.8)
pour tout p ∈ [2,+∞[ et u ∈ H 12 (T).
Demonstration. Il suffit d’adapter la methode dans Chemin–Xu [27] pour les injections de Sobolevgenerales. On suppose que ‖u‖
H12
= 1. Pour tout λ > 0, on decompose u comme u = u≤λ + u>λ,
ouu≤λ(x) =
∑|k|≤λ
ukeikx, u>λ(x) =
∑|k|>λ
ukeikx.
‖u≤λ‖2H1 =∑|k|≤λ(1 + k2)|uk|2 ≤ (1 + λ)2‖u≤λ‖2
H12
. l’inegalite de Brezis–Gallouet implique qu’il existe
une constante C > 0 telle que
‖u≤λ‖L∞ . ‖u≤λ‖2H
12
(ln(1 +
‖u≤λ‖H1
‖u≤λ‖H
12
)
) 12
≤ C(ln(1 + λ))12 .
On remplace λ par λ(t) := e( t2C )2−1, alors ‖u≤λ(t)‖L∞ ≤ t
2 , for every t > 0. On a donc
x ∈ T : |u(x)| > t ⊂ x ∈ T : |u>λ(t)(x)| > t
2, ∀t > 0.
On note σ comme la mesure de Haar classique sur la tore T. Alors pour tout p ≥ 1, on a
‖u‖pLp = p
∫ +∞
0
tp−1σx ∈ T : |u(x)| > tdt ≤ 4p
∫ +∞
0
tp−3‖u>λ(t)‖2L2dt.
Comme λ(t) < |n| si et seulement si t < 2C(ln(1 + |n|)) 12 , on applique le theoreme de Fubini,
∫ +∞
0
tp−3‖u>λ(t)‖2L2dt =∑k∈Z|uk|2
∫ 2C(ln(1+|n|))12
0
tp−3dt =(2C)p−2
p− 2
∑k∈Z|uk|2(ln(1 + |k|))
p−22 .
Comme ex ≥ xm
m! , ∀x ≥ 0, on a (ln(1 + |k|))m ≤ m!(1 + |k|) ≤ mm(1 + |k|), ∀m, k ∈ N. On choisit
m = bp2c >p−2
2 . Alors ‖u‖pLp ≤4pp−2 (2C)p−2‖u‖2
H12
(p2 )p2 . Comme la fonction p ∈ [2,+∞[7→ ( 4p
p−2 )1p ∈ R
est bornee, on a
‖u‖Lp ≤ 2C(4p
p− 2)
1p
√p
2.√p.
La demonstration de l’unicite globale de la solution faible est similaire a l’argument introduit par Yudovich[151] et est utilisee dans Vladimirov [144] et Ogawa [114].
Proposition 1.4.12. Soient u1, u2 ∈ C(R;H12+) deux solutions faibles de l’equation (1.4.3) telles que
u1(0) = u2(0), alors on a u1 = u2.
36 CHAPITRE 1. INTRODUCTION GENERALE
Demonstration. Comme on a i∂tu1 + ∂2xu1 = Π[|u1|2u1] et i∂tu2 + ∂2
xu2 = Π[|u2|2u2], on definit ladifference v := u1 − u2. Alors
i∂tv + ∂2xv = Π[|u1|2u1 − |u2|2u2], v(0) = 0.
Grace a la formule de Duhamel, on a
‖v(t)‖L2 ≤∫ t
0
‖Π[|u1|2u1 − |u2|2u2](τ)‖L2dτ ≤∫ t
0
‖|u1|2u1(τ)− |u2|2u2(τ)‖L2dτ =: Z(t)
pour presque partout t ∈ R+. On calcule la derivee de Z.
Z ′(t)2 ≤C∫T|v(t, x)|2(|u1(t, x)|+ |u2(t, x)|)4dx ≤ C
∫T|v(t, x)|2(1− 1
p )(|u1(t, x)|+ |u2(t, x)|dx)4+ 2p
pour tout p ≥ 2, parce que |v(t, x)| ≤ |u1(t, x)|+ |u2(t, x)|. En utilisant l’inegalite de Holder, on a∫T|v(t, x)|2(1− 1
p )(|u1(t, x)|+ |u2(t, x)|dx)4+ 2p ≤ ‖v(t)‖
2p−2p
L2 ‖|u1(t)|+ |u2(t)|‖4p+2p
L4p+2
Grace a l’inegalite de Trudinger (1.4.8), ‖|u1(t)| + |u2(t)|‖L4p+2 ≤ C√
4p+ 2(‖u1(t)‖H
12
+ ‖u2(t)‖H
12
).
On a donc
Z ′(t) ≤ ‖v(t)‖p−1p
L2 ‖|u1(t)|+ |u2(t)|‖2p+1p
L4p+2 ≤ CpZ(t)p−1p .
Alors Z(t) ≤ (Ct)p pour tout p ≥ 2. Alors Z(t) = 0 pour presque partout t ∈ [0, 12C ]. Alors u1 = u2 sur
[0, 12C ]. Ensuite on peut montrer que u1 = u2 sur [0, n
2C ] par recurrence par rapport a n ∈ N.
Corollary 1.4.13. Soit u ∈ Cw(R;H12+)⋂C1w(R;H
− 32
+ ) l’unique solution faible de l’equation (1.4.3)
associee a la donnee initiale u0 ∈ H12+ . Alors ‖u(t)‖
H12
= ‖u0‖H
12
, pour tout t ∈ R.
Demonstration. Soit t 6= 0, il existe une solution faible v ∈ Cw(R;H12+)⋂C1w(R;H
− 32
+ ) qui resout l’equation(1.4.3) associee a la donnee initiale u(t) et ‖v(−t)‖
H12≤ ‖v(0)‖
H12
= ‖u(t)‖H
12
par la proposition 1.4.10.
Mais v(t′) = u(t+ t′) pour tout t′ ∈ R, par la proposition 1.4.12. On a donc
‖u(0)‖H
12
= ‖v(−t)‖H
12≤ ‖v(−t)‖
H12.
Preuve du theoreme 1.4.2 lorsque s = 12 . Il suffit de verifier que la solution faible construite dans la pro-
position 1.4.10 est fortement continue en temps. En effet, soient tn −→ t ∈ R, on a u(tn) u(t) dans
l’espace H12+ . Comme ‖u(t)‖
H12
= ‖u0‖H
12
= ‖u(tn)‖H
12
, pour tout n ∈ N et H12+ est un espace d’Hilbert,
on a donc u(tn) → u(t) dans H12+ . Alors u ∈ C(R;H
12+). L’application du flot est une isometrie de H
12+
vers L∞(R;H12+)⋂C(R;H
12+) par rapport a la topologie uniforme.
Chapitre 2
Long time behavior of theNLS–Szego equation on the torus
Ce chapitre est une reprise de l’article Sun [133].
Resume Dans la premiere partie de la these, on s’interesse au comportement en grand temps de l’equationde NLS–Szego cubique sur le tore.
i∂tu+ εα∂2xu = Π(|u|2u), Π = ΠT = ΠS1
, 0 < ε < 1, α ≥ 0.
On obtient deux ensembles des resultats. D’abord, on etablit une estimation de Sobolev en grand tempspour les donnees petites a l’aide d’une transformation de forme normale de Birkhoff. Le deuxieme ensemblede resultats concerne la stabilite orbitale d’ondes planes. Grace a la propriete d’instabilite de l’equationde Szego cubique sur le tore decouverte par Gerard–Grellier [50], on met en evidence un phenomene deturbulence faible lorsque la dispersion est petite.
Mots− clefs : Equation de Schrodinger cubique, Projecteur de Szego, Dispersion petite, Stabilite,Turbulence d’onde, Forme normale de Birkhoff
Abstract We are interested in the influence of filtering the positive Fourier modes in the integrablenon linear Schrodinger equation. Equivalently, we want to study the effect of dispersion added to thecubic Szego equation, leading to the NLS-Szego equation on the circle S1
i∂tu+ εα∂2xu = Π(|u|2u), Π = ΠT = ΠS1
, 0 < ε < 1, α ≥ 0.
There are two sets of results in this paper. The first result concerns the long time Sobolev estimatesfor small data. The second set of results concerns the orbital stability of plane wave solutions. Someinstability results are also obtained, leading to some wave turbulence phenomenon.
Keywords Cubic Schrodinger equation, Szego projector, Small dispersion, Stability, Wave turbulence,Birkhoff normal form
37
38 CHAPITRE 2. LONG TIME BEHAVIOR OF THE NLS–SZEGO EQUATION ON THE TORUS
2.1 Introduction
We consider the NLS-Szego equation defined on the circle S1
i∂tu+ ∂2xu = Π(|u|2u), u(0, ·) = u0. (2.1.1)
Here Π : L2(S1)→ L2(S1) denotes the orthogonal projector from L2(S1) onto the space of L2 boundaryvalues of holomorphic functions on the unit disc,
Π :∑k∈Z
ukeikx 7−→
∑k≥0
ukeikx.
We denote by L2+ := Π(L2(S1)) ⊂ L2(S1), Hs
+ := Hs(S1)⋂L2
+, for all s ≥ 0, and C∞+ := C∞(S1)⋂L2
+.
2.1.1 Motivation
The NLS-Szego equation can be seen as the combination of two completely integrable systems : thedefocusing cubic Schrodinger equation
i∂tu+ ∂2xu = |u|2u, (t, x) ∈ R× S1, (2.1.2)
and the cubic Szego equation
i∂tV = Π(|V |2V ), (t, x) ∈ R× S1. (2.1.3)
They have both a Lax Pair structure and the action-angle coordinates, which can be used to obtain theirexplicit formulas with the inversed spectral method(see Zakharov–Shabat [152], Faddeev–Takhtajan [33],Grebert–Kappeler [65], Gerard [46], for the NLS equation and Gerard–Grellier [47, 49, 51, 52] for the cubicSzego equation). However, these two Lax pairs cannot be combined in order to give a Lax pair for (2.1.1).Moreover, the long time behaviors of these two equations are totally different.
The NLS equation (2.1.2) has a sequence of conservation laws controlling every Sobolev norms(seeFaddeev–Takhtajan [33], Grebert–Kappeler [65], Gerard [46]), so all the solutions are uniformly boundedin every Hs space. Moreover, Grebert and Kappeler [65] have proved the existence of the global Birkhoffcoordinates for the NLS equation. So the solutions of (2.1.2) are actually almost periodic on R valuedinto Hs(S1).
Compared to (2.1.2), the cubic Szego equation, which stands for a non-dispersive model, has both theLax pair structure and the wave turbulence phenomenon. Its long time behavior is extremely sensibleaccording to the different initial data. P.Gerard and S.Grellier have shown that(in [50, 51, 52]) for a Gδdense subset of initial data in C∞+ , the solutions may blow up inHs, for every s > 1
2 with super–polynomialgrowth on some sequence of times, while they go back to their initial data on another sequence of timestending to infinity. For another dense subset of initial data in C∞+ , the solutions are quasi-periodic. (seealso Theorem 2.2.3).
Remark 2.1.1. Consider the following equation without the Szego projector Π on S1 :i∂tV = |V |2V,V (0, ·) = V0.
(2.1.4)
2.1. INTRODUCTION 39
Then V (t, x) = eit|V0|2V0(x) and we have ‖V (t)‖Hs ' |t|s, for all s ≥ 0, if |V0| is not a constant function.Hence, the Szego projector both accelerates the energy transfer to high frequencies, and facilitates thetransition to low frequencies for (2.1.4).
One wonders about whether filtering the positive Fourier modes can change the long time Sobolev esti-mates of the cubic defocusing Schrodinger equation. So we introduce equation (2.1.1). On the other hand,it can also be obtained from the cubic Szego equation by adding the dispersive term ∂2
x to its linear part.In order to see the gradual change of the dispersion, we add the parameter εα in front of the Laplacian∂2x to get a more general model, the NLS-Szego equation (with small dispersion) :
i∂tu+ εα∂2xu = Π(|u|2u), u(0, ·) = u0, 0 < ε < 1, α ≥ 0. (2.1.5)
Equation (2.1.1) is the special case α = 0 for (2.1.5).
We endow L2+ with the canonical symplectic form ω(u, v) = Im
∫S1
uv2π . Equation (2.1.5) has the Hamil-
tonian formalism with the energy functional
Eα,ε(u) =εα
2‖∂xu‖2L2 +
1
4‖u‖4L4 , u ∈ H1
+. (2.1.6)
Besides Eα,ε, equation (2.1.5) has two other conservation laws,Q(u) = ‖u‖2L2 ,
I(u) = Im∫S1 u∂xu = ‖u‖2
H12,
which give the estimate of the solution for low frequencies :
supt∈R‖u(t)‖Hs ≤ ‖u0‖1−2s
L2 ‖u0‖2sH
12, ∀s ∈ [0,
1
2].
Proceeding as in the case of equation (2.1.2), one can prove the global existence and uniqueness ofthe solution of the NLS-Szego equation in high frequency Sobolev spaces, by using the Brezis–Gallouettype estimate [17], the Aubin–Lions–Simon theorem (see Theorem II.5.16 in Boyer–Fabrie [15]) and theTrudinger type inequality (see Yudovich [151], Vladimirov [144], Ogawa [114] and Gerard–Grellier [47]).Its well-posedness problem in low frequency Sobolev spaces can be dealt with Strichartz’s inequalityintroduced in Bourgain [13]. Only the high frequency Sobolev estimates are considered in this paper.
Proposition 2.1.2. For every s ≥ 12 , given u0 ∈ Hs
+, there exists a unique solution u ∈ C(R, Hs+) of
(2.1.5) such that u(0) = u0. For every T > 0, the mapping u0 ∈ Hs+ 7→ u ∈ C([−T, T ], Hs
+) is continuous.
2.1.2 Main results
The first result concerns the long time stability around the null solution of the NLS-Szego equation(2.1.5). If the initial data u0 is bounded by ε, we look for a time interval Iαε , in which the solution u(t)is still bounded by O(ε). Now we state the first result of this paper.
Theorem 2.1.3. For every s > 12 , there exist two constants as ∈ (0, 1) and Ks > 0 such that for all
0 < ε 1 and u0 ∈ Hs+, if ‖u0‖Hs = ε and u denotes the solution of (2.1.5) with u(0) = u0, then
sup|t|≤ asε4−α
‖u(t)‖Hs ≤ Ksε, if α ∈ [0, 2];
sup|t|≤ asε2‖u(t)‖Hs ≤ Ksε, if α > 2.
(2.1.7)
40 CHAPITRE 2. LONG TIME BEHAVIOR OF THE NLS–SZEGO EQUATION ON THE TORUS
Moreover, the time interval Iαε = [−asε2 ,asε2 ] is maximal for the case α > 2 and s ≥ 1 in the following
sense : for every 0 < ε 1, there exists uε0 ∈ C∞+ such that ‖uε0‖Hs ' ε and for every β > 0, we have
sup|t|≤ 1
ε2+β
‖u(t)‖Hs & ε| ln ε| 12 ε, u(0) = uε0.
Remark 2.1.4. In the case α ∈ [0, 2), the proof is based on the Birkhoff normal form method, similarlyto Bambusi [7], Grebert [64], Gerard–Grellier [48] and Faou–Gauckler–Lubich [34] for instance. However,the time interval [− as
ε4−α ,asε4−α ] may not be optimal. The resonant term of 6 indices in the homological
equation can not be cancelled by the Birkhoff normal form transform.(see subsubsection 2.3.2)
The second set of results concerns the long time Hs-estimates for the solutions of (2.1.5), if its initialdatum is a perturbation of the plane wave em : x 7→ eimx, for some m ∈ N and s ≥ 1. Let u = u(t, x)be the solution of equation (2.1.5) such that ‖u(0) − em‖Hs = ε. Its energy functional (2.1.6) gives thefollowing estimate :
supt∈R‖u(t)‖H1 .‖u0‖H1
ε−α2 , ∀0 < ε < 1, α ≥ 0. (2.1.8)
However, no information on the stability of the plane waves em is obtained from (2.1.8) during theprocess ε → 0+. Consider the super-polynomial growth of Sobolev norms in the cubic Szego equationcase (see Gerard–Grellier [50, 52] and Proposition 2.2.4 in this paper), the occurence of wave turbulencephenomenon for (2.1.5) depends on the level of its dispersion. We begin with three long time stabilityresults for the polynomial dispersion εα∂2
x case with 0 ≤ α ≤ 2. The following theorem indicates H1-orbital stability of the traveling waves em for equation (2.1.5).
Theorem 2.1.5. For all ε ∈ (0, 1), α ∈ [0, 2] and m ∈ N, there exists Cm > 0 such that if ‖u(0)−em‖H1 =ε, then we have
supt∈R
infθ∈R‖u(t)− eiθem‖H1 ≤ Cmε1−
α2 .
For each t ∈ R, the infimum can be attained when θ = arg um(t). A similar result is established byZhidkov [153, Sect. 3.3] and Gallay–Haragus [43, 44] for the 1D cubic Schrodinger equation. In smalldispersion case, Theorem 2.1.5 gives a significant improvement of estimate (2.1.8). We denote by Sα,ε the
non linear evolution group defined by (2.1.5) on H12+ . In other words, for every φ ∈ H
12+ , t 7→ Sα,ε(t)φ is
the solution u ∈ C(R, H12+) of equation (2.1.5) such that u(0) = φ.
Corollary 2.1.6. For every m ∈ N, we have
sup0<ε<10≤α≤2
sup‖φ−em‖H1≤ε
supt∈R‖Sα,ε(t)φ‖H1 <∞.
Compared to Proposition 2.2.4 (see Gerard–Grellier [47, 48, 51]), the dispersive term εα∂2x counteracts the
wave turbulence phenomenon in H1 norm for equation (2.1.5), if 0 ≤ α ≤ 2. After the change of variableu(t) = ei arg um(t)(em + ε1−
α2 v(t)), we use a bootstrap argument to get long time orbital stability of the
traveling waves em with respect to higher Sobolev norms.
Proposition 2.1.7. For all s ≥ 1 and m ∈ N, there exist two constants bm,s ∈ (0, 1) and Lm,s > 0 suchthat if 0 ≤ α < 2 and ‖u(0)− em‖Hs = ε ∈ (0, 1), then we have
sup|t|≤ bm,s
ε1−α
2
infθ∈R‖u(t)− eiθem‖Hs ≤ Lm,sε1−
α2 . (2.1.9)
2.1. INTRODUCTION 41
We also look for a larger time interval in which the estimate (2.1.9) holds, by using the Birkhoff normalform transformation. But the coefficients in front of the high frequency Fourier modes in the homologicalequation may be arbitrarily large, if α ∈ (0, 2). For this reason, we return to the case α = 0 and considerequation (2.1.1).
i∂tu+ ∂2xu = Π(|u|2u).
Then the time interval can be enlarged as [−dm,sε2 ,dm,sε2 ] in this case.
Theorem 2.1.8. In the case α = 0, for all s ≥ 1 and m ∈ N, there exist three constants dm,s, εm,s ∈ (0, 1)and Km,s > 0 such that if ‖u(0)− em‖Hs = ε ∈ (0, εm,s), then we have
sup|t|≤ dm,s
ε2
infθ∈R‖u(t)− eiθem‖Hs ≤ Km,sε.
A similar result is obtained in Faou–Gauckler–Lubich [34] for the focusing or defocusing cubic Schrodingerequation on the arbitrarily dimensional torus. (see Section 2.5 for the comparison between (2.1.1) and(2.1.2))
After stating the stability results, we turn to construct some large solutions for (2.1.5) with respect totheir initial data, if the level of dispersion is exponentially small with respect to the level of perturbationof the plane wave e1 : x 7→ eix. We state the last result of this paper.
Theorem 2.1.9. There exists a constant K > 0 such that for all 0 < δ 1, we denote by U the solutionof the following NLS-Szego equation with small dispersion
i∂tU + ν2∂2xU = Π(|U |2U), U(0, x) = eix + δ, (2.1.10)
where ν = e−πK2δ2 , then we have ‖U(tδ)‖H1 ' 1
δ with tδ := πδ√
4+δ2.
This H1-instability result indicates that the support of the energy functional of equation (2.1.10) is trans-ferred to higher Fourier modes. This phenomenon is similar to the cubic Szego equation case (see Gerard–Grellier [50, 51, 52]) and the 2D cubic NLS equation case (see Colliander–Keel–Staffilani–Takaoka–Tao
[29]). Compared to Theorem 2.1.5, adding the low-level dispersion e−πKδ2 ∂2
x fails to change the quality ofwave turbulence phenomenon (Proposition 2.2.4) for the cubic Szego equation.
The second part of Theorem 2.1.3 is a consequence of Theorem 2.1.9. Indeed, if α > 2 is fixed, we rescale
u(t, x) = εU(ε2t, x) with e−πK2δ2 = ν = ε
α−22 . Then u solves (2.1.5) with u(0, x) = ε(eix + δ) and
‖u(tδ
ε2)‖H1 = ε‖U(tδ)‖H1 ' ε
δ' ε√
(α− 2)| ln ε| ε,
while tδ
ε2 '√
(α−2)| ln ε|ε2 1
ε2+β , for all β > 0. However, this method does not work in the critical caseα = 2. If u solves
i∂tu+ ε2∂2xu = Π(|u|2u), u(0, x) = ε(eix + δ),
after rescaling U(t, x) = ε−1u(ε−2t, x), we get equation (2.1.10) with ν = 1, leading to (2.1.1) with initialdatum U(0, x) = eix + δ. Theorem 2.1.5 and Theorem 2.1.8 yield the following two estimates
supt∈R‖u(t)‖H1 = O(ε), sup
|t|≤ d1,s
ε2δ2
‖u(t)‖Hs = O(ε), ∀0 < δ 1, ∀0 < ε < 1,
42 CHAPITRE 2. LONG TIME BEHAVIOR OF THE NLS–SZEGO EQUATION ON THE TORUS
for every s > 12 . The problem of the optimal time interval in the case α = 2 of Theorem 2.1.3 remains open.
This paper is organized as follows. In Section 2.2, we recall some basic facts of the cubic Szego equationand its consequences. In Section 2.3, we study long time behavior for (2.1.5) with small data and proveTheorem 2.1.9 and Theorem 2.1.3. In Section 2.4, we study the orbital stability of the plane waves emfor (2.1.5) for every m ∈ N and give the proof of Theorem 2.1.5, Proposition 2.1.7 and Theorem 2.1.8.We compare the NLS-Szego equation with the NLS equation in Section 2.5.
2.2 The cubic Szego equation
In this section, we recall some results of the cubic Szego equation
i∂tV = Π(|V |2V ), V (0, ·) = V0. (2.2.1)
2.2.1 The Lax pair structure and L∞-estimate
Given V ∈ H12+ , the Hankel operator HV : L2
+ → L2+ is defined by
HV (h) = Π(V h).
Given b ∈ L∞(S1), the Toeplitz operator Tb : L2+ → L2
+ is defined by
Tb(h) = Π(bh).
Theorem 2.2.1. (Gerard–Grellier [47]) Set V ∈ C(R;Hs+) for some s > 1
2 . Then V solves the cubicSzego equation if and only if HV satisfies the following evolutive equation
∂tHV = [BV , HV ]. (2.2.2)
where BV := i2H
2V − iT|V |2 . In other words, (LV , BV ) is a Lax pair for the cubic Szego equation.
The equation (2.2.2) yields that the spectrum of the Hankel operator HV is invariant under the flow ofthe cubic Szego equation. Thus the quantity Tr|HV | is conserved. A theorem of Peller ([117] Theorem 2p. 454) states that
‖V ‖B11,1' Tr|HV |.
Using the embedding theorem Hs → B11,1 → L∞, for any s > 1, we have the following L∞ estimate of
the Szego flow.
Corollary 2.2.2. (Gerard–Grellier [47]) Assume V0 ∈ Hs+ for some s > 1, then we have
supt∈R‖V (t)‖L∞ .s ‖V0‖Hs
2.2.2 Wave turbulence
The following theorem indicates its chaotic long time behavior with turbulence phenomenon for generalinitial data.
2.2. THE CUBIC SZEGO EQUATION 43
Theorem 2.2.3. (Gerard–Grellier [50, 51, 52]) 1.There exists a Gδ−dense set U ⊂ C∞+ such that ifV0 ∈ U , then there exist two sequences (tn)n∈N and (t′n)n∈N tending to infinity such that
limn→+∞‖V (tn)‖Hs|tn|p = +∞, ∀s > 1
2 , ∀p ≥ 1,
limn→+∞ V (t′n) = V0.
2. If V0 is rational, then V (t) is also rational for every t ∈ R and the mapping t ∈ R 7→ V (t) ∈ C∞+ isquasi-periodic.
2.2.3 A special case
Set V0(x) = V δ0 (x) := δ + eix, we denote V δ the solution of (2.2.1). Refering to Gerard–Grellier [47 Sect.6.1, 6.2 ; 48 Sect. 3 ; 51 Sect. 4], we have the following explicit formula
V δ(t, x) =aδ(t)eix + bδ(t)
1− pδ(t)eix, (2.2.3)
where
aδ(t) = e−it(1+δ2), bδ(t) = e−it(1+ δ2
2 )(δ cos(ωt)− i 2 + δ2
√4 + δ2
sin(ωt)),
pδ(t) = − 2i√4 + δ2
sin(ωt)e−itδ2
2 , ω = δ
√1 +
δ2
4.
Proposition 2.2.4. (Gerard–Grellier [47, 48, 51]) For 0 < δ 1, set tδ := π2ω = π
δ√
4+δ2∼ π
2δ . Let V δ
be the solution of (2.2.1) with V δ(0, x) = eix + δ, then we have the following estimate
‖V δ(t)‖Hs .s ‖V δ(tδ)‖Hs 's1
δ2s−1, ∀t ∈ [0, tδ].
for every s > 12 .
Demonstration. Expanding formula (2.2.3) as Fourier series, we have
‖V δ(t)‖2Hs 's|aδ(t) + bδ(t)pδ(t)|2
(1− |pδ(t)|2)2s+1=
‖V δ(t)‖2H
12
(1− |pδ(t)|2)2s−1
with ‖V δ(t)‖2H
12
= ‖V δ(0)‖2H
12
= 1. By the explicit formula of pδ, we have
‖V δ(t)‖2Hs 's(
4 + δ2
4 cos2(ωt) + δ2
)2s−1
≤ ‖V δ(tδ)‖2Hs 'Csδ4s−2
with tδ := π2ω = π
δ√
4+δ2.
44 CHAPITRE 2. LONG TIME BEHAVIOR OF THE NLS–SZEGO EQUATION ON THE TORUS
2.3 Long time behavior for small data
2.3.1 The case α ≥ 2
For all s > 12 , consider the NLS-Szego equation with small dispersion and small data.
i∂tu+ εα∂2xu = Π(|u|2u), ‖u(0)‖Hs = ε, 0 < ε < 1, α ≥ 0. (2.3.1)
At first, we show how to find the time interval Iαε = [−asε2 ,asε2 ], in which the solutions of (2.3.1) are
bounded by O(ε), for all α ≥ 0. Then we prove the maximality of Iαε in the case α > 2.
The bootstrap argument
The time interval Iαε = [−asε2 ,asε2 ] is given by a bootstrap argument.
Lemma 2.3.1. Let a, b, T > 0, q > 1 and M : [0, T ] −→ R+ be a continuous function satisfying
M(τ) ≤ a+ bM(τ)q, for all τ ∈ [0, T ]
Assume that (qb)1q−1M(0) ≤ 1 and (qb)
1q−1 a ≤ q−1
q . Then
M(τ) ≤ q
q − 1a
for all τ ∈ [0, T ].
Demonstration. The function fq : z ∈ R+ 7−→ z − bzq attains its maximum at the critical point zc =
(qb)−1q−1 . fq(zc) = q−1
q (qb)−1q−1 . Since a ≤ maxz≥0 fq(z) = fq(zc), there exists z− ≤ zc ≤ z+ such that
z ≥ 0 : fq(z) ≤ a = [0, z−] ∪ [z+,+∞[
and fq(z±) = a. Since fq(M(τ)) ≤ a, ∀0 ≤ τ ≤ T and M(0) ≤ zc, we have M([0, T ]) ⊂ [0, z−]. By the
concavity of fq on [0,+∞[, we have fq(z) ≥ fq(zc)zc
z for all z ∈ [0, zc]. Consequently, M(τ) ≤ z− ≤ qq−1a,
for all 0 ≤ τ ≤ T .
Proof of of estimate (2.1.7). For all α ≥ 0 and ε ∈ (0, 1) fixed, we rescale u as u = εµ, equation (2.3.1)becomes
i∂tµ+ εα∂2xµ = ε2Π(|µ|2µ),
‖µ(0)‖Hs = 1.(2.3.2)
Duhamel’s formula of equation (2.3.2) gives the following estimate :
sup0≤τ≤t
‖µ(τ)‖Hs ≤ ‖µ(0)‖Hs + Csε2t sup
0≤τ≤t‖µ(τ)‖3Hs (2.3.3)
Here Cs denotes the Sobolev constant in the inequality ‖|µ|2µ‖Hs ≤ Cs‖µ‖3Hs . We choose as = 427Cs
andthe following estimate holds
sup|t|≤ as
ε2
‖u(t)‖Hs ≤3
2ε, (2.3.4)
by using Lemma 2.3.1 with q = 3, T = asε2 , a = M(0) = 1, b = Csε
2T and M(t) = sup0≤τ≤t ‖µ(τ)‖Hs .The case t < 0 is similar.
2.3. LONG TIME BEHAVIOR FOR SMALL DATA 45
Optimality of the time interval if α > 2
In order to prove the optimality of Iαε in which estimate (2.3.4) holds, we set u(0, x) = ε(eix + δ) andrescale u(t, x) = εU(ε2t, x). Then, we have
i∂tU + ν2∂2xU = Π(|U |2U), U(0, x) = eix + δ,
where ν := εα−2
2 . Since the optimality is a consequence of Theorem 2.1.9, we prove at first Theorem 2.1.9by comparing U to the solution of the cubic Szego equation with the same initial datum,
i∂tV = Π(|V |2V ), V (0, x) = eix + δ.
Proof of Theorem 2.1.9. We shall estimate their difference r(t, x) := U(t, x)− V (t, x), which satisfies thefollowing equation
i∂tr + ν2∂2xr = −ν2∂2
xV + Π(V 2r + 2|V |2r) +Q(r), r(0) = 0, (2.3.5)
with Q(r) := Π(V r2 + 2V |r|2 + |r|2r). Thus, we can calculate the derivative of ‖r(t)‖2H1 ,
∂t‖r(t)‖2H1
=∂t‖r(t)‖2L2 + ∂t‖∂xr(t)‖2L2
=2Im〈i∂tr(t), r(t)〉L2 + 2Im〈∂x(i∂tr(t)), ∂xr(t)〉L2
=2Im
∫S1
ν2∂xV ∂xr + V 2r2 − V |r|2r
+ 2Im
∫S1
−ν2∂3xV ∂xr + V 2(∂xr)
2 + 2V ∂xV r∂xr + 4Re(V ∂xV )r∂xr
+ 2Im
∫S1
∂xV r2∂xr + 2V r|∂xr|2 + 2∂xV |r|2∂xr + 4V Re(r∂xr)∂xr + r2(∂xr)
2.
Then, we have the following estimate∣∣∣∂t‖r(t)‖2H1
∣∣∣≤2ν2‖∂xr‖L2(‖∂xV ‖L2 + ‖∂3
xV ‖L2) + 2‖V ‖2L∞‖r‖2H1 + 2‖V ‖L∞‖r‖L∞‖r‖2L2
+ 12‖V ‖L∞‖r‖L∞‖∂xV ‖L2‖∂xr‖L2 + 6‖r‖2L∞‖∂xV ‖L2‖∂xr‖L2
+ 12‖V ‖L∞‖r‖L∞‖∂xr‖2L2 + 2‖r‖2L∞‖∂xr‖2L2
≤ν2(2‖∂xr‖2L2 + ‖∂xV ‖2L2 + ‖∂3xV ‖2L2) + 2‖V ‖2L∞‖r‖2H1
+ 12‖V ‖L∞‖∂xV ‖L2‖r‖L∞‖∂xr‖L2 +O(‖r‖3H1).
L∞-estimate of V is given by Corollary 2.2.2 and Hs-growth of V is given by Proposition 2.2.4, for alls > 1
2 . Thus, we have
M∞ := sup0<δ<1
supt∈R‖V (t)‖L∞ < +∞,
and there exist C1, C3 > 0 such that
‖∂xV (t)‖L2 ≤ C1
δ, ‖∂3
xV (t)‖L2 ≤ C3
δ5,
46 CHAPITRE 2. LONG TIME BEHAVIOR OF THE NLS–SZEGO EQUATION ON THE TORUS
for all 0 ≤ t ≤ tδ = πδ√
4+δ2. We use a bootstrap argument to estimate the term O(‖r‖3H1). Set
T := supt > 0 : sup0≤τ≤t
‖r(τ)‖H1 ≤ 1,
then we havesup
0≤t≤T‖r(t)‖L∞ ≤ C sup
0≤t≤T‖r(t)‖H1 ≤ C,
where C denotes the Sobolev constant H1(S1) → L∞. Consequently, for all 0 ≤ t ≤ min(T, tδ), we have∣∣∣∂t‖r(t)‖2H1
∣∣∣≤ν2(‖∂xV ‖2L2 + ‖∂3
xV ‖2L2)
+ ‖r‖2H1(2ν2 + 2‖V ‖2L∞ + 12C‖V ‖L∞‖∂xV ‖L2 + 6C‖r‖L∞‖∂xV ‖L2 + 12‖V ‖L∞‖r‖L∞ + 2‖r‖2L∞)
≤ν2(C2
1
δ2+C2
3
δ10) + (2 + 2M2
∞ + 12CM∞ + 2C2 + (12CM∞ + 6C2)C1
δ)‖r‖2H1
≤K(ν2
δ10+‖r(t)‖2H1
δ), ∀0 < δ, ν < 1,
with K := max(C21 + C2
3 , 2 + 2M2∞ + 12CM∞ + 2C2 + (12CM∞ + 6C2)C1). We set
ν = εα−2
2 = e−πK2δ2 ⇐⇒ δ =
√πK
(α−2)| ln ε| .
Using Gronwall’s inequality, we deduce that
‖r(t)‖2H1 ≤ν2
δ9eπK2δ2 = δ−9e−
πK2δ2 1 δ−2, ∀0 ≤ t ≤ tδ, ∀0 < δ 1.
Since ‖V (tδ)‖H1 ' 1δ by Theorem 2.2.4, we have ‖U(tδ)‖H1 = ‖V (tδ) + r(tδ)‖H1 ' 1
δ .
Fix α > 2, for every 0 < ε 1, we set
δ = δα,ε :=√
πK(α−2)| ln ε| 1, Tα,ε := tδ
ε2 = π
2ε2δ
√1+ δ2
4
'√
(α−2)| ln ε|ε2 .
Then we have ‖u(Tα,ε)‖H1 ' ε√
(α− 2)| ln ε| ε, while u(0, x) = ε(eix + δ). Then the optimality ofIαε = [−asε2 ,
asε2 ] is obtained.
2.3.2 The case 0 ≤ α < 2
We assume at first that u(0) ∈ C∞+ so that the energy functional of (2.3.1)
Eα,ε(u) =εα
2‖∂xu‖2L2 +
1
4‖u‖4L4
is well defined. For general initial data u(0) ∈ Hs+, if s ∈ ( 1
2 , 1), we use the density argument C∞+ = Hs+
and the continuity of the mapping u(0) ∈ Hs+ 7→ C([− as
ε4−α ,asε4−α ];Hs
+) .
We rescale u(t, x) 7→ ε−α2 u(ε−αt, x), then (2.3.1) is reduced to the case α = 0. It suffices to prove the
following estimatesup|t|≤ as
ε4
‖u(t)‖Hs = O(ε)
for the NLS-Szego equation i∂tu+ ∂2xu = Π(|u|2u) with ‖u(0)‖Hs = ε.
2.3. LONG TIME BEHAVIOR FOR SMALL DATA 47
Identifying the resonances
The study of the resonant set of the NLS-Szego equation is necessary before the Birkhoff normal formtransformation. We refer to Eliasson–Kuksin [32] to see the analysis of resonances for a more general nonlinear term and the KAM theorem for the NLS equation.
We use again the change of variable u = εµ and the Duhamel’s formula of µ with η(t) =∑k≥0 ηk(t)eikx :=
e−it∂2xµ(t). Then we have
ηk(t) = µ0(t)− iε2∑
k1−k2+k3−k=0
∫ t
0
e−iτ(k21−k
22+k2
3−k2)ηk1
(τ)ηk2(τ)ηk3
(τ)dτ,
for all k ≥ 0. Recall the classical identification of the resonant setk1 − k2 + k3 − k4 = 0
k21 − k2
2 + k23 − k2
4 = 0.⇐⇒
k1 = k2
k3 = k4
or
k1 = k4
k2 = k3
.
In order to cancel all the resonances, we apply the transformation v(t) := e2itε2‖µ(0)‖2L2µ(t). As ‖µ‖L2 is
a conservation law, we have the following nonlinear Schrodinger equation without resonances
i∂tv(t) + ∂2xv(t) = ε2
[Π(|v(t)|2v(t))− 2‖v(t)‖2L2v(t)
], ∀t ∈ R. (2.3.6)
Its energy functional is
Hε(v) =1
2‖∂xv‖2L2 +
ε2
4
(‖v‖4L4 − 2‖v‖4L2
)=: H0(v) + ε2R(v). (2.3.7)
Consider the Fourier modes of v =∑n≥0 vne
inx, then we have
4R(v) =∑
k1−k2+k3−k4=0k2
1−k22+k2
3−k24 6=0
vk1vk2
vk3vk4−∑k≥0
|vk|4.
The Birkhoff normal form
Equation (2.3.6) is transferred to another Hamiltonian equation which is closer to the linear Schrodingerequation by the Birkhoff normal form transformation. We look for a symplectomorphism Ψε such thatthe energy functional Hε is reduced to the following Hamiltonian
Hε Ψε(v) = H0(v) + ε2R(v) +O(ε4−α),
where R(v) = − 14
∑k≥0 |vk|4. Ψε is chosen as the value at time 1 of the Hamiltonian flow of some energy
ε2F .
We fix the value s > 12 . Recall that, given a smooth real valued function H, we denote XH the Hamiltonian
vector field, i.e,dH(v)(h) = ω(h,XH(v)).
Given two smooth real-valued functions F and G on Hs+, their Poisson bracket F,G is defined by
F,G(v) := ω(XF (v), XG(v)) =2
i
∑k≥0
(∂vkF∂vkG− ∂vkG∂vkF ) (v), (2.3.8)
48 CHAPITRE 2. LONG TIME BEHAVIOR OF THE NLS–SZEGO EQUATION ON THE TORUS
for all v =∑k≥0 vke
ikx ∈ Hs+. In particular, if F and G are respectively homogeneous of order p and q,
then their Poisson bracket is homogeneous of order p+ q − 2.
Lemma 2.3.2. Set F (v) :=∑k1−k2+k3−k4=0 fk1,k2,k3,k4
vk1vk2
vk3vk4
, with the coefficients
fk1,k2,k3,k4=
i
4(k21−k2
2+k23−k2
4), if k2
1 − k22 + k2
3 − k24 6= 0,
0, otherwise.
Thus, F is real-valued and its Hamiltonian field XF is smooth on Hs+ such that F,H0 + R = R and
the following estimates hold for all v ∈ Hs+.
‖XF (v)‖Hs .s ‖v‖3Hs ,‖dXF (v)‖B(Hs) .s ‖v‖2Hs .
Demonstration. F is well defined because sup(k1,k2,k3,k4)∈Z4 |fk1,k2,k3,k4| ≤ 1
4 , the Sobolev embedding
yields that |F (v)| ≤ 14
(∑k≥0 |vk|
)4
.s ‖v‖4Hs . The Sobolev estimates of ‖XF (v)‖Hs and ‖dXF (v)‖B(Hs)
are given by the Young’s convolution inequality l1 ∗ l1 ∗ l2 → l2. Using (2.3.8) and the definition offk1,k2,k3,k4
, we have
F,H0(v) =i∑
k1−k2+k3−k4=0
(k21 − k2
2 + k23 − k2
4)fk1,k2,k3,k4vk1
vk2vk3
vk4
=− 1
4
∑k1−k2+k3−k4=0k2
1−k22+k2
3−k24 6=0
vk1vk2vk3vk4
=−R(v) + R(v).
Set χσ := exp(ε2σXF ) the Hamiltonian flow of ε2F , i.e.,
d
dσχσ(v) = ε2XF (χσ(v)), χ0(v) = v.
We perform the canonical transformation Ψε := χ1 = exp(ε2XF ). The next lemma will prove the localexistence of χσ, for |σ| ≤ 1 and give the estimate of the difference between v and Ψ−1
ε (v)
Lemma 2.3.3. For s > 12 , there exist two constants ρs, Cs > 0 such that for all v ∈ Hs
+, if ε‖v‖Hs ≤ ρs,then χσ(v) is well defined on the interval [−1, 1] and the following estimates hold :
supσ∈[−1,1]
‖χσ(v)‖Hs ≤3
2‖v‖Hs ,
supσ∈[−1,1]
‖χσ(v)− v‖Hs ≤ Csε2‖v‖3Hs ,
‖dχσ(v)‖B(Hs) ≤ exp(Csε2‖v‖2Hs |σ|), ∀σ ∈ [−1, 1].
2.3. LONG TIME BEHAVIOR FOR SMALL DATA 49
Demonstration. The Lipschitz coefficient of the mapping v 7−→ ε2XF (v) is bounded by Csε2‖v‖2Hs ≤
Csρ2s, by Lemma 2.3.2. If ρs is sufficiently small, the Hamiltionian flow (σ, v) 7→ χσ(v) exists on the
maximal interval (−σ∗, σ∗), by the Picard-Lindelof theorem. Assume that σ∗ < 1, then Lemma 2.3.2 andthe following integral formula
χσ(v) = v + ε2∫ σ
0
XF (χτ (v))dτ, ∀0 ≤ σ < σ∗. (2.3.9)
yield thatsup
0≤τ≤σ‖χτ (v)‖Hs ≤ ‖v‖Hs + Csσε
2 sup0≤τ≤σ
‖χτ (v)‖3Hs , ∀0 ≤ σ < σ∗ < 1.
By Lemma 2.3.1 with M(t) = sup0≤τ≤t ‖χτ (v)‖Hs , q = 3, a = M(0) = ‖v‖Hs and b = Csε2, we have
sup|σ|≤σ∗
‖χσ(v)‖Hs ≤3
2‖v‖Hs ,
if ε‖v‖Hs ≤ 23√
3Cs. This is a contradiction to the blow-up criterion. Hence σ∗ ≥ 1, and we have
sup|σ|≤1 ‖χσ(v)‖Hs ≤ 32‖v‖Hs , if ε‖v‖Hs ≤ ρs := 2
3√
3Cs. For all σ ∈ [−1, 1], by using Lemma 2.3.2,
we have
‖χσ(v)− v‖Hs ≤ |σ|ε2 sup0≤t≤|σ|
‖XF (χt(v))‖Hs ≤ Csε2 sup0≤t≤|σ|
‖χt(v)‖3Hs ≤ Csε2‖v‖3Hs .
if ε‖v‖Hs ≤ ρs. We differentiate equation (2.3.9) and use again Lemma 2.3.2 to obtain
‖dχσ(u)‖B(Hs) =‖IdHs + ε2∫ σ
0
dXF (χt(u))dχt(u)dt‖B(Hs)
≤1 + ε2∣∣∣ ∫ σ
0
‖dXF (χt(v))‖B(Hs)‖dχt(v)‖B(Hs)dt∣∣∣
≤1 + Csε2‖v‖2Hs
∣∣∣ ∫ σ
0
‖dχt(v)‖B(Hs)dt∣∣∣
≤eCsε2|σ|‖v‖2Hs , ∀σ ∈ [−1, 1].
Gronwall’s inequality is used in the last step.
Composed with the symplectomorphism Ψε = χ1, the energy functional Hε can be reduced to the normalform.
Lemma 2.3.4. For s > 12 , there exists a smooth mapping Y : Hs
+ −→ Hs+ and a constant C ′s > 0 such
that XHεΨε = XH0 + ε2XR + ε4Y,
‖Y (v)‖Hs ≤ C ′s‖v‖5Hs ,
for all v ∈ Hs+ such that ε‖v‖Hs ≤ ρs. Set w(t) := Ψ−1
ε (v(t)), then we have∣∣∣ d
dt‖w(t)‖2Hs
∣∣∣ ≤ C ′sε4‖w(t)‖6Hs , (2.3.10)
if ε‖w(t)‖Hs ≤ ρs.
50 CHAPITRE 2. LONG TIME BEHAVIOR OF THE NLS–SZEGO EQUATION ON THE TORUS
Demonstration. We expand the energy Hε χ1 = H0 χ1 + ε2R χ1 with Taylor’s formula at time σ = 1around 0. Since χ0 = IdHs+ , we have
Hε χ1 =
(H0 +
d
dσ[H0 χσ]|σ=0 +
∫ 1
0
(1− σ)d2
dσ2[H0 χσ]dσ
)+ ε2R+ ε2
∫ 1
0
d
dσ[R χσ]dσ
=H0 + ε2 [F,H0+R] + ε4∫ 1
0
(1− σ)F, F,H0 χσ + F,R χσdσ
=H0 + ε2R+ ε4∫ 1
0
[(1− σ)F, R+ σF,R
] χσdσ
We set G(σ) := (1 − σ)F, R + σF,R, ∀σ ∈ [0, 1]. Since XF,R and XF,R(u) are homogeneous ofdegree 5 with uniformly bounded coefficients, we have
‖XG(σ)(v)‖Hs ≤ (1− σ)‖XF,R(v)‖Hs + σ‖XF,R(v)‖Hs .s ‖v‖5Hs , ∀v ∈ Hs+,
By the chain rule of Hamiltonian vector fields :
XG(σ)χσ (v) = dχ−σ(χσ(v)) XG(σ)(χσ(v)), ∀v ∈ Hs+, ∀σ ∈ [0, 1], (2.3.11)
and Lemma 2.3.3, we have
‖XG(σ)χσ (v)‖Hs ≤ ‖dχ−σ(χσ(v))‖B(Hs)‖XG(σ)(χσ(v))‖Hs .s eCsε2‖v‖2Hs ‖χσ(v)‖5Hs .s ‖v‖5Hs ,
for all v ∈ Hs+ such that ε‖v‖Hs ≤ ρs. Thus we define Y :=
∫ 1
0XG(σ)χσdσ and we have
XHεΨε = XHεχ1= XH0
+ ε2XR + ε4Y.
If ε‖v‖Hs ≤ ρs, then ‖Y (v)‖Hs .s ‖v‖5Hs .
Since XR(w)(k) = −i|wk|2wk, ∀k ≥ 0 and w(t) = Ψ−1ε (v(t)), w =
∑k≥0 wke
ikx solves the followinginfinite dimensional Hamiltonian system of Fourier modes :
i∂twk(t)− k2wk(t)− ε2|wk(t)|2wk(t) = iε4 Y (w(t))(k), ∀k ≥ 0. (2.3.12)
If ε‖w(t)‖Hs ≤ ρs, then we have∣∣∣∂t‖w(t)‖2Hs∣∣∣ ≤ 2ε4‖Y (w(t))‖Hs‖w(t)‖Hs .s ε4‖w(t)‖6Hs .
End of the proof of the case 0 ≤ α < 2
Demonstration. Recall that w(t) = χ−1(v(t)) and ‖v(0)‖Hs = 1. Lemma 2.3.3 yields that if ε ≤ ρs, thenwe have
‖v(0)− w(0)‖Hs = ‖v(0)− χ−1(v(0))‖Hs ≤ Csε2‖v(0)‖3Hs ≤ Cs.
2.3. LONG TIME BEHAVIOR FOR SMALL DATA 51
Set Ks := 3Cs + 1. Then ‖w(0)‖Hs ≤ Ks3 . We define
εs := min
(ρs
3Ks,
1√8CsK2
s
)and
T := supt ≥ 0 : sup0≤τ≤t
‖v(τ)‖Hs ≤ 2Ks.
For all ε ∈ (0, εs) and t ∈ [0, T ], we have ε‖v(t)‖Hs ≤ ρs. Hence Lemma 2.3.3 gives the following estimate
‖w(t)‖Hs ≤‖v(t)‖Hs + ‖χ−1(v(t))− v(t)‖Hs≤‖v(t)‖Hs + Csε
2‖v(t)‖3Hs≤2Ks + 8CsK
3s ε
2
≤3Ks.
So we have ε sup0≤t≤T ‖w(t)‖Hs ≤ ρs and∣∣∣ d
dt‖w(t)‖2Hs∣∣∣ ≤ C ′sε
4‖w(t)‖6Hs , by Lemma 2.3.4. Set as :=1
37K4sC′s. We can precise the estimate of ‖w(t)‖Hs by limiting 0 ≤ t ≤ asε−4 :
‖w(t)‖2Hs ≤‖w(0)‖2Hs + C ′stε4‖w(t)‖6Hs ≤
K2s
9+ 36C ′sK
6s tε
4 ≤ 4K2s
9,
for all 0 ≤ t ≤ min(T, asε4 ). Then we have
‖v(t)‖Hs ≤ ‖w(t)‖Hs + ‖χ1(w(t))− w(t)‖Hs ≤ ‖w(t)‖Hs + Csε2‖w(t)‖3Hs ≤ Ks,
for all t ∈ [0, asε4 ]. Consequently, we have
sup0≤t≤ as
ε4
‖u(t)‖Hs = ε sup0≤t≤ as
ε4
‖v(t)‖Hs ≤ Ksε.
In the case t < 0, we use the same procedure and we replace t by −t.
The open problem of optimality
Recall that Hε = H0 + ε2R is the energy functional of the equation
i∂tu+ ∂2xu = Π(|u|2u), ‖u(0, ·)‖Hs = ε. (2.3.13)
with H0(v) = 12‖∂xv‖
2L2 and R(v) = 1
4 (‖v‖4L4 − 2‖v‖4L2). In order to get a longer time interval in whichthe solution is uniformly bounded by O(ε), we expand the Hamiltonian Hε χ1 by using the Taylorexpansion of higher order to see whether the resonances can be cancelled by the Birkhoff normal formmethod.
Hε χ1
=H0 χ1 + ε2R χ1
=H0 + ∂σ(H0 χσ)∣∣σ=0
+1
2∂2σ(H0 χσ)
∣∣σ=0
+1
2
∫ 1
0
(1− σ)2∂3σ(H0 χσ)dσ
+ ε2(R+ ∂σ(R χσ)
∣∣σ=0
+
∫ 1
0
(1− σ)∂2σ(H0 χσ)dσ
)=H0 + ε2R+
ε4
2F,R+ R+
ε6
2
∫ 1
0
(1− σ)F, F, (1− σ)R+ (1 + σ)R χσdσ
52 CHAPITRE 2. LONG TIME BEHAVIOR OF THE NLS–SZEGO EQUATION ON THE TORUS
We try to cancel the term ε4
2 F,R+R by using a canonical transform to Hε1 = Hεχ1 with the following
functional
G(v) =∑
k1−k2+k3−k4+k5−k6=0
gk1,k2,k3,k4,k5,k6vk1
vk2vk3
vk4vk5
vk6.
Then we should solve the homological equation G,H0+ 12F,R+ R = 0.
G,H0(v)
=i∑
k1−k2+k3−k4+k5−k6=0
(k21 − k2
2 + k23 − k2
4 + k25 − k2
6)gk1,k2,k3,k4,k5,k6vk1
vk2vk3
vk4vk5
vk6.
We can calculate that
1
2F,R+ R(v)
=2Im∑
k1−k2+k3−k4+k5−k6=0ki,k:=k1−k2+k3≥0
fk1,k2,k3,kvk1vk2vk3vk4vk5vk6
− 4Im∑
k1−k2+k3−k4+k5−k6=0ki,k:=k1−k2+k3≥0,k5=k6
fk1,k2,k3,kvk1vk2
vk3vk4
vk5vk6
− 2Im∑
k1−k2+k3−k4+k5−k6=0ki,k:=k1−k2+k3≥0,k4=k5=k6
fk1,k2,k3,kvk1vk2
vk3vk4
vk5vk6
In the first term of the preceding formula, there is a resonant set k21 − k2
2 + k23 − k2
4 + k25 − k2
6 = 0 thatcannot be cancelled by the other two terms. Thus the resonant subset
k1 − k2 + k3 − k4 + k5 − k6 = 0
k21 − k2
2 + k23 − k2
4 + k25 − k2
6 = 0
should be cancelled before the Birkhoff normal form transform, just like the step µ 7→ v = e2itε2‖µ(0)‖2L2µ(t),
which can cancel all the resonances k1 − k2 + k3 − k4 = 0
k21 − k2
2 + k23 − k2
4 = 0
before we do the canonical transformation Hε 7→ Hε χ1. We only know that fk1,k2,k3,k1−k2+k3=
fk4,k5,k6,k4−k5+k6if
k1 − k2 + k3 − k4 + k5 − k6 = 0
k21 − k2
2 + k23 − k2
4 + k25 − k2
6 = 0.
This resonant subset contains the case k5 6= k6. The optimality of the time interval for the case 0 ≤ α < 2remains open. We can see Grebert–Thomann [66] and Haus–Procesi [71] for instance to analyse theresonant set for 6 indices for the quintic NLS equation.
2.4. ORBITAL STABILITY OF THE PLANE WAVE 53
2.4 Orbital stability of the plane wave
Consider the following NLS-Szego equation
i∂tu+ εα∂2xu = Π(|u|2u), 0 < ε < 1, 0 ≤ α ≤ 2. (2.4.1)
We shall prove at first H1-orbital stability of the traveling waves em, for all m ∈ N. Then, we study theirlong time Hs-stability, for all s ≥ 1.
2.4.1 The proof of Theorem 2.1.5
We follow the idea of using conserved quantities mentioned in Gallay–Haragus [43] for equation (2.4.1).
Demonstration. For all m ∈ N, 0 < ε < 1 and 0 ≤ α ≤ 2, we denote u(0, x) = eimx + εf(x) with‖f‖H1 ≤ 1. The NLS-Szego equation has three conservation laws :
Q(u(t)) := ‖u(t)‖2L2 = ‖u(0)‖2L2 ;
P (u(t)) := (Du(t), u(t)) = P (u(0));
Eα,ε(u(t)) := εα
2 ‖∂xu(t)‖2L2 + 14‖u(t)‖4L4 = Eα,ε(u(0)),
with D = −i∂x and (u, v) := Re∫S1 uv. Thus the following quantity is conserved,
εα
2‖Du(t)−mu(t)‖2L2 +
1
4‖|u(t)|2 − 1‖2L2
=Eα,ε(u(t))− εαmP (u(t)) +|m|2εα − 1
2Q(u(t)) +
1
4
=ε2∫S1
|Ref(x)e−imx|2dx+ε2+α
2‖Df −mf‖2L2 + ε3
∫S1
|f(x)|2Re(f(x)e−imx)dx+ ε4‖f‖4L4
.mε2.
Then, we have supt∈R ‖Du(t)−mu(t)‖L2 .m ε1−α2 . Recall that em(x) = eimx, then the following estimate
holds,
‖u(t)− um(t)em‖2H1 =∑n 6=m
(1 + n2)|un(t)|2 .m ‖Du(t)−mu(t)‖2L2 .m ε2−α.
We have
infθ∈R‖u(t)− um(0)eiθem‖2H1
=‖u(t)− um(0)ei(arg um(t)−arg um(0))em‖2H1
=(1 +m2)∣∣∣|um(t)| − |um(0)|
∣∣∣2 + ‖u(t)− um(t)em‖2H1
and by the conservation of ‖u(t)‖L2 , we have∣∣∣|um(t)| − |um(0)|∣∣∣2 ≤∣∣∣|um(t)|2 − |um(0)|2
∣∣∣=∣∣∣ ∑n6=m
|un(0)|2 −∑n 6=m
|un(t)|2∣∣∣
= max(‖u(0)− um(0)em‖2L2 , ‖u(t)− um(t)em‖2L2)
.mε2−α.
54 CHAPITRE 2. LONG TIME BEHAVIOR OF THE NLS–SZEGO EQUATION ON THE TORUS
Thus supt∈R ‖u(t) − um(0)ei(arg um(t)−arg um(0))em‖H1 .m ε1−α2 . The proof can be finished by um(0) =
1 + εfm = 1 +O(ε).
The preceding theorem also holds for the defocusing NLS equation on Td, for d = 1, 2, 3 (in the energysub-critical case) with T = R/Z ' S1.(see Gallay–Haragus [43, 44]) We refer to Zhidkov [153 Sect. 3.3]for a detailed analysis of the stability of plane waves.
Remark 2.4.1. Obtaining the estimate supt∈R ‖u(t)− um(t)em‖L∞ .m ε1−α2 by the Sobolev embedding
H1(S1) → L∞, we can also proceed by using the following estimate, which is uniform on x and t,
|u(t, x)|2 − 1 = |um(t)|2 − 1 +Om(ε1−α2 ).
The notation Om means that sup(t,x)∈R×S1
∣∣∣|u(t, x)|2−|um(t)|2∣∣∣ .m ε1−
α2 . Integrating the preceding term
with respect to x, we have
||um(t)|2 − 1| ≤ ‖|u(t)|2 − 1‖L2 + ‖Om(ε1−α2 )‖L2 .m ε1−
α2
Thus |um(t)| = 1 +Om(ε1−α2 ) and um(t) = ei arg um(t) +Om(ε1−
α2 ). Then we have
supt∈R‖u(t)− ei arg um(t)em‖2H1 .m ε2−α. (2.4.2)
Recall that if z = 1 +O(ε) then ei arg z = 1 +O(ε).
2.4.2 Long time Hs-stability
For every s ≥ 1, we suppose that ‖u(0)− em‖Hs ≤ ε. Thanks to estimate (2.4.2), we change the variableu 7→ v = vm,α,ε(t, x) =
∑n≥0 vn(t)einx ∈ C∞(R× S1) such that
u(t, x) = ei arg um(t)(eimx + ε1−α2 v(t, x)) (2.4.3)
to study Hs-stability of plane waves em and we have
Im := sup0≤α≤2
sup0<ε<1
supt∈R‖v(t)‖H1 < +∞. (2.4.4)
Proposition 2.4.2. For every s ≥ 1, m ∈ N, ε ∈ (0, 1) and α ∈ [0, 2), if u is smooth and solves (2.4.1)with u(0, x) = eimx + εf(x) and ‖f‖Hs ≤ 1, v is defined by formula (2.4.3), then we have
vm(t) ∈ R, ∀t ∈ R,supt∈R |vm(t)| .m εmin(α2 ,1−
α2 ),
supt∈R |∂tvm(t)| .m ε1−α2 ,
‖v(0)‖Hs .m,s εα2 .
Moreover, there exists a smooth function ϕ = ϕm : R→ R/2πZ ' S1 and ε∗m ∈ (0, 1) such that for every1 < 2− < 2, we have
arg um(t) = −(1 +m2εα)t+ εmin(1,2−α)ϕ(t),
sup0≤α≤2− sup0<ε<ε∗msupt∈R |ϕ′(t)| < +∞.
2.4. ORBITAL STABILITY OF THE PLANE WAVE 55
The parameter v satisfies the following equation
i∂tv + εα∂2xv −He2imx(v)− (1−m2εα + εmin(1,2−α)ϕ′(t))v
=εmin(α2 ,1−α2 )ϕ′(t)eimx + ε1−
α2 Π(e−imxv2 + 2eimx|v|2) + ε2−αΠ(|v|2v),
(2.4.5)
where He2imx(v) := Π[e2imxv] denotes the Hankel operator of symbol e2m.
Demonstration. Since um(t) = ei arg um(t)(1 + ε1−α2 vm(t)), we have 1 + ε1−
α2 vm(t) = |um(t)| ∈ R. So
vm(t) ∈ R, for all t ∈ R. By using the conservation law ‖ · ‖L2 and estimate (2.4.4), we have
1 + 2εRefm + ε2 = ‖u(0)‖2L2 = ‖u(t)‖2L2 = 1 + 2ε1−α2 vm(t) + ε2−α‖v(t)‖2L2 ,
which yields that supt∈R |vm(t)| .m εmin(α2 ,1−α2 ). Recall that
u(0, x) =∑n≥0
un(0)einx = eimx + εf(x).
Then we have um(0) = 1 + εfm = 1 +O(ε) and |ei arg um(0) − 1| . ε. Thus we have
ε1−α2 ‖v(0)‖Hs . (1 +m2)
s2 |ei arg um(0) − 1|+ ε‖f‖Hs .m,s ε.
We define θ(t) := arg(um(t)). Combing the following two formulas
i∂tu+ εα∂2xu = eiθ(t)
[ε1−
α2 (i∂tv + εα∂2
xv − θ′(t)v)− (m2εα + θ′(t))eimx]
Π[|u|2u] = eiθ(t)[eimx + ε1−
α2 (2v + Π(e2imxv)) + ε2−αΠ(e−imxv2 + 2eimx|v|2) + ε3(1−α2 )Π(|v|2v)
]we obtain that
ε1−α2 [i∂tv + εα∂2
xv −He2imx(v)− (2 + θ′(t))v]
=(1 +m2εα + θ′(t))eimx + ε2−αΠ(e−imxv2 + 2eimx|v|2) + ε3(1−α2 )Π(|v|2v),(2.4.6)
where He2imx(v) := Π[e2imxv]. The Fourier mode vm(t) satisfies the following equation
ε1−α2
[i∂tvm(t)−m2εαvm(t)− vm(t)− (2 + θ′(t))vm(t)
]=1 +m2εα + θ′(t) + ε2−αΠ(e−imxv2 + 2eimx|v|2)m(t) + ε3(1−α2 )Π(|v|2v)m(t).
Estimate (2.4.4) yields that
supt∈R|ε2−αΠ(e−imxv2 + 2eimx|v|2)m(t) + ε3(1−α2 )Π(|v|2v)m(t)| .m ε2−α.
Thus, we have
ε1−α2
[i∂tvm(t)− (3 +m2εα + θ′(t))vm(t)
]= 1 +m2εα + θ′(t) +Om(ε2−α). (2.4.7)
The imaginary part and the real part of (2.4.7) give respectively the two following estimates :
supt∈R|∂tvm(t)| .m ε1−
α2 ;
1 +m2εα + θ′(t) =−2ε1−
α2 vm(t) +Om(ε2−α)
1 + ε1−α2 vm(t)
=Om(εmin(1,2−α))
1 +Om(εmin(1,2−α))= Om(εmin(1,2−α)).
56 CHAPITRE 2. LONG TIME BEHAVIOR OF THE NLS–SZEGO EQUATION ON THE TORUS
for all 0 < ε 1. Then we define ϕ(t) := (1+m2εα)t+θ(t)εmin(1,2−α) . Consequently, there exists ε∗m ∈ (0, 1) such that
arg um(t) = −(1 +m2εα)t+ εmin(1,2−α)ϕ(t)
sup0≤α≤2− sup0<ε<ε∗msupt∈R |ϕ′(t)| < +∞.
We replace θ′(t) by −1−m2εα + εmin(1,2−α)ϕ′(t) in (2.4.6) and we obtain (2.4.5).
Proof of Proposition 2.1.7
For every n ∈ N, we define the projector Pn : L2+ → L2
+ such that
Pn(∑k≥0
vkeikx) =
n∑j=0
vkeikx.
Now we prove Proposition 2.1.7 by using a bootstrap argument.
Demonstration. At the beginning, we suppose that u(0) ∈ C∞+ . In the general case u(0) ∈ Hs+, the proof
can be completed by using the continuity of the mapping u(0) 7→ u from Hs+ to C([− bs,m
ε1−α2,bs,m
ε1−α2
], Hs+).
We use the same transformation u 7→ v as (2.4.3). Proposition 2.4.2 yields that there exists Am,s ≥ 1such that ‖v(0)‖Hs .m,s ε
α2 ≤ Am,s. By using estimate (2.4.4), we have
supε∈(0,1)
supt∈R‖P2m(v(t))‖Hs ≤ (1 + 4m2)
s2 Im
We define that Lm,s := max(2(1 + 4m2)s2 Im, 2Am,s + 1) and
T := supt > 0 : sup0≤τ≤t
‖v(τ)‖Hs ≤ 2Lm,s.
Rewrite equation (2.4.5) with Fourier modes and we have
i∂tvn − (1 + (n2 −m2)εα + εmin(1,2−α)ϕ′(t))vn = ε1−α2 [Z(v(t))]n, ∀n ≥ 2m+ 1,
with Z(v) =∑n≥0[Z(v)]ne
inx = Π(e−imxv2 + 2eimx|v|2) + ε1−α2 Π(|v|2v). Then we have
‖Z(v)− P2m(Z(v))‖Hs .s ‖v‖2Hs + ε1−α2 ‖v‖3Hs . (2.4.8)
Then we have
|∂t‖v(t)− P2m(v(t))‖2Hs | ≤2ε1−α2
∑n≥2m+1
(1 + n2)s|vn(t)||[Z(v(t))]n|
≤2ε1−α2 ‖v(t)‖Hs‖Z(v(t))− P2m(Z(v(t)))‖Hs
.sε1−α2 ‖v(t)‖3Hs + ε2−α‖v(t)‖4Hs .
For all t ∈ [0, T ], we have
‖v(t)‖2Hs =‖P2mv(t)‖2Hs + ‖v(t)− P2m(v(t))‖2Hs≤(1 + 4m2)sI2
m + Cs(ε1−α2 ‖v(t)‖3Hs + ε2−α‖v(t)‖4Hs)t+ ‖v(0)‖2Hs
≤1
4L2m,s + 32CsL
4m,sε
1−α2 t+A2m,s.
Define bm,s = 164CsL2
m,sand we have ‖v(t)‖Hs ≤ Lm,s, for all t ∈ [0,
bs,m
ε1−α2
]. The case t < 0 is similar.
2.4. ORBITAL STABILITY OF THE PLANE WAVE 57
Homological equation
We try to improve Proposition 2.1.7 and get a longer time interval in which the solution v is stillbounded by Om,s(1), by using the Birkhoff normal form transformation. Recall the symplectic formω(u, v) = Im
∫S1 uv
dθ2π on the energy space H1
+ and the Poisson bracket for two smooth real-valuedfunctionals F,G : C∞+ → R
F,G(v) =2
i
∑k≥0
(∂vkF∂vkG− ∂vkG∂vkF ) (v),
for all v =∑k≥0 vke
ikx ∈ C∞+ . For all 0 ≤ α < 2 and 0 < ε 1, equation (2.4.5) is a non autonomousHamiltonian equation. Its energy functional is
Hm,α,ε(t, v)
=Hm,α,ε0 (v) + ε1−α2Hm1 (v) +
ε2−α
4N4(v) + εmin(α2 ,1−
α2 )ϕ′(t)(Lm(v) +
ε1−α2
2N2(v)),
with Hm,α,ε0 (v) = εα
2 ‖∂xv‖2L2 + 1−m2εα
2 ‖v‖2L2 + 12
∫S1 Re(e2imxv2),
Hm1 (v) = Re(∫S1 e−imx|v|2v),
Np(v) = ‖v‖pLp , p = 2 or 4,
Lm(v) = Revm.
We want to cancel all the high frequencies in the term Hm1 (v) by composing Hm,α,ε with the Hamiltonianflow of some auxiliary functional ε1−
α2 Fm. In order to get the appropriate Fm, we need to solve the
following homological systemFm,Hm,α,ε0 (v) +Hm1 (v) = Rm(v)
Fm,Lm(v) = −Nm2 (v) := −
∑n≥2m+1 |vn|2
such that Rm depends only on finitely many Fourier modes of v. The remaining coefficient in front ofε1−
α2 would be Rm + ϕ′(t)εmin(α2 ,1−
α2 )(−Nm
2 + N2
2 ). One can prove the following proposition.(see alsoProposition 2.4.4 and Appendix for the proof in the special case α = 0)
Proposition 2.4.3. For every m ∈ N, the homogenous functional Fm of degree 3 is defined as
Fm(v) =∑
j−l+k=mj,k,l∈N
Re(aj,l,kvjvlvk), ∀v ∈⋃n≥0
Pn(C∞+ ),
for some ak,l,j = aj,l,k ∈ C. Then we have the following formula
Fm,Hm,α,ε0 (v) +Hm1 (v) = Re(
Resonlow(v) + Reson≥2m+1(v)),
where
Resonlow(v)
=∑
0≤j,k≤2m
cj,j+k−m,kvjvj+k−mvk −∑
j−l+k=mj,l,k∈N,l≤2m
iaj,l,kvjvk[(1 + (l2 −m2)εα)vl + v2m−l
],
58 CHAPITRE 2. LONG TIME BEHAVIOR OF THE NLS–SZEGO EQUATION ON THE TORUS
for some cj,j+k−m,k = ck,j+k−m,j ∈ C and
Reson≥2m+1(v)
=∑
k≥2m+1
m−1∑j=0
2 (−ia2m−j,m+k−j,k + (1 + 2(m− j)(k − j)εα)iaj,k,k+m−j + 1) vjvkvk+m−j
+∑
k≥2m+1
m−1∑j=0
2((1− 2(k −m)(m− j)εα)ia2m−j,m+k−j,k − iaj,k,k+m−j + 1)v2m−jvm+k−jvk
+∑
k≥2m+1
(2(iam,k,k − iam,k,k + 1)vm|vk|2 + ((1− 2(k −m)2εα)iak,2k−m,k + 1)v2
kv2k−m)
+∑
k≥2m+1
∑j≥k+1
2((1− 2(j −m)(k −m)εα)iak,j+k−m,j + 1)vkvj+k−mvj .
The term Resonlow depends only on the small Fourier modes v1, v2, · · · , v3m. We try to find a boundedsequence (aj,l,k)j−l+k=m such that Reson≥2m+1 = 0 in order to cancel the term Hm1 . However, the coeffi-cient (1−2(j−m)(k−m)εα) in front of the parameter ak,j+k−m,j may have an arbitrarily small absolutevalue if α > 0. Such sequence does not exist if ε−α ∈ 2N
⋂[2(m+ 1)2,+∞).
We suppose that ε−α /∈ Q, then Reson≥2m+1 = 0 is equivalent to a linear system, which has a uniquesolution
a2m−j,m+k−j,k = ak,m+k−j,2m−j = i(k−j)(m−j)(1−2(k−m)(k−j)εα) , ∀0 ≤ j ≤ m− 1,
aj,k,k+m−j = ak+m−j,k,j = i(m−k)(m−j)(1−2(k−m)(k−j)εα) , ∀0 ≤ j ≤ m− 1,
am,k,k = ak,k,m = i2 ,
ak,k+j−m,j = aj,k+j−m,k = i1−2(j−m)(k−m)εα , ∀j ≥ 2m+ 1,
(2.4.9)
for all k ≥ 2m+ 1. In the case m = 0, (2.4.9) has only the last two formulas. When α > 0, the sequence(aj,l,k)j−l+k=m can be arbitrarily large, for 0 < ε 1. We suppose that εα is an irrational algebraicnumber of degree d ≥ 2. Then we have the Liouville estimate [100]
|aj,j+k−m,k| ≤1
cε,α(2(j −m)(k −m))d−1, ∀j, k ≥ 2m+ 1,
which loses the regularity of v in the estimates of XFm(v). It is difficult to find the same kind of estimatefor the transcendental numbers, which can preserve the regularity. So we return to the case α = 0.
2.4.3 Long time Hs-stability in the case α = 0
For α = 0 and every m ∈ N and s ≥ 1, assume that u is the smooth solution of the NLS-Szego equation
i∂tu+ ∂2xu = Π(|u|2u), u(0, x) = eimx + εf(x), ‖f‖Hs ≤ 1,
and u(t, x) = ei arg um(t)(eimx + εv(t, x)). Then v is solves the following Hamiltonian equation
i∂tv + ∂2xv −He2imx(v)− (1−m2 + εϕ′(t))v
=ϕ′(t)eimx + εΠ(e−imxv2 + 2eimx|v|2) + ε2Π(|v|2v).
2.4. ORBITAL STABILITY OF THE PLANE WAVE 59
Its energy functional is
Hm,ε(t, v) = Hm0 (v) + ϕ′(t)Lm(v) + ε(Hm1 (v) +
ϕ′(t)
2N2(v)) +
ε2
4N4(v),
with Hm0 (v) = 1
2‖∂xv‖2L2 + 1−m2
2 ‖v‖2L2 + 12
∫S1 Re(e2imxv2)dx,
Lm(v) = Revm,
Hm1 (v) = Re(∫S1 e−imx|v|2v)dx,
Np(v) = ‖v‖pLp , p = 2 or 4.
We define Nm2 (v) := ‖v − P2mv‖2L2 =
∑n≥2m+1 |vn|2 and the following proposition holds.
Proposition 2.4.4. For every s > 12 and m ∈ N, there exists a sequence (aj,l,k)j−l+k=m such that
aj,l,k = ak,l,j, supm≥1 supj−l+k=m |aj,l,k| = 12 and the functional Fm : Hs
+ → R, defined by
Fm(v) =∑
j−l+k=mj,k,l∈N
Re(aj,l,kvjvlvk), ∀v ∈ C∞+ ,
satisfies that Fm,Lm = −Nm2 and Rm := Fm,Hm0 + Hm1 is a finite sum of the Fourier modes
v1, · · · , v3m. Moreover, for all v, h ∈ Hs+, we have
‖XFm(v)‖Hs .m,s ‖v‖2Hs , ‖dXFm(v)h‖Hs .m,s ‖v‖Hs‖h‖Hs .
Demonstration. For the convenience of the reader, the detailed calculus for Rm = Fm,Hm0 + Hm1and formula (2.4.9) in the case α = 0 are postponed in Appendix. We define aj,j+k−m,k = 0, for all0 ≤ j, k ≤ 2m and an,m+1+n,2m+1 = a2m+1,m+1+n,n = 0, for all 0 ≤ n ≤ m − 1. Combing Proposition2.4.3 and (2.4.9) with α = 0, we have
Reson≥2m+1(v)∣∣∣α=0
= 0, Re(
Resonlow(v)∣∣∣α=0
)= Rm(v),
Fm,Lm(v) = 2Im(∑k≥0 am,k,k|vk|2 + 1
2
∑j+k=2m aj,m,kvjvk) = −
∑n≥2m+1 |vn|2.
By (2.4.9) with α = 0, we have |aj,j+k−m,k| ≤ 12 , for all j, k ≥ 0. By the definition of Fm, we have
[XFm(v)](n) = −2i∑k−l+n=m ak,l,nvkvl − i
∑k−n+l=m ak,n,lvkvl
[dXFm(v)h](n) = −2i∑k−l+n=m ak,l,n(vkhl + vlhk)− 2i
∑k−n+l=m ak,n,lvkhl,
for all n ≥ 0. The last two estimates are obtained by Young’s convolution inequality for l1 ∗ l2 → l2.
The Birkhoff normal form
Set χmσ := exp(εσXFm) the Hamiltonian flow of εFm, i.e.,
d
dσχmσ (v) = εXFm(χmσ (v)), χm0 (v) = v.
60 CHAPITRE 2. LONG TIME BEHAVIOR OF THE NLS–SZEGO EQUATION ON THE TORUS
We perform the canonical transformation Ψm,ε := χm1 . We want to reduce the energy functional Hm,ε tothe following normal form
Hm,ε(t) Ψm,ε = Hm0 + ϕ′(t)Lm(v) + ε
(Rm + ϕ′(t)(−Nm
2 +N2
2)
)+O(ε2).
Since Rm depends only on low-frequency Fourier modes v1, · · · , v3m, the high-frequency filtering Hs-norm of Ψ−1
m,ε(v) is appropriately estimated by the Birkhoff normal form transformation. The estimateof ‖P3m(v)‖Hs is given by (2.4.4). The next lemma will give the local existence of χmσ , for |σ| ≤ 1 andestimate the difference between v and Ψ−1
m,ε(v).
Lemma 2.4.5. For every s > 12 and m ∈ N, there exist two constants γm,s, Cm,s > 0 such that for all
v ∈ Hs+, if ε‖v‖Hs ≤ γm,s, then χmσ (v) is well defined on the interval [−1, 1] and the following estimates
hold :
supσ∈[−1,1]
‖χmσ (v)‖Hs ≤ 2‖v‖Hs ,
supσ∈[−1,1]
‖χmσ (v)− v‖Hs ≤ Cm,sε‖v‖2Hs ,
‖dχmσ (v)‖B(Hs) ≤ exp(Cm,sε‖v‖Hs |σ|), ∀σ ∈ [−1, 1].
The proof is based on a bootstrap argument, which is similar to Lemma 2.3.3, given by Lemma 2.3.1with q = 2. The energy functional Hm,ε is reduced to the normal form in the following lemma.
Lemma 2.4.6. For all s > 12 , m ∈ N and 0 < ε < ε∗m, there exists a smooth mapping Ym : R×Hs
+ −→Hs
+ and a constant C ′m,s > 0 such that for all t ∈ R, we have
XHm,ε(t)Ψm,ε = XHm0 + ϕ′(t)XLm + ε
(XRm + ϕ′(t)(−XNm2 +
1
2XN2)
)+ ε2Ym(t),
and supt∈R ‖Ym(t, v)‖Hs ≤ C ′m,s‖v‖2Hs(1 + ‖v‖Hs), for all v ∈ Hs+ such that ε‖v‖Hs ≤ γm,s. Set w(t) :=
Ψ−1m,ε(v(t)), then we have∣∣∣ d
dt‖w(t)− P3m(w(t))‖2Hs
∣∣∣ ≤ C ′m,sε2‖w(t)‖3Hs(1 + ‖w(t)‖Hs), (2.4.10)
if ε‖w(t)‖Hs ≤ γm,s.
Demonstration. For every t ∈ R, we expand the energy Hm,ε(t) Ψm,ε = Hm,ε(t) χm1 with Taylor’sformula at time σ = 1 around 0. Since χm0 = IdHs+ , we have
(Hm0 + ϕ′(t)Lm) χm1
=Hm0 + ϕ′(t)Lm +d
dσ[(Hm0 + ϕ′(t)Lm) χmσ ]|σ=0 +
∫ 1
0
(1− σ)d2
dσ2[(Hm0 + ϕ′(t)Lm) χmσ ]dσ
=Hm0 + ϕ′(t)Lm + εFm,Hm0 + ϕ′(t)Lm+ ε2∫ 1
0
(1− σ)Fm, Fm,Hm0 + ϕ′(t)Lm χmσ dσ
and (Hm1 +
ϕ′(t)
2N2
) χm1 =Hm1 +
ϕ′(t)
2N2 +
∫ 1
0
d
dσ[(Hm1 +
ϕ′(t)
2N2) χmσ ]dσ
=Hm1 +ϕ′(t)
2N2 + ε
∫ 1
0
Fm,Hm1 +ϕ′(t)
2N2 χmσ dσ.
2.4. ORBITAL STABILITY OF THE PLANE WAVE 61
Since Fm solves the homological system
Fm,Hm0 +Hm1 = RmFm,Lm+ Nm
2 = 0in Proposition 2.4.4, we have
Hm,ε(t) χm1
=Hm0 + ϕ′(t)Lm + ε
(Fm,Hm0 +Hm1 + ϕ′(t)(Fm,Lm+
1
2N2)
)+ ε2
[∫ 1
0
Fm, (1− σ)Fm,Hm0 + ϕ′(t)Lm+Hm1 +ϕ′(t)
2N2 χmσ dσ +
N4 χm14
]=Hm0 + ϕ′(t)Lm + ε
(Rm + ϕ′(t)(−Nm
2 +1
2N2)
)+ ε2
[∫ 1
0
Gm(t, σ) χmσ dσ +N4 χm1
4
],
where Gm(t, σ) = Fm, (1− σ)Rm + σHm1 + ϕ′(t)((σ − 1)Nm2 + 1
2N2). We set
Ym(t, v) :=
∫ 1
0
XGm(t,σ)χmσ (v)dσ +1
4XN4χm1 (v),
then we get
XHm,ε(t)χm1 = XHm0 + ϕ′(t)XLm + ε
(XRm + ϕ′(t)(−XNm2 +
1
2XN2
)
)+ ε2Ym(t).
Since Fm, H1m and Rm are homogeneous series of order 3 with uniformly bounded coefficients, N2 and
Nm2 are homogeneous series of order 2 with uniformly bounded coefficients, we have
‖XFm,H1m(v)‖Hs + ‖XFm,Rm(v)‖Hs .s ‖v‖3Hs ,
‖XFm,N2(v)‖Hs + ‖XFm,Nm2 (v)‖Hs .s ‖v‖2Hs ,
because for Jm(v) =∑j−l+k=m Re(bj,l,kvjvlvk) with supj−l+k=m |bj,l,k| < +∞, we have
Fm,Jm(v)
=4Im∑n≥0
∂vnFm(v)∂vnJm(v)
=∑
k1+k2=l1+l2
Im(4ak1,l1,m+l1−k1bl2,k2,m+k2−l2 + al1,l1+l2−m,l2bk1,k1+k2−m,k2)vk1vk2vl1vl2
+∑
k1+k2+l1−l2=2m
2Im(ak1,k1+l1−m,l1bk2,l2,m+l2−k2− ak2,l2,m+l2−k2
bk1,k1+l1−m,l1)vk1vk2
vl1vl2
and Fm,N2(v) = −2Im(∑j−l+k=m aj,l,kvjvlvk). Recall that sup0<ε<ε∗m
supt∈R |ϕ′(t)| < +∞ andXN4(v) =
−4iΠ(|v|2v), then we have
sup0≤σ≤1
supt∈R‖XGm(t,σ)(v)‖Hs + ‖XN4
(v)‖Hs .m,s ‖v‖2Hs(1 + ‖v‖Hs).
By using Lemma 2.4.5 and (2.3.11), for all v ∈ Hs+ such that ε‖v‖Hs ≤ γm,s, we have
sup0≤σ≤1
supt∈R‖XGm(t,σ)χmσ (v)‖Hs ≤ sup
0≤σ≤1supt∈R‖dχm−σ(χmσ (v))‖B(Hs)‖XGm(t,σ)(χ
mσ (v))‖Hs
.m,s sup0≤σ≤1
eCm,sε‖χmσ (v)‖Hs ‖χmσ (v)‖2Hs(1 + ‖χmσ (v)‖Hs)
.m,s‖v‖2Hs(1 + ‖v‖Hs)
62 CHAPITRE 2. LONG TIME BEHAVIOR OF THE NLS–SZEGO EQUATION ON THE TORUS
and supt∈R ‖XN4χm1 (v)‖Hs .m,s ‖v‖2Hs(1 + ‖v‖Hs). Then we obtain the estimate of Ym.
Since w(t) = χm−1(v(t)), w =∑n≥0 wne
inx solves the following infinite dimensional Hamiltonian systemof Fourier modes :
i∂twn(t) = (1 + n2 −m2 − εϕ′(t))wn(t) + iε2 Ym(t, w(t))(n), ∀n ≥ 3m+ 1.
because for all n ≥ 3m+ 1, we haveHe2imx(w(t))(n) = XRm(w(t))(n) = XLm(w(t))(n) = 0,
XN2(w(t))(n) = XNm2(w(t))(n) = −2iwn(t).
Consequently, if ε‖w(t)‖Hs ≤ γm,s, then we have∣∣∣∂t‖w(t)− P3m(w(t))‖2Hs∣∣∣ ≤2ε2
∑n≥3m+1
(1 + n2)s| Ym(t, w(t))(n)||wn(t)|
≤2ε2‖Ym(t, w(t))‖Hs‖w(t)‖Hs.m,sε
2‖w(t)‖3Hs(1 + ‖w(t)‖Hs).
End of the proof of Theorem 2.1.8
The proof is completed by a bootstrap argument and estimate (2.4.10), obtained by the Birkhoffnormal form transformation. It suffices to prove the case u(0) ∈ C∞+ by the same density argument inthe proof of Proposition 2.1.7.
Demonstration. For all m ∈ N and s ≥ 1, we recall that ∂tv(t) = XHm,ε(t)(v(t)) and w(t) = χm−1(v(t)). InProposition 2.4.2, there exists Am,s ≥ 1 such that sup0<ε<1 ‖v(0)‖Hs ≤ Am,s. By using (2.4.4), we have
sup0<ε<1
supt∈R‖P3m(v(t))‖Hs ≤ (1 + 9m2)
s2 Im.
Set Km,s := max(6Am,s, 6(1 + 9m2)s2 Im), εm,s := min(ε∗m,
γm,s3Km,s
, 148Cm,sKm,s
) and
Tm,s := supt ≥ 0 : sup0≤τ≤t
‖v(τ)‖Hs ≤ 2Km,s.
We choose ε ∈ (0, εm,s). Since ε = ε‖v(0)‖Hs ≤ εAm,s ≤ γm,s, Lemma 2.4.5 yields that
‖v(0)− w(0)‖Hs = ‖v(0)− χm−1(v(0))‖Hs ≤ εA2m,sCm,s ≤ Am,s.
So we have ‖w(0)‖Hs ≤ Km,s3 . For all t ∈ [0, Tm,s], we have ε‖v(t)‖Hs ≤ γm,s. Hence Lemma 2.4.5 gives
the following estimate
‖w(t)‖Hs ≤‖v(t)‖Hs + ‖χm−1(v(t))− v(t)‖Hs≤‖v(t)‖Hs + Cm,sε‖v(t)‖2Hs≤2Km,s + 4Cm,sK
2m,sε
≤3Km,s.
2.5. COMPARISON TO THE NLS EQUATION 63
So we have ε sup0≤t≤Tm,s ‖w(t)‖Hs ≤ γm,s, which implies that∣∣∣ d
dt‖w(t)− P3m(w(t))‖2Hs
∣∣∣ ≤ C ′m,sε2‖w(t)‖3Hs(1 + ‖w(t)‖Hs),
in Lemma 2.4.6. Set dm,s := 1486K2
m,sC′m,s
. We can obtain the following estimate :
‖w(t)− P3m(w(t))‖2Hs≤‖w(0)‖2Hs + C ′m,s|t|ε2 sup
0≤τ≤Tm,s‖w(τ)‖3Hs(1 + sup
0≤τ≤Tm,s‖w(τ)‖Hs)
≤K2m,s
9+ 162C ′m,sK
4m,s|t|ε2
≤4K2
m,s
9,
for all 0 ≤ t ≤ min(Tm,s,dm,sε2 ). We use Lemma 2.4.5 again to obtain that
‖v(t)‖Hs ≤2‖w(t)− v(t)‖Hs + ‖w(t)− P3m(w(t))‖Hs + ‖P3m(v(t))‖Hs
≤2Cm,sε‖v(t)‖2Hs +2Km,s
3+Km,s
6≤Km,s,
for all t ∈ [0,dm,sε2 ]. In the case t < 0, we use the same procedure and we replace t by −t. Consequently,
we have
sup|t|≤ dm,s
ε2
‖u(t)− ei(·+arg um(t))‖Hs = ε sup|t|≤ dm,s
ε2
‖v(t)‖Hs ≤ Km,sε.
2.5 Comparison to the NLS equation
Although we have some similar results for the NLS equation, there are still some differences between theNLS equation and the NLS-Szego equation. We denote by u = u(t, x) =
∑n≥0 un(t)einx the solution of
the NLS equation
i∂tu+ ∂2xu = |u|2u. (2.5.1)
In Fourier modes, we have i∂tun = n2un+∑k1−k2+k3=n uk1
uk2uk3 . Fix m ∈ Z, for every n ∈ Z, we define
vn := un+mei(m2+2nm)t. Then ‖v(t)‖L2 = ‖u(t)‖L2 and we have
i∂tvn = n2vn +∑
k1−k2+k3=n
vk1vk2
vk3. (2.5.2)
If u is localized in the m-th Fourier mode, then v is localized on the zero mode. Thus the orbital stabilityof the traveling wave em can be reduced to the case m = 0. In Faou–Gauckler–Lubich [34], long timeHs-orbital stability of plane wave solutions is established by limiting the mass of the initial data to a cer-tain full measure subset of (0,+∞) for the defocusing cubic Schrodinger equation with the time interval[−ε−N , ε−N ], for all N ≥ 1 and s 1. However, this above transformation u 7→ v does not preserve the
64 CHAPITRE 2. LONG TIME BEHAVIOR OF THE NLS–SZEGO EQUATION ON THE TORUS
L2 norm for the NLS-Szego equation and formula (2.5.2) fails too.
On the other hand, the approach that we use to prove Theorem 2.1.8 does not work for (2.5.1). In fact,the negative high frequency Fourier modes in the term Hm1 (v) = Re
∫S1 e−imx|v|2v can not be cancelled
by the homological equation. The energy functional of the equation of v can not be reduced as Hm0 +O(ε2)by using the same method in this paper.
The Szego filtering to (2.5.1) makes it possible to cancel all the high frequency resonances in the termRm = Fm,Hm0 + Hm1 . Then we use a bootstrap argument to deal with the equation ∂tw(t) =XHm,ε(t)χm1 (w(t)) after the Birkhoff normal form transformation.
2.6 Appendix
We give the details of the homological equation in Proposition 2.4.4 and prove (2.4.9) and Proposition2.4.3 in the case α = 0.
For all v ∈⋃n∈N Pn(C∞+ ), Hm0 (v) = 1
2‖∂xv‖2L2 + 1−m2
2 ‖v‖2L2 + 12
∫S1 Re(e2imxv2)dx and Fm(v) =∑
j−l+k=mj,k,l∈N
Re(aj,l,kvjvlvk). With the convention vn = 0, for all n < 0, we have
∂vnHm0 (v) =∂vnHm0 (v) = 1+n2−m2
2 vn + 12v2m−n,
∂vnFm(v) =∂vnFm(v) =∑
k−l+n=m
ak,l,nvkvl + 12
∑k−n+l=m
ak,n,lvkvl.
Combing (2.3.8), we have the Poisson bracket of Fm and Hm0 ,
Fm,Hm0 (v)
=− 2i∑n≥0
(∂vnFm∂vnHm0 − ∂vnHm0 ∂vnFm) (v)
=Im(∑
k−l+n=mk,l,n∈N
2(1 + n2 −m2)ak,l,nvkvlvn +∑
k−n+l=mk,l,n∈N
(1 + n2 −m2)ak,n,lvkvnvl)
+ Im(∑
k−l+n=mk,l,n∈N,n≤2m
2ak,l,nvlvkv2m−n +∑
k−n+l=mk,l,n∈N,n≤2m
ak,n,lvkvlv2m−n)
=Im(A1 +A2 +A3 +A4).
The term A4 =∑
k−n+l=mk,l,n∈N,n≤2m
ak,n,lvkvlv2m−n has only finite terms depending on low-frequency Fou-
rier modes v1, · · · , v3m. We divide A1,A2,A3 into two parts. The first part consists of low-frequencyresonances, the second part consists of high-frequency resonances.
A1 = −∑
k−l+n=mk,l,n∈N
2(1 + n2 −m2)ak,l,nvkvlvn = Alow1 +A≥2m+11 ,
2.6. APPENDIX 65
where Alow1 consists of all the resonances vjvj+k−mvk such that j, k ≤ 2m,
Alow1 =− (
bm−12 c∑j=0
2m∑k=m−j
+
2m−1∑j=bm+1
2 c
2m∑k=j+1
)2(2 + j2 + k2 − 2m2)aj,j+k−m,kvjvj+k−mvk
−m−1∑j=0
2m∑k=j+m+1
2(2 + j2 + (k +m− j)2 − 2m2)aj,k,k+m−jvjvkvk+m−j
−2m∑
k=bm+12 c
2(1 + k2 −m2)ak,2k−m,kv2kv2k−m,
and A≥2m+11 contain other resonances vjvj+k−mvk such that at least one of j, k is strictly larger than
2m.
A≥2m+11 =−
m∑j=0
∑k≥2m+1
2(2 + j2 + (k +m− j)2 − 2m2)aj,k,k+m−jvjvkvk+m−j
−2m∑
j=m+1
∑k≥2m+1
2(2 + j2 + k2 − 2m2)aj,j+k−m,kvjvj+k−mvk
−∑
k≥2m+1
∑j≥k+1
2(2 + k2 + j2 − 2m2)ak,k+j−m,jvkvk+j−mvj
−∑
k≥2m+1
2(1 + k2 −m2)ak,2k−m,kv2kv2k−m.
Then, we calculate A2 =∑
j−l+k=mj,l,k∈N
(1 + l2 −m2)aj,l,kvjvlvk = Alow2 +A≥2m+12 . Alow2 consists of all the
resonances vjvlvk such that j, k ≤ 2m or l = j + k −m ≤ 2m.
Alow2 =∑
j−l+k=mj,l,k∈N,l≤2m
(1 + l2 −m2)aj,l,kvjvlvk +∑
j−l+k=mm+1≤j≤2m−1
j+1≤k≤2m,l≥2m+1
2(1 + l2 −m2)aj,l,kvjvlvk
+
2m∑k=b 3m
2 c+1
(1 + (2k −m)2 −m2)ak,2k−m,kv2kv2k−m;
A≥2m+12 plays the same role as A≥2m+1
1 .
A≥2m+12 =
m∑j=0
∑k≥2m+1
2(1 + k2 −m2)aj,k,m+k−jvjvkvm+k−j
+
2m∑j=m+1
∑k≥2m+1
2(1 + (j + k −m)2 −m2)aj,j+k−m,kvjvj+k−mvk
+∑
k≥2m+1
∑j≥k+1
2(1 + (k + j −m)2 −m2)ak,k+j−m,nvkvk+j−mvj
+∑
k≥2m+1
(1 + (2k −m)2 −m2)ak,2k−m,kv2kv2k−m.
66 CHAPITRE 2. LONG TIME BEHAVIOR OF THE NLS–SZEGO EQUATION ON THE TORUS
At last, A3 =∑
k−l+n=mk,l,n∈N,n≤2m
2ak,l,nvlvkv2m−n =∑
j−l+k=mj,k,l,n∈N,j≤2m
2al,k,2m−jvkvlvj . Using the same idea,
we have A3 = Alow3 +A≥2m+13 , with
Alow3 =
m∑j=0
2m∑k=0
2a2m−j,m+k−j,kvjvkvm+k−j +
2m∑j=m+1
2m∑k=0
2a2m−j,k,k+j−mvjvk+j−mvk;
and
A≥2m+13 =
m∑j=0
∑k≥2m+1
2a2m−j,m+k−j,kvjvkvm+k−j +
2m∑j=m+1
∑k≥2m+1
2a2m−j,k,k+j−mvjvk+j−mvk.
Recall that Hm1 (v) = Re(∫S1 e−imx|v|2v) =
∑k−l+n=m Re(vkvlvn). A similar calculus as in the case of
A1 shows that Hm1 (v) = Re(Blow +B≥2m+1) with
Blow
=(
bm−12 c∑j=0
2m∑k=m−j
+
2m−1∑j=bm+1
2 c
2m∑k=j+1
)2vjvj+k−mvk +
2m∑k=bm+1
2 c
v2kv2k−m +
m−1∑j=0
2m∑k=j+m+1
2vjvkvk+m−j ,
B≥2m+1 =∑
k≥2m+1
m∑j=0
2vjvkvk+m−j +
2m∑j=m+1
2vjvj+k−mvk + v2kv2k−m +
∑j≥k+1
2vkvk+j−mvj
.
At last we define Resonlow(v)∣∣∣α=0
= −i(Alow1 +Alow2 +Alow3 +A4) +Blow and
Reson≥2m+1(v)∣∣∣α=0
= −i(A≥2m+11 +A≥2m+1
2 +A≥2m+13 ) +B≥2m+1.
Then we have Fm,Hm0 (v)+Hm1 (v) = Re(Resonlow(v)∣∣∣α=0
+Reson≥2m+1(v)∣∣∣α=0
). Since Resonlow(v)∣∣∣α=0
contains only finite terms and depends only on v1, · · · , v3m, so is Rm = Re(Resonlow(v)∣∣∣α=0
). For high-
frequency resonances, we compute
Reson≥2m+1(v)∣∣∣α=0
=∑
k≥2m+1
m−1∑j=0
2 (−ia2m−j,m+k−j,k + (1 + 2(m− j)(k − j))iaj,k,k+m−j + 1) vjvkvk+m−j
+∑
k≥2m+1
2m∑j=m+1
2((1− 2(k −m)(j −m))iaj,j+k−m,k − ia2m−j,k,k+j−m + 1)vjvj+k−mvk
+∑
k≥2m+1
[2(iam,k,k − iam,k,k + 1)vm|vk|2 + ((1− 2(k −m)2)iak,2k−m,k + 1)v2
kv2k−m]
+∑
k≥2m+1
∑j≥k+1
2((1− 2(j −m)(k −m))iak,j+k−m,j + 1)vkvj+k−mvj .
After replacing j by 2m−j in the summ+1 ≤ j ≤ 2m, we have the equivalence between Reson≥2m+1
∣∣∣α=0
=
0 and (2.4.9) in the case α = 0.
Chapitre 3
Traveling waves of NLS–Szegoequation on the line
Ce chapitre est une reprise de l’article Sun [134].
Resume Dans la seconde partie, afin de comprendre mieux l’action du projecteur de Szego sur l’equationde Schrodinger uni-dimensionnelle, on introduit l’equation de NLS–Szego quintique focalisante sur ladroite,
i∂tu+ ∂2xu+ Π(|u|4u) = 0, (t, x) ∈ R× R, u(0, ·) = u0.
On constate trois differences entre l’equation de NLS–Szego quintique et l’equation de SchrodingerL2-critique focalisante. La quantite de mouvement controle la norme H
12 (R) de la solution, de sorte
que l’equation de NLS–Szego quintique est globalement bien posee dans l’espace d’energie H1+(R) =
ΠR(H1(R)), pour des donnees initiales arbitraires. Ensuite, en adaptant l’argument de concentration–compacite de Lions [101, 102] precise par Gerard [45] a travers le theoreme de decomposition en profils,on peut obtenir une version faible de la stabilite des ondes progressives et classifier un certain type d’etatsfondamentaux. Enfin, le seuil inferieur de de masse pour la diffusion de l’equation de NLS–Szego quin-tique est strictement plus petit que la masse de l’etat fondamental associe, ce qui est en contraste avecle resultat de Dodson [31] pour l’equation de Schrodinger L2-critique focalisante.
Mots− clefs : Equation de Schrodinger L2-critique focalisante, Projecteur de Szego, Stabilite orbitale,Seuil de diffusion, Principe de concentration–compacite
Abstract We study the influence of Szego projector on the L2−critical one-dimensional non linearfocusing Schrodinger equation, leading to the quintic focusing NLS–Szego equation
i∂tu+ ∂2xu+ Π(|u|4u) = 0, (t, x) ∈ R× R, u(0, ·) = u0.
This equation has no Galilean invariance but the momentum P (u) = 〈−i∂xu, u〉L2 becomes the H12−norm.
Thus this equation is globally well-posed in H1+ = Π(H1(R)), for every initial datum u0. The solu-
tion L2-scatters both forward and backward in time if u0 has sufficiently small mass. By using theconcentration–compactness principle, we prove the orbital stability of some weak type of the traveling
67
68 CHAPITRE 3. TRAVELING WAVES OF NLS–SZEGO EQUATION ON THE LINE
wave : uω,c(t, x) = eiωtQ(x + ct), for some ω, c > 0, where Q is a ground state associated to Gagliardo–Nirenberg type functional
I(γ)(f) =‖∂xf‖2L2‖f‖4L2 + γ〈−i∂xf, f〉2L2‖f‖2L2
‖f‖6L6
, ∀f ∈ H1+\0,
for some γ ≥ 0. Its Euler–Lagrange equation is a non local elliptic equation. The ground states arecompletely classified in the case γ = 2, leading to the actual orbital stability for appropriate travelingwaves. As a consequence, the scattering mass threshold of the focusing quintic NLS–Szego equation isstrictly below the mass of ground state associated to the functional I(0), unlike the recent result byDodson [31] on the usual quintic focusing non linear Schrodinger equation.
Keywords L2−critical focusing Schrodinger equation, Szego projector, orbital stability, scatteringthreshold, concentration–compactness principle
3.1 Introduction
We consider the following L2−critical 1-dimensional NLS–Szego equation on the line R
i∂tu+ ∂2xu = −Π(|u|4u), (t, x) ∈ R× R, u(0, ·) = u0, (3.1.1)
where Π : L2(R)→ L2(R) denotes the Szego projector that cancels all negative Fourier modes
Πf(ξ) = 1[0,+∞)(ξ)f(ξ), ∀ξ ∈ R, ∀f ∈ L2(R). (3.1.2)
Set L2+ = Π(L2(R)), then L2
+ can be identified as the Hardy space that consists of all the holomorphicfunctions on the Poincare half-plane H+ := z ∈ C : Imz > 0 with L2 boundary
L2+ = f holomorphic on H+ : ‖f‖2L2
+:= sup
y>0
∫R|f(x+ iy)|2dx < +∞.
Since Π = id+iH2 , where H = −isign(D) is the Hilbert transform with D = −i∂x, Π : Lp(R) → Lp(R)
is a bounded operator, for every 1 < p < +∞ (Stein [129]). We define the filtered Sobolev spacesHs
+ = Hs(R)⋂L2
+, for every s ≥ 0.
The motivation to study this equation is based on the following two results. On the one hand, theL2−critical focusing non linear Schrodinger equation
i∂tU + ∂2xU = −|U |4U, (t, x) ∈ R× R, U(0, ·) = U0. (3.1.3)
marks the transition between the global existence (see Cazenave–Weissler [25, 26] for small data case) andthe blow-up phenomenon (see Glassey [61] for viriel identity method, Perelman [118] and Merle–Raphael[106] for blow-up dynamics). The instability of traveling waves U(t, x) = eiωtR(x) of equation (3.1.3) andthe classification of its ground states R associated to the Gagliardo–Nirenberg inequality
‖f‖6L6 . ‖∂xf‖2L2‖f‖4L2 , ∀f ∈ H1(R)
are established in Weinstein [145]. The ground states are unique up to scaling, phase rotation and spatialtranslation. It has been proved that the scattering mass threshold of equation (3.1.3) is equal to the mass
3.1. INTRODUCTION 69
of ground state ‖R‖L2 in Dodson [31].
On the other hand, it has been shown in Gerard–Grellier [47, 52] that filtering the positive Fourier modescould accelerate the transition to high frequencies in a Hamiltonian evolution PDE, leading to the super-polynomial growth of Sobolev norms of solutions of the cubic Szego equation on the torus S1. So weintroduce the cubic defocusing NLS–Szego equation on the torus S1 in Sun [133] in order to understandhow applying a filter keeping only positive Fourier modes modifies the long time dynamics of the nonlinear Schrodinger equation.
We continue this topic in this paper and we put the NLS–Szego equation on the line R. The travelingwaves and the classification of ground states of the cubic Szego equation on the line R
i∂tV = Π(|V |2V ), (t, x) ∈ R× R, V (0, ·) = V0 (3.1.4)
are studied in Pocovnicu [119]. Now we consider the quintic focusing NLS–Szego equation on the line Rin order to understand how Π modifies the global wellposedness result, the scattering mass threshold andthe stability result of traveling waves of the L2−critical non linear Schrodinger equation.
Definition 3.1.1. Fix s ≥ 0, a global solution u ∈ C(R;Hs+) of equation (3.1.1) is said to Hs−scatter
forward in time if there exists u+ ∈ Hs+ such that
limt→+∞
‖eit∂2xu+ − u(t)‖Hs = 0.
A global solution u ∈ C(R;Hs+) of equation (3.1.1) is said to Hs−scatter backward in time if there exists
u− ∈ Hs+ such that
limt→−∞
‖eit∂2xu− − u(t)‖Hs = 0.
In the small mass case, equation (3.1.1) is globally well-posed in L2+ and the solution L2−scatters both
forward and backward in time. The proof is similar to Cazenave–Weissler [25, 26], by using Strichartzestimates.
Proposition 3.1.2. There exists ε0 > 0 such that if ‖u0‖L2 ≤ ε0, then there exists a unique functionu ∈ C(R;L2
+) solving equation (3.1.1) and the solution u L2−scatters both forward and backward in time.
There are three conservation laws for (3.1.1) and (3.1.3) : the mass, the momentum and the Hamiltonian
M(u) = ‖u‖2L2 , P (u) = 〈Du, u〉L2 , E(u) =‖∂xu‖2L2
2−‖u‖6L6
6,
where D = −i∂x and u ∈ H1+ for (3.1.1), u ∈ H1(R) for (3.1.3). If u ∈ C(R;H1
+) solves equation (3.1.1),
then the momentum P (u) = ‖|D| 12u‖2L2 and the mass M(u) control H12−norm of the solution, leading
to the global wellposedness of equation (3.1.1). By using Gagliardo–Nirenberg’s interpolation inequality
‖u‖L2m+2 .m ‖|D|12u‖
mm+1
L2 ‖u‖1
m+1
L2 , ∀m ≥ 0, ∀u ∈ H 12 (R), (3.1.5)
one can solve the problem of global wellposedness for all L2−supercritical NLS–Szego equations forall large initial data u0 ∈ H1
+. On the other hand, when U0 ∈ H1(R) such that E(U0) < 0 andU0 ∈ L2(R, x2dx), then the solution U of equation (3.1.3) associated to initial datum U0 blows up
70 CHAPITRE 3. TRAVELING WAVES OF NLS–SZEGO EQUATION ON THE LINE
in finite time by the viriel identity (Glassey [61], Cazenave [22, 23], Ogawa–Tsutsumi [115]). We refer toPerelman [118] and Merle–Raphael [106] to see the asymptotic representation of the blow-up dynamicsof equation (3.1.3) in details. Now we state the first result of this paper.
Theorem 3.1.3. For all m ≥ 0, λ = ±1 and u0 ∈ H1+, there exists a unique function u ∈ C(R;H1
+)solving the following L2−supercritical NLS-Szego equation
i∂tu+ ∂2xu = λΠ(|u|2mu), u(0, x) = u0(x), (t, x) ∈ R× R. (3.1.6)
The local well-posedness of equation (3.1.6) is established by the fixed-point theorem and Sobolev esti-mates. In the focusing case λ = −1, since the mass, the momentum and the Hamiltonian are conservedunder the flow of equation (3.1.6), inequality (3.1.5) yields that
supt∈R‖∂xu(t)‖2L2 .m ‖∂xu0‖2L2 + ‖|D| 12u0‖2mL2 ‖u0‖2L2 .
Thus every solution of (3.1.6) is global. Besides the global well-posedness problem, there are still otherdifferences between equation (3.1.1) and equation (3.1.3).
We consider the traveling waves of equation (3.1.1), u(t, x) = eiωtQ(x + ct), for some ω, c ∈ R. u solves(3.1.1) if and only if Q solves the following non local elliptic equation
∂2xQ+ Π(|Q|4Q) = ωQ+ cDQ. (3.1.7)
It suffices to identity equation (3.1.7) to the Euler–Lagrange equation of some functional associated toGagliardo–Nirenberg inequality (3.1.5) in order to obtain the traveling waves. For all m ≥ 2 and γ ≥ 0,we define
I(γ)m (f) :=
‖∂xf‖mL2‖f‖m+2L2 + γ‖|D| 12 f‖2mL2 ‖f‖2L2
‖f‖2m+2L2m+2
, ∀f ∈ H1(R)\0. (3.1.8)
This functional is invariant by space-translation, phase-translation, interior and exterior scaling.
I(γ)m (f) = I(γ)
m (fλ,µ,y,θ), where fλ,µ,y,θ(x) = λeiθf(µx+ y), ∀x, y, θ ∈ R, ∀λ, µ > 0.
We denote its greatest lower bound by J(γ)m = inff∈H1
+\0 I(γ)m (f) and all its minimizers by
G(γ)m = f ∈ H1
+\0 : I(γ)m (f) = J (γ)
m =⋃a,b>0
G(γ)m (a, b), (3.1.9)
where G(γ)m (a, b) = f ∈ G(γ)
m : ‖f‖L2 = a, ‖f‖L2m+2 = b. Then we have
G(γ)m (a, b) = λf(µ·) ∈ H1
+\0 : f ∈ G(γ)m (1, 1), λ = a−
1m b
m+1m and µ = b
2m+2m a−
2m+2m .
A concentration-compactness argument shows that the functional I(γ)m attains its minimum in H1
+\0.We shall follow the idea of profile decomposition of minimizing sequence introduced in Gerard [45] andBahouri–Gerard [6], which is a refinement of the concentration-compactness principle (see Lions [101, 102]and Cazenave–Lions [24] for orbital stability of traveling waves of L2−subcritical NLS equation), in orderto establish the existence of minimizers.
3.1. INTRODUCTION 71
Theorem 3.1.4. For all m ≥ 2 and γ ≥ 0, if (fn)n∈N ∈ H1+ is a minimizing sequence for I
(γ)m such
that ‖fn‖L2 = ‖fn‖L2m+2 = 1 and limn→+∞ I(γ)m (fn) = J
(γ)m , then there exists a profile U ∈ G(γ)
m (1, 1), astrictly increasing function ψ : N→ N and a real-valued sequence (xn)n∈N such that
limn→+∞
‖fψ(n) − U(· − xn)‖H1 = 0. (3.1.10)
Remark 3.1.5. If H1+ is replaced by H1(R), then there exists U ∈ H1(R)\0 such that ‖U‖L2 =
‖U‖L2m+2 = 1, I(γ)m (U) = minf∈H1(R)\0 I
(γ)m (f) and the limit (3.1.10) also holds.
Thus G(γ)m (a, b) is not empty, for all a, b > 0. We refer to Gerard–Lenzmann–Pocovnicu–Raphael [56] to
see the asymptotic dynamics and long time behavior in two different regimes of the two-soliton solutionsof the cubic focusing half-wave equation on R. Similarly, one obtains the existence of ground states oftraveling waves for L2−critical Schrodinger equation on Rd (see Hmidi–Keraani [73]), for cubic Szegoequation (see Pocovnicu [119]) etc. Furthermore, the profile decomposition can be used to prove glo-bal regularity, scattering and a priori bounds for the energy critical Maxwell–Klein–Gordon equation inKrieger–Luhrmann [92].
Given m ≥ 2 and γ ≥ 0, let f ∈ H1+\0 be a minimizer of I
(γ)m , then d
dε
∣∣∣ε=0
log I(γ)m (f + εh) = 0, for all
h ∈ H1+. f solves the following Euler–Lagrange equation
m‖f‖m+2L2 ‖∂xf‖m−2
L2 ∂2xf + 2(m+ 1)J (γ)
m Π(|f |2mf)
=((m+ 2)‖f‖mL2‖∂xf‖mL2 + 2γ‖|D| 12 f‖2mL2 )f + 2γm‖f‖2L2‖|D|12 f‖2m−2
L2 Df.(3.1.11)
From now on, we restrict ourselves to the case m = 2. We want to identify equation (3.1.7) to equation
‖Q‖4L2
3J(γ)2
∂2xQ+ Π(|Q|4Q) =
2‖Q‖2L2‖∂xQ‖2L2 + γ‖|D| 12Q‖4L2
3J(γ)2
Q+2γ‖Q‖2L2‖|D|
12Q‖2L2
3J(γ)2
DQ.
A minimizer Q(γ) ∈ G(γ)2 is called the ground state of functional I
(γ)2 , if ‖Q(γ)‖4L2 = 3J
(γ)2 . If u(t, x) =
eiωtQ(γ)(x+ ct) solves equation (3.1.1) and Q(γ) ∈ G(γ)2 (
4
√3J
(γ)2 , b) for some b, ω > 0 and c, γ ≥ 0, then
c = 0 if and only if γ = 0.
If γ = 0, then we have ‖∂xQ(γ)‖2L2 = ω2
√3J
(0)2 and ‖Q(γ)‖6L6 = 3ω
2
√3J
(0)2 .
If γ > 0, then we have
‖∂xQ(γ)‖2L2 =
√3J
(γ)2
8γ(4γω − c2), ‖|D| 12Q(γ)‖2L2 =
√3J
(γ)2 c
2γ, ‖Q(γ)‖6L6 = 3
√3J
(γ)2 (
ω
2+c2
8γ).
Furthermore, the interpolation inequality ‖|D| 12Q(γ)‖2L2 ≤ ‖Q(γ)‖L2‖∂xQ(γ)‖L2 yields that c2 ≤ 4γ2ωγ+2 .
Even though we do not know how to classify all the ground states of I(γ)2 , for general γ ≥ 0, the
H1−orbital stability with scaling can be established by using theorem 3.1.4 and the conservation lawP (u) = 〈−i∂xu, u〉L2 = ‖|D| 12u‖2L2 .
72 CHAPITRE 3. TRAVELING WAVES OF NLS–SZEGO EQUATION ON THE LINE
Theorem 3.1.6. For every ε, b > 0 and γ ≥ 0, there exists δ = δ(b, ε, γ) > 0 such that if
inff∈G(γ)
2 (4√
3J(γ)2 ,b)
‖u0 − f‖H1 < δ,
then we have supt∈R infΨ∈
⋃C(γ)−1≤θ≤C(γ) G
(γ)2 (
4√
3J(γ)2 ,θb)
‖u(t)−Ψ‖H1 < ε, where u is the solution of equa-
tion (3.1.1) with initial datum u(0) = u0 and C(γ) :=
(inff∈H1
+\0‖|D|
13 f‖L2
‖f‖L6
)−16
√J
(γ)2
1+γ .
The problem of uniqueness of ground states of a non-local elliptic equation is difficult to solve (SeeFrank–Lenzmann [40] and Frank–Lenzmann–Silvestre [41] for the fractional Laplacians in R and alsoLenzmann–Sok [99] for a strict rearrangement principle in Fourier space). We refer to subsection 3.4.2
to discuss the classification of ground states of I(γ)2 . For general γ ≥ 0, we only have the H1−orbital
stability with scaling. Since we do not know the uniqueness of ground states of I(γ)2 , the L6−norm of the
ground state Ψ that approaches u(t) is unknown. We can only give a range‖f‖L6
C(γ) ≤ ‖Ψ‖L6 ≤ C(γ)‖f‖L6 ,
where f denotes the ground state that approaches the initial datum u0.
On the other hand, all the ground states can be completely classified in the case γ = 2, by using
Cauchy–Schwarz inequality. The ground state of I(2)2 is unique up to scaling, phase rotation and spatial
translation.
Proposition 3.1.7. In the case γ = m = 2, we have
J(2)2 = min
f∈H1+\0
I(2)2 (f) =
8π2
3, G
(2)2 = x 7→ λeiθ
µx+ y + i∈ H1
+ : ∀λ, µ > 0 ∀θ, y ∈ R. (3.1.12)
If u(t, x) = eiωtQ(x+ ct) is a traveling wave of (3.1.1) and Q ∈ G(2)2 , then we have
3c2 = 8ω, ‖Q‖4L2 = 8π2, ‖Q‖6L6 =3πc2√
2, ‖|D| 12Q‖2L2 =
πc√2, ‖∂xQ‖2L2 =
πc2
2√
2,
thanks to the classification of G(2)2 . In this case, the traveling wave uc(t, x) = e
3c2it8 Qc(x + ct) is
H1−orbitally stable, for every c > 0, where Qc ∈ G(2)2 (
4√
8π2, 6
√3πc2√
2).
Theorem 3.1.8. For every ε, c > 0, there exists δε,c > 0 such that if inff∈G(2)
2 (4√
8π2, 6√
3πc2√2
)‖u0−f‖H1 <
δ, then we have supt∈R inff∈G(2)
2 (4√
8π2, 6√
3πc2√2
)‖u(t)−f‖H1 < ε, where u solves equation (3.1.1) with initial
datum u(0) = u0.
Remark 3.1.9. In the case γ = 2, since all the ground states are completely classified and unique up toscaling, phase rotation and spatial translation by proposition 3.1.7, we obtain the H
12−norm of the ground
state that approaches u(t) by the conservation law P (u) = 〈Du, u〉L2 and formula (3.1.10). Then we knowalso the L6−norm of the very ground state and we have the actual orbital stability without scaling, whichis a refinement of theorem 3.1.6.
Similar results on the classification of ground states of traveling waves by using Cauchy–Schwarz inequa-lity can be found in Foschi [39] for linear Schrodinger equation and linear wave equation on Rd, for d ≥ 1,
3.1. INTRODUCTION 73
Gerard–Grellier [47, 52] for the cubic Szego equation, etc.
In the case γ = 0, we have
I(0)2 (f) :=
‖∂xf‖2L2‖f‖4L2
‖f‖6L6
, ∀f ∈ H1(R)\0.
All of the ground states in H1(R) of I(0)2 have been completely classified in Weinstein [145]. We know
minf∈H1(R)\0 I(0)2 (f) = π2
4 and there exists a unique real-valued, positive, spherically symmetric and
decreasing function R(x) =4√3√
cosh(2x)such that
(I(0)2 )−1(
π2
4) = λeiθR(µ · −y) : λ, µ > 0 θ, y ∈ R.
The traveling wave U(t, x) = eiωtR(x) is an unstable solution of the L2−critical focusing Schodinger
equation (3.1.3) in the following sense : there exists a sequence u(n)0 = (1 + 1
n )R ⊂ H1(R) such that
u(n)0 → R, as n→ +∞, but the corresponding maximal solution u(n) blows up in finite time. We denote
by Q(0) ∈ G(0)2 (
4
√3J
(0)2 , ‖Q(0)‖L6) one of the ground states of I
(0)2 in Hardy space H1
+. Since R /∈ H1+ and
Q+ : x 7→ 1x+i ∈ H
1+, we have π2
4 = I(0)2 (R) < J
(0)2 = I
(0)2 (Q(0)) ≤ I(0)
2 (Q+) = 4π2
3 .
In proposition 3.1.2, we know that the solution of equation (3.1.1) or equation (3.1.3) scatters if theinitial datum has sufficiently small mass. Furthermore, Dodson [31] has proved that if ‖U0‖ < ‖R‖L2 ,then equation (3.1.3) is globally well-posed and the solution L2−scatters both forward and backwardin time. Together with the instability result of traveling waves by Weinstein [145], the scattering mass
threshold of equation (3.1.3) is equal to the mass of ground state R ∈ H1(R) of I(0)2 .
On the other hand, adding the Szego projector in front of the non linear term of the L2−critical focusingSchrodinger equation makes the scattering mass threshold strictly smaller than the mass of ground state
Q(0) ∈ H1+ of I
(0)2 , thanks to the orbital stability theorem 3.1.6. We define E ⊂ R∗+ to be all ε > 0 such
that if ‖u0‖L2 < ε, the corresponding solution of (3.1.1) L2−scatters both forward and backward in time.If an H1−solution of equation (3.1.1) L2−scatters, then it also H1−scatters and its Lr−norm decays,with 2 < r ≤ +∞ (See proposition 3.4.1 and 3.4.2). Thus traveling waves do not L2−scatter and neitherdo the solutions that approach the traveling waves.
Corollary 3.1.10. sup E < ‖Q(0)‖L2 =4
√3J
(0)2 . Precisely, there exists u ∈ C(R;H1
+) solving equation
(3.1.1) such that ‖u(0)‖L2 < ‖Q(0)‖L2 and u does not L2−scatter neither forward nor backward in time.
Remark 3.1.11. The mass of ground state of I(2)2 is strictly larger than the mass of ground state of I
(0)2 .
‖Q(2)‖4L2 = 8π2 > 4π2 ≥ ‖Q(0)‖4L2 .
E(Q(2)) = − πc2
4√
2< 0 = E(Q(0)).
The value of scattering mass threshold of equation (3.1.1) remains as an open problem. The Galileaninvariance is necessary to use interaction Morawetz estimate (see Dodson [31], Killip–Visan [88], Killip–Visan–Zhang [89]). However, the NLS–Szego equation has no Galilean invariance. The next step to study
74 CHAPITRE 3. TRAVELING WAVES OF NLS–SZEGO EQUATION ON THE LINE
is to compare the value of the scattering mass threshold of equation (3.1.3) and the scattering massthreshold of the original L2−critical NLS equation.
This paper is organized as follows. In section 3.2, we recall profile decomposition theorem prove theorem3.1.4. In section 3.3, the orbital stability of traveling wave u(t, x) = eiωtQ(x+ ct) is proved at first. Thenwe give the details of the special case γ = 2. In the first appendix, we prove the persistence of regularityof scattering. In the second appendix, we discuss the open problem of uniqueness of ground states of the
functional I(γ)2 , for general γ ≥ 0. It suffices to study the uniqueness of ground states modulo positive
Fourier transformation.
3.2 Profile decomposition
At first, we recall the result of profile decomposition theorem in Gerard [45], which is a refinement ofconcentration-compactness argument of Sobolev embedding introduced in Lions [101, 102]. Every boundedsequence in H1
+ has a subsequence which can be written as a nearly orthogonal sum of a superposition ofsequence of shifted profiles and a sequence tending to zero in Lp(R), for every 2 < p ≤ +∞. It will be usedto find the minimizers of some functionals in calculus of variation and establish the orbital stability ofsome traveling waves. We shall use the version of Hmidi–Keraani [74] and construct the profiles withoutscaling.
Theorem 3.2.1 (Gerard 45, Hmidi–Keraani 74). If (fn)n∈N+is a bounded sequence in H1
+, then there
exists a subsequence of (fn)n∈N+, denoted by (fφ(n))n∈N+
, a sequence of profiles (U (j))j∈N+⊂ H1
+ and
a double-indexed sequence (x(j)n )n,j∈N+
⊂ R such that if j 6= k, then |x(j)n − x(k)
n | → +∞ and for everyl ∈ N+, we have
fφ(n)(x) =
l∑j=1
U (j)(x− x(j)n ) + r(l)
n (x), ∀x ∈ R, (3.2.1)
where lim supn→+∞ ‖r(l)n ‖Lp → 0, as l→ +∞, ∀2 < p ≤ +∞. For every l ∈ N+ and s ∈ [0, 1], we have
∣∣∣‖|D|sfφ(n)‖2L2 −l∑
j=1
‖|D|sU (j)‖2L2 − ‖|D|sr(l)n ‖2L2
∣∣∣→ 0, as n→ +∞. (3.2.2)
Remark 3.2.2. According to Gerard [45] and Hmidi–Keraani [74], we may construct the profiles (U (j))j∈N+⊂
H1+ such that if U (l) = 0 for some l ∈ N+, then U (j) = 0, for every j ≥ l.
Then we shall use this theorem to establish the existence of minimizers in H1+ of I
(γ)m , for every m ≥ 2
and γ ≥ 0.
Proof of theorem 3.1.4. Since supn∈N ‖fn‖H1 < +∞, theorem 3.2.1 gives a subsequence of (fn)n∈N+, de-
noted by (fφ(n))n∈N+, a sequence of profiles (U (j))j∈N+
⊂ H1+ and a double-indexed sequence (x
(j)n )n,j∈N+
⊂R such that |x(j)
n −x(k)n | → +∞, if j 6= k, (3.2.1) and (3.2.2) hold and liml→+∞ lim supn→+∞ ‖r
(l)n ‖L2m+2 →
0.
3.2. PROFILE DECOMPOSITION 75
For all 0 ≤ s ≤ 1, l ∈ N+ and δ > 0, there exists N = N(s, l, δ) ∈ N+ such that
‖|D|sfφ(n)‖2L2 ≥l∑
j=1
‖|D|sU (j)‖2L2 − δ, ∀n > N.
Taking n → +∞, δ → 0 and l → +∞, we have∑lj=1 ‖|D|sU (j)‖2L2 ≤ lim infn→+∞ ‖|D|sfφ(n)‖2L2 , for
every 0 ≤ s ≤ 1. Then, there exists a subsequence of (fφ(n))n∈N, denoted by (fφφ(n))n∈N such that both
sequences (‖|D| 12 fφφ(n)‖L2)n∈N and (‖∂xfφφ(n)‖L2)n∈N converge and we have∑+∞j=1 ‖∂xU (j)‖2L2 ≤ limn→+∞ ‖∂xfφφ(n)‖2L2 ,∑+∞j=1 ‖|D|
12U (j)‖2L2 ≤ limn→+∞ ‖|D|
12 fφφ(n)‖2L2 ,∑+∞
j=1 ‖U (j)‖2L2 ≤ limn→+∞ ‖fφ(n)‖2L2 = 1.
(3.2.3)
Thus 0 ≤ ‖U (j)‖L2 ≤ 1, for every j ∈ N+. We set ψ = φ φ : N→ N. Since m ≥ 2 and
J (γ)m = lim
n→+∞I(γ)m (fψ(n)) = lim
n→+∞‖∂xfψ(n)‖mL2 + γ lim
n→+∞‖|D| 12 fψ(n)‖2mL2 ,
estimates (3.2.3) yields that
J (γ)m ≥(
+∞∑j=1
‖∂xU (j)‖2L2)m2 + γ(
+∞∑j=1
‖|D| 12U (j)‖2L2)m
≥+∞∑j=1
(‖∂xU (j)‖mL2‖U (j)‖m+2L2 + γ‖|D| 12U (j)‖2mL2 ‖U (j)‖2L2)
≥J (γ)m
+∞∑j=1
‖U (j)‖2m+2L2m+2 .
(3.2.4)
We claim that+∞∑j=1
‖U (j)‖2m+2L2m+2 = 1. (3.2.5)
In fact, each profile U (j) ∈ L∞(R). More precisely, we have∑∞j=1 ‖U (j)‖2L∞ .m 1 by Sobolev embedding
H1 → L∞ and estimate (3.2.3). One can easily check that
‖l∑
j=1
U (j)‖2m+2L2m+2 =
l∑j=1
‖U (j)‖2m+2L2m+2 +R(l)
n , ∀l ∈ N+,
where |R(l)n | .m,l
∑1≤j<k≤l
∫R |U
(j)(x−x(j)n )||U (k)(x−x(k)
n )|dx. Since U (j) ∈ L2+, limn→+∞ |x(j)
n −x(k)n | =
+∞, for all 1 ≤ j < k ≤ l, we have |U (j)(· − x(j)n + x
(k)n )| 0 in L2
+, as n→ +∞. So limn→+∞R(l)n = 0.
The profile decomposition theorem (3.2.1) implies that
‖r(l)n ‖L2m+2 ≥
∣∣∣‖fψ(n)‖L2m+2 − ‖l∑
j=1
U (j)‖L2m+2
∣∣∣ =∣∣∣1− (
l∑j=1
‖U (j)‖2m+2L2m+2 +R(l)
n )1
2m+2
∣∣∣.
76 CHAPITRE 3. TRAVELING WAVES OF NLS–SZEGO EQUATION ON THE LINE
Taking n → +∞, we have∣∣∣1 − (
∑lj=1 ‖U (j)‖2m+2
L2m+2)1
2m+2
∣∣∣ ≤ lim supn→+∞ ‖r(l)n ‖L2m+2 → 0, as l → +∞
and we obtain (3.2.5).
Combining (3.2.4), we have J(γ)m ≥ J
(γ)m∑+∞j=1 ‖U (j)‖2m+2
L2m+2 = J(γ)m . All inequalities in (3.2.3) and (3.2.4)
are actually equalities. In particular, we have
‖∂xU (1)‖mL2‖U (1)‖2m+2L2 = ‖∂xU (1)‖mL2 , I(γ)
m (U (1))‖U (1)‖2m+2L2m+2 = J (γ)
m ‖U (1)‖2m+2L2m+2 .
If U (1) ≡ 0 a.e. in R, then so is U (j), for every j ≥ 2 by the construction of profiles in remark 3.2.2.It contradicts formula (3.2.5). Thus Gagliardo–Nirenberg inequality yields that ‖∂xU (1)‖L2 > 0 and‖U (1)‖L2 = 1. Since
∑+∞j=1 ‖U (j)‖2L2 ≤ 1, we have U (j) = 0 a.e., for every j ≥ 2. Formula (3.2.5) implies
that ‖U (1)‖L2m+2 = 1 and I(γ)m (U (1)) = J
(γ)m .
Estimate (3.2.3) is also equality, so ‖∂xU (1)‖2L2 = limn→+∞ ‖∂xfψ(n)‖2L2 . We set
ε(l)n,s := ‖|D|sfψ(n)‖2L2 −l∑
j=1
‖|D|sV (j)‖2L2 − ‖|D|sr(l)
φ(n)‖2L2 , ∀n ∈ N, l ∈ N+, 0 ≤ s ≤ 1.
Thus limn→+∞ ε(l)n,s = 0, for all l ∈ N+, 0 ≤ s ≤ 1. Then
‖fψ(n) − U (1)(· − y(1)n )‖2H1 = ‖r(1)
φ(n)‖2H1 = ‖∂xfψ(n)‖2L2 − ‖∂xU (1)‖2L2 − ε(1)
n,0 − ε(1)n,1 → 0,
as n→ +∞.
3.3 Orbital stability
At first, we prove the H1−orbital stability with scaling for traveling wave u(t, x) = eiωtQ(x+ ct) of theL2−critical focusing NLS-Szego equation,
i∂tu+ ∂2xu = −Π(|u|4u), (t, x) ∈ R× R, u(0, ·) = u0,
with ω, c > 0, γ ≥ 0 and Q ∈ G(γ)2 (
4
√3J
(γ)2 , ‖Q‖L6). Then we focus on the case γ = 2 by classifying all
ground states and we refine theorem 3.1.6 as theorem 3.1.8.
3.3.1 Proof of theorem 3.1.6
Demonstration. Fix b > 0 and γ ≥ 0, for every n ∈ N, we choose un0 ∈ H1+ and ϕn ∈ G(γ)
2 (4
√3J
(γ)2 , b)
such that ‖un0 − ϕn‖H1 → 0, as n → +∞. Let un solve (3.1.1) with initial datum un(0) = un0 . We shallprove that
infΨ∈
⋃C(γ)−1≤θ≤C(γ) G
(γ)2 (
4√
3J(γ)2 ,θb)
‖un(tn)−Ψ‖H1 → 0, as n→ +∞, (3.3.1)
3.3. ORBITAL STABILITY 77
up to a subsequence, for every temporal sequence (tn)n∈N ⊂ R. We use the three conservation laws
E(u) =‖∂xu‖2L2
2−‖u‖6L6
6, P (u) = 〈Du, u〉L2 = ‖|D| 12u‖2L2 , M(u) = ‖u‖2L2 .
to construct another conservation law
Kγ(u) := E(u) +γP (u)2
M(u)=‖∂xu‖2L2
2−‖u‖6L6
6+γ‖|D| 12u‖4L2
2‖u‖2L2
=‖u‖6L6
6‖u‖4L2
(3I(γ)2 (u)− ‖u‖4L2). (3.3.2)
Since I(γ)2 (ϕn) = J
(γ)2 and ‖un0 − ϕn‖H1 → 0, as n → +∞, we have limn→+∞ ‖un0‖4L2 = 3J
(γ)2 and
limn→+∞ I(γ)2 (un0 ) = J
(γ)2 . Thus
Kγ(un(tn)) = Kγ(un0 )→ 0, as n→ +∞. (3.3.3)
In order to construct a minimizing sequence of I(γ)2 , we need to prove the following inequalities
C(γ)−1b ≤ lim infn→+∞
‖un(tn)‖L6 ≤ lim supn→+∞
‖un(tn)‖L6 ≤ C(γ)b. (3.3.4)
In fact, we denote by C61 := 1+γ
J(γ)2
C(γ)6, then Sobolev embedding ‖f‖L6 ≤ C1‖|D|13 f‖L2 yields that
‖∂xun(tn)‖2L2
2+γ‖|D| 12un(tn)‖4L2
2‖un(tn)‖2L2
≥(1 + γ)‖|D| 12un0‖4L2
2‖un0‖2L2
≥(1 + γ)‖|D| 13un0‖6L2
2‖un0‖4L2
≥(1 + γ)‖un0‖6L6
2C61‖un0‖4L2
.
Thus ‖un(tn)‖L6 ≥ 6
√3(1+γ)‖un0 ‖6L6
C61‖un0 ‖4L2
− 6Kγ(un(tn))→ bC(γ) , as n→ +∞. Similarly, we have
‖un(tn)‖L6 ≤ C1‖|D|13un(tn)‖L2 ≤ C1‖|D|
12un(tn)‖
23
L2‖un(tn)‖13
L2 = C1‖|D|12un0‖
23
L2‖un0‖13
L2 ,
and ‖un(tn)‖L6 ≤ C1‖∂xun0‖13
L2‖un0‖23
L2 . Thus we have ‖un(tn)‖L6 ≤ C1‖un0‖L66
√I
(γ)2 (un0 )1+γ → C(γ)b, as
n→ +∞. Thus estimates (3.3.4) are proved and we have limn→+∞ I(γ)2 (un(tn)) = J
(γ)2 .
Rescaling vn(x) := λnun(tn, µnx) such that ‖vn‖L2 = ‖vn‖L6 = 1, with λn, µn > 0. Then
√µnλ
−1n = ‖un0‖L2 , 6
√µnλ
−1n = ‖un(tn)‖L6 , lim
n→+∞I
(γ)2 (vn) = J
(γ)2 (3.3.5)
‖∂xvn‖2L2 = I(vn) = I(un(tn)) =‖∂xun(tn)‖2L2
‖un(tn)‖6L6
‖un0‖4L2 → J, as n→ +∞.
Theorem 3.1.4 yields that there exists a profile U ∈ G(γ)2 (1, 1) and a sequence of real numbers (xn)n∈N
such that ‖vn−U(·−xn)‖H1 → 0, as n→ +∞ up to a subsequence, still denoted by (vn)n∈N+. Moreover,
we assume that ‖un(tn)‖L6 → θb, as n→ +∞ in the same subsequence. Then (3.3.4) yields that C−1 ≤θ ≤ C. We denote by λ∞ = limn→+∞ λn and µ∞ = limn→+∞ µn. Then we have
λ∞ > 0, µ∞ > 0,√µ∞λ
−1∞ =
4
√3J
(γ)2 and 6
√µ∞λ
−1∞ = θb. (3.3.6)
78 CHAPITRE 3. TRAVELING WAVES OF NLS–SZEGO EQUATION ON THE LINE
Then
limn→∞
‖λnun(tn, µn·)− U(· − xn)‖H1 = 0 ⇐⇒ limn→∞
‖un(tn)− 1
λnU(· − µnxn
µn)‖H1 = 0. (3.3.7)
Since U ∈ H1+, we have limn→∞ ‖ 1
λnU( ·µn )− 1
λ∞U( ·µ∞ )‖H1 = 0. Together with (3.3.7), we have
‖un(tn)−Ψ(· − µnxn)‖H1 → 0, as n→ +∞, (3.3.8)
where Ψ(x) := 1λ∞
U( xµ∞
). Since ‖U‖L2 = ‖U‖L6 = 1, we have Ψ ∈ G(γ)2 (
4
√3J
(γ)2 , θb) by (3.3.6), leading
to (3.3.1) up to a subsequence.
3.3.2 The special case γ = 2
We prove proposition 3.1.7 in order to classify all the ground states of the functional
I(2)2 =
‖∂xf‖2L2‖f‖4L2 + 2‖|D| 12 f‖4L2‖f‖2L2
‖f‖6L6
, ∀f ∈ H1+\0.
as G(2)2 = x 7→ λeiθ
µx+y+i ∈ H1+ : ∀λ, µ > 0 ∀θ, y ∈ R. The idea of using Cauchy–Schwarz inequality to
classify ground states follows from Foschi [39] for linear Schrodinger equation and linear wave equationon Rd, for d ≥ 1, Gerard–Grellier [47, 52] for the cubic Szego equation on the torus S1, Pocovnicu [119]for the cubic Szego equation on the line R.
Proof of proposition 3.1.7. Plancherel formula gives that ‖f‖6L6 = 132π5
∫ξ>0|(f ∗ f ∗ f)(ξ)|2dξ. By using
Cauchy–Schwarz inequality, for every ξ > 0, we have
|(f ∗ f ∗ f)(ξ)|2 =∣∣∣ ∫η1>0,η2>0,η1+η2≤ξ
f(η1)f(η2)f(ξ − η1 − η2)dη1dη2
∣∣∣2≤ξ
2
2|∫η1>0,η2>0,η1+η2≤ξ
∣∣∣f(η1)f(η2)f(ξ − η1 − η2)∣∣∣2dη1dη2.
Thus we have
‖f‖6L6 ≤1
64π5
∫η1>0,η2>0,η3>0
(η1 + η2 + η3)2∣∣∣f(η1)f(η2)f(η3)
∣∣∣2dη1dη2dη3
=3
8π2(‖∂xf‖2L2‖f‖4L2 + 2‖|D| 12 f‖4L2‖f‖2L2).
If I(2)2 (f) = 8π2
3 , then f(η1)f(η2)f(η3) = f(0)2f(η1 + η2 + η3), for all η1, η2, η3 > 0. Since f ∈ H1+\0,
we have f(0) 6= 0. Thus
f(η1)f(η2) = f(η1 + η2)f(0), ∀η1, η2 ≥ 0.
This is true if and only if f(η) = e−ipη f(0), for some p ∈ C such that Imp < 0. Thus we have f(x) = Ax−p ,
for some A ∈ C. We conclude by I(2)2 (f) = I
(2)2 (fλ,µ,θ,y), with fλ,µ,θ,y(x) = λeiθf(µx+ y).
3.4. APPENDICES 79
For every c > 0, we prove the orbital stability of traveling wave u(t, x) = e3c2it
8 Q∗c(x + ct) of equation
(3.1.1), with Q∗c(x) = 24√
2c2
cx+2i . Proposition 3.1.7 yields that G(2)2 (
4√
8π2, 6
√3πc2√
2) = x 7→ 2
4√2c2eiθ
cx+2i+y ∈ H1+ :
∀θ, y ∈ R. If Qc ∈ G(2)2 (
4√
8π2, 6
√3πc2√
2), then we have
‖Qc‖4L2 = 8π2, ‖∂xQc‖2L2 =πc2
2√
2, ‖|D| 12Qc‖2L2 =
πc√2, ‖Qc‖6L6 =
3πc2√2
(3.3.9)
Proof of theorem 3.1.8. Fix c > 0, for every n ∈ N, we choose un0 ∈ H1+ and Qnc ∈ G
(2)2 (
4√
8π2, 6
√3πc2√
2)
such that ‖un0 −Qnc ‖H1 → 0, as n → +∞. Let un solve (3.1.1) with initial datum un(0) = un0 . We shallprove that
infΨ∈G(2)
2 (4√
8π2, 6√
3πc2√2
)
‖un(tn)−Ψ‖H1 → 0, as n→ +∞, (3.3.10)
up to a subsequence, for every temporal sequence (tn)n∈N ⊂ R. We use the same procedure as the proof of
theorem 3.1.6 to obtain that (un(tn))n∈N is a minimizing sequence of I(2)2 . We set vn(x) := λnu
n(tn, µnx)
such that ‖vn‖L2 = ‖vn‖L6 = 1, with λn, µn > 0 and limn→+∞ I(2)2 (vn) = limn→+∞ I
(2)2 (un(tn)) = J
(2)2 .
I(vn) = I(un(tn))→ 1, as n→ +∞.
Theorem 3.1.4 yields that there exists a profile V ∈ G(2)2 (1, 1) and a sequence of real numbers (yn)n∈N+
such that ‖vn−V (·−yn)‖H1 → 0, as n→ +∞ up to a subsequence, still denoted by (vn)n∈N+. Similarly,
we have (3.3.7) for V .
limn→∞
‖λnun(tn, µn·)− V (· − yn)‖H1 = 0 ⇐⇒ limn→∞
‖un(tn)− 1
λnV (· − µnynµn
)‖H1 = 0. (3.3.11)
Since all the ground states are completely classified by proposition 3.1.7, we obtain the values of λ∞ and
µ∞ from (3.3.9) and the conservation law P (u) = ‖|D| 12u‖2L2 . Since λ2n‖|D|
12
1λnV ( ·−µnynµn
)‖2L2 =√
23π,
‖|D| 12un(tn)‖2L2 = ‖|D| 12un0‖2L2 → πc√2
and∣∣∣‖|D| 12un(tn)‖L2 − ‖|D| 12 1
λnV (· − µnynµn
)‖L2
∣∣∣→ 0, as n→ +∞,
we have λ2∞ := limn→+∞ λ2
n = 2√3c
and µ∞ := limn→+∞ µn =√
8π2 limn→+∞ λ2n = 4
√2π√3c
. Together with
(3.3.11), we have‖un(tn)− Ψ(· − µnyn)‖H1 → 0, as n→ +∞,
where Ψ(x) := 1λ∞
V ( xµ∞
). We have Ψ ∈ G(2)2 (
4√
8π2, 6
√3πc2√
2), leading to (3.3.10) up to a subsequence.
3.4 Appendices
In the first appendix, we prove that if a H1−solution of equation (3.1.1) L2−scatters, then it also
H1−scatters. Then we discuss the problem of uniqueness of ground states of the functional I(γ)2 , for
general γ ≥ 0.
80 CHAPITRE 3. TRAVELING WAVES OF NLS–SZEGO EQUATION ON THE LINE
3.4.1 Persistence of regularity for scattering
The persistence of regularity for scattering can be established by Strichartz estimates and a bootstrapargument.
Proposition 3.4.1. If u0 ∈ H1+ and there exists u+ ∈ L2
+ such that limt→+∞ ‖u(t) − eit∂2xu+‖L2 = 0,
where u is the unique solution of (3.1.1), then we have u+ = u0 + i∫ +∞
0e−iτ∂
2xΠ(|u(τ)|4u(τ))dτ ∈ H1
+
and limt→+∞ ‖u(t)− eit∂2xu+‖H1 = 0.
Demonstration. We claim that u ∈ L6(0,+∞;L6(R)). In fact (6, 6) is 1-admissible. Set v(t, x) :=
eit∂2xu+(x) and r(t, x) = u(t, x)− v(t, x), then we have v ∈ L6(Rt × Rx) by Strichartz inequality and
i∂tr + ∂2xr = −Π(|u|4u). (3.4.1)
r(t) = ei(t−σ)∂2xr(σ) + i
∫ t
σ
ei(t−τ)∂2xΠ(|u(τ)|4u(τ))dτ. (3.4.2)
Since u ∈ L∞(R;H1+), we have r ∈ L6
loc(R+, L6(R)). Recall that Π is bounded Lp → Lp, for all 1 < p <
+∞. Applying Strichartz inequality to formula (3.4.2), then there exists a constant C∗ > 0 such that
‖r‖L6(T,T ′;L6(R)) ≤C∗(‖r(T )‖L2 + ‖|u|4u‖
L65 (T,T ′;L
65 (R))
)≤C∗
(‖r(T )‖L2 +
5∑k=0
‖v‖kL6(T,T ′;L6(R))‖r‖5−kL6(T,T ′;L6(R))
)
holds for all 0 ≤ T < T ′. Set δ = min1, 12C∗(2+C∗)4 , there exists T > 0 such that ‖v‖L6(T,+∞;L6(R)) ≤ δ
and ‖r(T )‖L2 ≤ δ. We choose T1 := supS ∈ [T,+∞) : ‖r‖L6(T,S;L6(R)) ≤ 12+C∗ . For every T ′ ∈ (T, T1),
we have ‖r‖L6(T,T ′;L6(R)) ≤ C∗(δ + δ5 + (2 + C∗)−5 + 4δ‖r‖L6(T,T ′;L6(R))
). Then we have
‖r‖L6(T,T ′;L6(R)) ≤ 2δC∗ +1
(2 + C∗)4≤ 1
(2 + C∗)3<
1
2 + C∗.
Thus T1 = +∞ and ‖r‖L6(T,+∞;L6(R)) <1
2+C∗ , which yields that u ∈ L6(0,+∞, L6(R)).
Set u+ := u0 + i∫ +∞
0e−iτ∂
2xΠ(|u(τ)|4u(τ))dτ . Since u ∈ L6(Rt × Rx), Strichartz inequality implies that
‖u(t)− eit∂2x u+‖L2 . ‖Π(|u|4u)‖
L65 (t,+∞;L
65 (R))
. ‖u‖5L6(t,+∞;L6(R)) → 0, as t→ +∞. (3.4.3)
Thus u+ = u+. Since the momentum P (u) = 〈−i∂xu, u〉L2 = ‖|D| 12u‖2L2 is conserved, we have
supt∈R‖∂xu(t)‖2L2 . E(u0) + ‖|D| 12u0‖4L2‖u0‖2L2 . (3.4.4)
Thus ∂x Π(|u|4u) ∈ L∞(R;L2+) → L1
loc(R;L2+). The Strichartz estimate yields that ∀T > 0, we have
‖∂xu‖L6(0,T ;L6(R)) . ‖∂xu0‖L2 +
∫ T
0
‖∂xΠ(|u(τ)|4u(τ))‖L2dτ < +∞. (3.4.5)
Since u ∈ L6(Rt × Rx), there exists T0 > 0 such that ‖u‖4L6(T0,+∞;L6(R)) ≤1
2C∗ . For all T > T0, we have
‖∂xu‖L6(T0,T ;L6(R)) ≤C∗(‖∂xu0‖L2 + ‖|∂xu||u|4‖L
65 (T0,T ;L
65 (R))
)
≤C∗(‖∂xu0‖L2 + ‖u‖4L6(T0,+∞;L6(R))‖∂xu‖L6(T0,T ;L6(R)))
3.4. APPENDICES 81
Thus ‖∂xu‖L6(T0,+∞;L6(R)) ≤ 2C∗‖∂xu0‖L2 and ∂xu ∈ L6(0,+∞;L6(R)). Consequently, we use Strichartzestimate to obain
‖∂x(u(t)− eit∂2xu+)‖L2 . ‖u‖4L6(t,+∞;L6(R))‖∂xu‖L6(t,+∞;L6(R)) → 0,
as t→ +∞.
Similarly, we have the following proposition for scattering backward in time.
Proposition 3.4.2. If u0 ∈ H1+ and there exists u− ∈ L2
+ such that limt→−∞ ‖u(t) − eit∂2xu−‖L2 = 0,
where u is the unique solution of (3.1.1), then we have u− = u0 − i∫ 0
−∞ e−iτ∂2xΠ(|u(τ)|4u(τ))dτ ∈ H1
+
and limt→+∞ ‖u(t)− eit∂2xu−‖H1 = 0.
The Lr−norm of a solution of linear Schrodinger equation decays as t→ ±∞, for all 2 < r ≤ +∞.
Lemma 3.4.3. If f ∈ H1(R) and 2 < r ≤ +∞, then ‖eit∂2xf‖Lr → 0, as |t| → +∞.
Demonstration. We set w(t) := eit∂2xf and q = 4r
r−2 , then 2q + 1
r = 12 and w ∈ L∞(R, H1
+)⋂Lq(R;Lr(R))
by Strichartz estimate. Gagliardo–Nirenberg inequality yields that
‖w(t)− w(s)‖Lr ≤ ‖∂xw(t)− ∂xw(s)‖2q
L2‖w(t)− w(s)‖1−2q
L2 ≤ 2‖∂xf‖2q
L2‖w(t)− w(s)‖1−2q
L2 .
Since supt∈R ‖∂tw(t)‖H−1(R) = supt∈R ‖∂2xw(t)‖H−1(R) ≤ ‖f‖H1 , the mapping t → w(t) is Lipschitz
continuous R→ H−1(R). Therefore,
‖w(t)− w(s)‖L2 ≤√‖w(t)− w(s)‖H1‖w(t)− w(s)‖H−1 . ‖f‖H1 |t− s| 12
and ‖w(t) − w(s)‖Lr . ‖f‖H1 |t − s|q−22q . The mapping t → w(t) is uniformly continuous R → Lr(R).
Since w ∈ Lq(R;Lr(R)), we have ‖eit∂2xf‖Lr = ‖w(t)‖Lr → 0, as |t| → +∞.
This lemma yields that a traveling wave u(t, x) = eiωtQ(x) does not H1−scatter neither L2−scatter,
with ω > 0 and Q(0) ∈ G(0)2 (
4
√3J
(0)2 , (
3
√3J
(0)2 ω
2 )16 ). Together with theorem 3.1.6, we can prove corollary
3.1.10.
3.4.2 The open problem of uniqueness of ground states
The problem of classification of ground states of I(γ)2 remains open for general γ, since it is difficult
to solve the non local equation (3.1.7). However, if f is a ground state of I(γ)2 , then so is P (f), where
P (f)(ξ) = |f(ξ)|. Precisely, we have the following proposition.
Proposition 3.4.4. For every m ∈ N⋂
[2,+∞) and γ ≥ 0, if f ∈ G(γ)m , then P (f) ∈ G(γ)
m and there
exist two parameters a, b ∈ R such that f(ξ) = |f(ξ)|ei(aξ+b), for every ξ ≥ 0.
82 CHAPITRE 3. TRAVELING WAVES OF NLS–SZEGO EQUATION ON THE LINE
Demonstration. Parseval identity implies that
‖|D|sP (f)‖2L2 =1
2π
∫ +∞
0
|ξ|2s|f(ξ)|2dξ = ‖|D|sf‖2L2 , ∀0 ≤ s ≤ 1.
However, since P (f)(ξ) = |f(ξ)|, for every ξ > 0, we have P (f)m+1 ≥ |fm+1|, for every m ∈ N and
‖P (f)‖2m+2L2m+2 =
1
2π
∫ +∞
0
| P (f)m+1(ξ)|2dξ ≥ 1
2π
∫ +∞
0
|fm+1(ξ)|2dξ = ‖f‖2m+2L2m+2
Thus J(γ)m ≤ I
(γ)m (P (f)) ≤ I
(γ)m (f) = J
(γ)m , for all γ ≥ 0 and m ∈ N. So P (f) ∈ G(γ)
m and all precedent
inequalities become equalities. In particular, P (f)m+1 = |fm+1|. We set h(ξ) = f(ξ), then the Euler–Lagrange equation (3.1.11) reads in Fourier modes as
(Am(f) +Bm,γ(f)ξ + Cm(f)ξ2)h(ξ) = Dm,γ1ξ≥0T (h, h, · · · , h)(ξ), (3.4.6)
where Am(f), Bm,γ(f), Cm(f), Dm,γ > 0 and T : L1(R+)2m+1 → L1(R+) is (2m+ 1)-linear defined as
Tm(h1, h2, · · · , h2m+1)(ξ) =
∫Sm(ξ)
Π2m+1j=1 hj(ηj)dη1 · · · dη2m
with Sm(ξ) = (η1, · · · , η2m+1) ∈ R2m+1+ :
∑m+1j=1 ηj =
∑2m+1j=m+2 ηk + ξ.
We claim that if h(ξ) = 0 for some ξ ∈ R+ then h ≡ 0. In fact, assume by contradiction that h(ζ) 6= 0,
for some ζ ≥ 0. Since P (f) ∈ G(γ)m , we replace h by |h| in equation (3.4.6) in order to get the following
implication :
h(ξ) = 0 =⇒ h(mζ + ξ
m+ 1) = 0.
We construct an iterative sequence ξ0 = ξ and ξn+1 = mζ+ξnm+1 , ∀n ∈ N. Then we have h(ξn) = 0 and
limn→+∞ ξn = ζ. The continuity of h gives that h(ζ) = 0, contradiction. Thus h ≡ 0.
Since h = f and f 6= 0, then h is continuous R→ C∗. Thus there exists a continuous function α : R+ → S1
such that
Πm+1j=1 h(ξj) = α(
m+1∑j=1
ξj)Πm+1j=1 |h(ξj)|.
The lifting theorem yields that there exists a unique continuous function ϕ : R+ → R such that
h(ξ) = |h(ξ)|eiϕ(ξ),
m+1∑j=1
ϕ(ξj) = β(
m+1∑j=1
ξj)
for some continuous function β : R+ → R. We set φ(ξ) = ϕ(ξ)− ϕ(0) then we have
φ(ξ1) + φ(ξ2) = 2φ(ξ1 + ξ2
2), ∀ξ1, ξ2 ≥ 0.
Consequently, ϕ(ξ) = φ(1)ξ + ϕ(0) = (ϕ(1)− ϕ(0))ξ + ϕ(0), for every ξ ≥ 0.
3.4. APPENDICES 83
Thus it suffices to study the uniqueness of ground states modulo the positive Fourier transformation.
We compare theorem 3.1.6 and theorem 3.1.8. Since we do not know the uniqueness of ground states of
the functional I(γ)2 , the conservation law P (u) = ‖|D| 12u‖2L2 can not be completely used to determine the
L6−norm of the final profile that approaches u(t). However, if the ground state is unique up to scaling,phase rotation and spatial translation, then we can determine the L6−norm of the profile that approachesu(t). So we have the actual orbital stability in the case γ = 2.
84 CHAPITRE 3. TRAVELING WAVES OF NLS–SZEGO EQUATION ON THE LINE
Chapitre 4
Integrability of the BO equation onthe line
Ce chapitre est une reprise de l’article Sun [135].
Resume Le but de cette troisieme contribution est de construire des coordonnees action–angle pourl’equation de Benjamin–Ono (BO) sur la droite restreinte a la variete des N -solitons, notee par UN , pourtout entier N strictement positif. Il existe un symplectomorphisme reel analytique de UN vers un cer-tain sous-ensemble ouvert convexe de R2N , pour lequel l’energie de BO ne depend que des N premieresvariables, les variables d’actions. Cette application permet de lineariser l’equation de BO et de decrirecompletement l’orbite a conjugaison reel analytique pres. On peut resoudre cette equation par quadra-ture. Inspire du travail de Gerard–Kappeler [54], on caracterise la variete UN par une representation faconpolynomiale et en terme spectral. Un ingredient important est l’introduction d’une fonction generatriceauxiliaire, qui engendre l’ensemble des hierarchies de BO. Une autre consequence est que la variete UN estle revetement universel de la variete des potentiels a N “gap” pour l’equation de BO sur le tore etudieedans [54].
Mots− clefs : Equation de Benjamin–Ono, Coordonnees d’action–angle, Paire de Lax, Espace deHardy, Transformee inverse spectrale, Multi-soliton, Revetement universel
Abstract This chapter is dedicated to proving the complete integrability of the Benjamin–Ono (BO)equation on the line when restricted to every N -soliton manifold, denoted by UN . We construct gene-ralized action–angle coordinates which establish a real analytic symplectomorphism from UN onto someopen convex subset of R2N and allow to solve the equation by quadrature for any such initial datum. As aconsequence, UN is the universal covering of the manifold of N -gap potentials for the BO equation on thetorus as described by Gerard–Kappeler [54]. The global well-posedness of the BO equation in UN is givenby a polynomial characterization and a spectral characterization of the manifold UN . Besides the spectralanalysis of the Lax operator of the BO equation and the shift semigroup acting on some Hardy spaces,the construction of such coordinates also relies on the use of a generating functional, which encodes theentire BO hierarchy.
Keywords Benjamin–Ono equation, Action–angle coordinates, Lax pair, Hardy space, Inverse spectraltransform, Multi-soliton, Universal covering manifold
85
86 CHAPITRE 4. INTEGRABILITY OF THE BO EQUATION ON THE LINE
4.1 Introduction
The Benjamin–Ono (BO) equation on the line reads as
∂tu = H∂2xu− ∂x(u2), (t, x) ∈ R× R, (4.1.1)
where u is real-valued and H = −isign(D) : L2(R)→ L2(R) denotes the Hilbert transform, D = −i∂x,
Hf(ξ) = −isign(ξ)f(ξ), ∀f ∈ L2(R). (4.1.2)
sign(±ξ) = ±1, for all ξ > 0 and sign(0) = 0, f ∈ L2(R) denotes the Fourier–Plancherel transformof f ∈ L2(R). We adopt the convention Lp(R) = Lp(R,C). Its R-subspace consisting of all real-valuedLp-functions is specially emphasized as Lp(R,R) throughout this paper. Equipped with the inner product(f, g) ∈ L2(R)× L2(R) 7→ 〈f, g〉L2 =
∫R f(x)g(x)dx ∈ C, L2(R) is a C-Hilbert space.
Derived by Benjamin [8] and Ono [116], this equation describes the evolution of weakly nonlinear internallong waves in a two-layer fluid. The BO equation is globally well-posed in every Sobolev spaces Hs(R,R),s ≥ 0. (see Tao [137] for s ≥ 1, Burq–Planchon [21] for s > 1
4 , Ionescu–Kenig [80], Molinet–Pilod [108]and Ifrim–Tataru [79] for s ≥ 0, etc.) Recall the scaling and translation invariances of equation (4.1.1) :if u = u(t, x) is a solution, so is uc,y : (t, x) 7→ cu(c2t, c(x− y)). A smooth solution u = u(t, x) is called asolitary wave of (4.1.1) if there exists R ∈ C∞(R) solving the following non local elliptic equation
HR′ +R−R2 = 0, R(x) > 0 (4.1.3)
and u(t, x) = Rc(x − y − ct), where Rc(x) = cR(cx), for some c > 0 and y ∈ R. The unique (up totranslation) solution of equation (4.1.3) is given by the following formula
R(x) =2
1 + x2, ∀x ∈ R, (4.1.4)
in Benjamin [8] and Amick–Toland [4] for the uniqueness statement. Inspired from the complete classifi-cation of solitary waves of the BO equation, we introduce the main object of this paper.
Definition 4.1.1. A function of the form u(x) =∑Nj=1Rcj (x − xj) is called an N -soliton, for some
positive integer N ∈ N+ := Z⋂
(0,+∞), where cj > 0 and xj ∈ R, for every j = 1, 2, · · · , N . LetUN ⊂ L2(R,R) denote the subset consisting of all the N -solitons.
In the point of view of topology and differential manifolds, the subset UN is a simply connected, realanalytic, embedded submanifold of the R-Hilbert space L2(R,R). It has real dimension 2N . The tangentspace to UN at an arbitrary N -soliton is included in an auxiliary space
T := h ∈ L2(R, (1 + x2)dx) : h(R) ⊂ R,∫Rh = 0, (4.1.5)
in which a 2-covector ω ∈ Λ2(T ∗) is well defined by ω(h1, h2) = i2π
∫Rh1(ξ)h2(ξ)
ξ dξ, for every h1, h2 ∈ T ,
by Hardy’s inequality. We define a translation-invariant 2-form ω : u ∈ UN 7→ ω ∈ Λ2(T ∗), endowed with
4.1. INTRODUCTION 87
which UN is a symplectic manifold. The tangent space to UN at u ∈ UN is denoted by Tu(UN ). For everysmooth function f : UN → R, its Hamiltonian vector field Xf ∈ X(UN ) is given by
Xf : u ∈ UN 7→ ∂x∇uf(u) ∈ Tu(UN ),
where ∇uf(u) denotes the Frechet derivative of f , i.e. df(u)(h) = 〈h,∇uf(u)〉L2 , for every h ∈ Tu(UN ).The Poisson bracket of f and another smooth function g : UN → R is defined by
f, g : u ∈ UN 7→ ωu(Xf (u), Xg(u)) = 〈∂x∇uf(u),∇ug(u)〉L2 ∈ R.
Then the BO equation (4.1.1) in the N -soliton manifold (UN , ω) can be written in Hamiltonian form
∂tu = XE(u), where E(u) =1
2〈|D|u, u〉
H−12 ,H
12− 1
3
∫Ru3. (4.1.6)
The Cauchy problem of (4.1.6) is globally well-posed in the manifold UN (see proposition 4.4.9). Inspiredfrom the construction of Birkhoff coordinates of the space-periodic BO equation discovered by Gerard–Kappeler [54], we want to show the complete integrability of (4.1.6) in the Liouville sense.
Let ΩN := (r1, r2, · · · , rN ) ∈ RN : rj < rj+1 < 0, ∀j = 1, 2, · · · , N − 1 denote the subset of actions
and ν =∑Nj=1 drj ∧ dαj denotes the canonical symplectic form on ΩN × RN . The main result of this
paper is stated as follows.
Theorem 1. There exists a real analytic symplectomorphism ΦN : (UN , ω)→ (ΩN × RN , ν) such that
E Φ−1N (r1, r2, · · · , rN ;α1, α2, · · · , αN ) = − 1
2π
N∑j=1
|rj |2. (4.1.7)
Remark 4.1.2. A consequence of theorem 1 is that UN is simply connected. In fact the manifold UNcan be interpreted as the universal covering of the manifold of N -gap potentials for the Benjamin–Onoequation on the torus as described by Gerard–Kappeler in [54]. We refer to section 4.6 for a direct proofof these topological facts, independently of theorem 1.
Remark 4.1.3. Then ΦN : u ∈ UN 7→ (I1(u), I2(u), · · · , IN (u); γ1(u), γ2(u), · · · , γN (u)) ∈ ΩN × RNintroduces the generalized action–angle coordinates of the BO equation in the N -soliton manifold, i.e.
Ik, E(u) = 0, γk, E(u) =Ik(u)
π, ∀u ∈ UN . (4.1.8)
Theorem 1 gives a complete description of the orbit structure of the flow of equation (4.1.6) up to realbi-analytic conjugacy. Let u : t ∈ R 7→ u(t) ∈ UN denote the solution of equation (4.1.6), rk(t) = Ik u(t)denotes action coordinates and αk(t) = γk u(t) denotes the generalized angle coordinates, then we have
rk(t) = rk(0), αk(t) = αk(0)− rk(0)t
π, ∀k = 1, 2, · · · , N. (4.1.9)
We refer to definition 4.5.1 and theorem 4.5.2 for a precise description of ΦN .
In order to establish the link between the action–angle coordinates and the translation–scaling parametersof an N -soliton, we introduce the inverse spectral matrix associated to ΦN , denoted by
M : u ∈ UN 7→ (Mkj(u))1≤j,k≤N ∈ CN×N , Mkj(u) =
2πiIk(u)−Ij(u)
√Ik(u)Ij(u) , if j 6= k,
γj(u) + πiIj(u) , if j = k,
(4.1.10)
where Ik, γk : U → R is given by remark 4.1.3. Then UN has the following polynomial characterization.
88 CHAPITRE 4. INTEGRABILITY OF THE BO EQUATION ON THE LINE
Proposition 4.1.4. A real-valued function u ∈ UN if and only if there exists a monic polynomial
Qu ∈ C[X] of degree N , whose roots are contained in the lower half-plane C− and u = −2ImQ′uQu
. Precisely,
Qu is unique and is the characteristic polynomial of the matrix M(u) ∈ CN×N defined by (4.1.10).
An N -soliton is expressed by u(x) =∑Nj=1Rcj (x − xj) if and only if its translation–scaling parameters
xj − c−1j i1≤j≤N ⊂ CN− are the roots of the characteristic polynomial Qu(X) = det(X −M(u)), whose
coefficients are expressed in terms of the action–angle coordinates (Ij(u), γj(u))1≤j≤N ∈ ΩN × RN .Proposition 4.1.4 is restated with more details in proposition 4.4.1, formula (4.5.11) and theorem 4.4.8which gives a spectral characterization of UN . If u : t ∈ R 7→ u(t) ∈ UN solves the BO equation (4.1.1),then we have the following explicit formula
u(t, x) = 2Im〈(M(u0)− (x+ t
πV(u0)))−1
X(u0), Y (u0)〉CN , (t, x) ∈ R× R, (4.1.11)
where the inner product of CN is 〈X,Y 〉CN = XTY , for every u ∈ UN , the matrix V(u) ∈ CN×N and thevectors X(u), Y (u) ∈ CN are defined by
√2πX(u)T = (
√|I1(u)|,
√|I2(u)|, · · · ,
√|IN (u)|),
√2π−1Y (u)T = (
√|I1(u)|−1,
√|I2(u)|−1, · · · ,
√|IN (u)|−1),
V(u) =
I1(u)I2(u)
. . .IN (u)
.
4.1.1 Notation
Before outlining the construction of action–angle coordinates, we introduce some notations used in thispaper. The indicator function of a subset A ⊂ X is denoted by 1A, i.e. 1A(x) = 1 if x ∈ A and1A(x) = 0 if x ∈ X\A. Recall that H : L2(R) → L2(R) denotes the Hilbert transform given by (4.1.2).Set IdL2(R)(f) = f , for every f ∈ L2(R). Let Π : L2(R)→ L2(R) denote the Szego projector, defined by
Π :=IdL2(R) + iH
2⇐⇒ Πf(ξ) = 1[0,+∞)(ξ)f(ξ), ∀ξ ∈ R, ∀f ∈ L2(R). (4.1.12)
If O is an open subset of C, we denote by Hol(O) all holomorphic functions on O. Let the upper half-plane and the lower half-plane be denoted by C+ = z ∈ C : Imz > 0 and C− = z ∈ C : Imz < 0respectively. For every p ∈ (0,+∞], we denote by Lp+ to be the Hardy space of holomorphic functions onC+ such that Lp+ = g ∈ Hol(C+) : ‖g‖Lp+ < +∞, where
‖g‖Lp+ = supy>0
(∫R|g(x+ iy)|pdx
) 1p
, if p ∈ (0,+∞), (4.1.13)
and ‖g‖L∞+ = supz∈C+|g(z)|. A function g ∈ L∞+ is called an inner function if |g| = 1 on R. When p = 2,
the Paley–Wiener theorem yields the identification between L2+ and Π[L2(R)] :
L2+ = F−1[L2(0,+∞)] = f ∈ L2(R) : suppf ⊂ [0,+∞) = Π(L2(R)),
where F : f ∈ L2(R) 7→ f ∈ L2(R) denotes the Fourier–Plancherel transform. Similarly, we setL2− = (IdL2(R) − Π)(L2(R)). Let the filtered Sobolev spaces be denoted as Hs
+ := L2+
⋂Hs(R) and
Hs− := L2
−⋂Hs(R), for every s ≥ 0.
4.1. INTRODUCTION 89
The domain of definition of an unbounded operator A on some Hilbert space E is denoted by D(A) ⊂ E .Given another operator B on D(B) ⊂ E such that A(D(A)) ⊂ D(B) and B(D(B)) ⊂ D(A), their Liebracket is an operator defined on D(A)
⋂D(B) ⊂ E , which is given by
[A,B] := AB − BA. (4.1.14)
If the operator A is self-adjoint, let σ(A) denote its spectrum, σpp(A) denotes the set of its eigenva-
lues and σcont(A) denotes its continuous spectrum. Then σcont(A)⋃σpp(A) = σ(A) ⊂ R. Given two
C-Hilbert spaces E1 and E2, let B(E1, E2) denote the C-Banach space of all bounded C-linear transforma-tions E1 → E2, equipped with the uniform norm.
Given a smooth manifold M of real dimension N , let C∞(M) denote all smooth functions f : M → Rand the set of all smooth vector fields is denoted by X(M). The tangent (resp. cotangent) space to M atp ∈M is denoted by Tp(M) (resp. T ∗p (M)). Given k ∈ N, the R-vector space of smooth k-forms on M is
denoted by Ωk(M). Given a R-vector space V, we denote by Λk(V∗) the vector space of all k-covectorson V. Given a smooth covariant tensor field A on M and X ∈ X(M), the Lie derivative of A with respectto X is denoted by LX(A), which is also a smooth tensor field on M. If N is another smooth manifold,F : N→M is a smooth map and A is a smooth covariant k-tensor field on M, the pullback of A by Fis denoted by F∗A, which is a smooth k-tensor field on N defined by ∀p ∈ N, ∀j = 1, 2, · · · , k,
(F∗A)p(v1, v2, · · · , vk) = AF(p) (dF(p)(v1),dF(p)(v2), · · · ,dF(p)(vk)) , ∀vj ∈ Tp(N). (4.1.15)
Given a positive integer N , let C≤N−1[X] denote the C-vector space of all polynomials with complexcoefficients whose degree is no greater than N − 1 and CN [X] = C≤N [X]\C≤N−1[X] consists of allpolynomials of degree exactly N . R+ = [0,+∞) and R∗+ = (0,+∞). D(z, r) ⊂ C denotes the open discof radius r > 0, whose center is z ∈ C.
4.1.2 Organization of this paper
The construction of action–angle coordinates for the BO equation (4.1.6) mainly relies on the Lax pairformulation ∂tLu = [Bu, Lu], discovered by Nakamura [110] and Bock–Kruskal [12]. Section 4.2 is dedica-ted to the spectral analysis of the Lax operator Lu : h ∈ H1
+ 7→ −i∂xh−Π(uh) ∈ L2+ given by definition
4.2.1 for general symbol u ∈ L2(R,R), where Π denotes the Szego projector given in (4.1.12) and theHardy space L2
+ is defined in (4.1.13). Lu is an unbounded self-adjoint operator on L2+ that is bounded
from below, it has essential spectrum σess(Lu) = [0,+∞). If x 7→ xu(x) ∈ L2(R) in addition, everyeigenvalue is negative and simple, thanks to an identity firstly found by Wu [146]. Then we introduce agenerating function which encodes the entire BO hierarchy,
Hλ(u) = 〈(Lu + λ)−1Πu,Πu〉L2 , if λ ∈ C\σ(−Lu), (4.1.16)
in definition 4.2.9. It provides a sequence of conservation laws controlling every Sobolev norms.
In section 4.3, we study the shift semigroup (S(η)∗)η≥0 acting on the Hardy space L2+, where S(η)f = eηf
and eη(x) = eiηx. Then a weak version of Beurling–Lax theorem can be obtained by solving a linear dif-ferential equation with constant coefficients. Every N -dimensional subspace of L2
+ that is invariant under
its infinitesimal generator G = i ddη
∣∣η=0+S(η)∗ is of the form
C≤N−1[X]
Q , for some monic polynomial Q
whose roots are contained in the lower half-plane C−.
90 CHAPITRE 4. INTEGRABILITY OF THE BO EQUATION ON THE LINE
In section 4.4, the real analytic structure and symplectic structure of the N -soliton subset UN are es-tablished at first. Then we continue the spectral analysis of the Lax operator Lu, ∀u ∈ UN . Lu has Nsimple eigenvalues λu1 < λu2 < · · · < λuN < 0 and the Hardy space L2
+ splits as
L2+ = Hcont(Lu)
⊕Hpp(Lu), Hcont(Lu) = Hac(Lu) = ΘuL
2+, Hpp(Lu) =
C≤N−1[X]
Qu. (4.1.17)
where Qu denotes the characteristic polynomial of u given by proposition 4.1.4 and Θu = QuQu
is an in-
ner function on the upper half-plane C+. Proposition 4.1.4 is proved by identifying M(u) in (4.1.10)as the matrix of the restriction G|Hpp(Lu) associated to the spectral basis ϕu1 , ϕu2 , · · · , ϕuN, whereϕuj ∈ Ker(λuj − Lu) such that ‖ϕuj ‖L2 = 1 and
∫R uϕ
uj > 0. The generating function Hλ in (4.1.16)
can be identified as the Borel–Cauchy transform of the spectral measure of Lu associated to the vectorΠu, which yields the invariance of UN under the BO flow in H∞(R,R). Hence (4.1.6) is a globally well-posed Hamiltonian system on UN .
Section 4.5 is dedicated to completing the proof of theorem 1. The generalized angle-variables are the realparts of the diagonal elements of the matrix M(u), i.e. γj : u ∈ UN 7→ Re〈Gϕuj , ϕuj 〉L2 ∈ R and the action-
variables are Ij : u ∈ UN 7→ 2πλuj ∈ R. Thanks to the Lax pair formulation dL(u)(XHλ(u)) = [Bλu , Lu],
where L : u ∈ UN 7→ Lu ∈ B(H1+, L
2+) is R-affine and Bλu is some skew-adjoint operator on L2
+, we havethe following formulas of Poisson brackets,
2πλj , γk = 1j=k, γj , γk = 0 on UN , 1 ≤ j, k ≤ N. (4.1.18)
which implies that ΦN : u ∈ UN 7→ (I1(u), I2(u), · · · , IN (u); γ1(u), γ2(u), · · · , γN (u)) ∈ ΩN × RN isa real analytic immersion. The diffeomorphism property of ΦN is given by Hadamard’s global inverse
theorem. The inverse spectral formula Πu = iQ′uQu
with Qu(X) = det(X − G|Hpp(Lu)), which is restated
as formula (4.5.11), implies the explicit formula (4.1.11) of all multi-soliton solutions of the BO equa-tion (4.1.1) and (4.5.11) provides an alternative proof of the injectivity of ΦN . Finally, we show thatΦN : (UN , ω) → (ΩN × RN , ν) is a symplectomorphism by restricting the 2-form ω − Φ∗Nν to a special
Lagrangian submanifold ΛN :=⋂Nj=1 γ
−1j (0) ⊂ UN .
In appendix 4.6, we establish the simple connectedness of UN and a covering map from UN to the manifoldof N -gap potentials from their constructions without using the integrability theorems.
4.1.3 Related work
The BO equation has been extensively studied for nearly sixty years in the domain of partial differentialequations. We refer to Saut [128] for an excellent account of these results. Besides the global well-posednessproblem, various properties of its multi-soliton solutions has been investigated in details. Matsuno [104]has found the explicit expression of multi-soliton solutions of (4.1.1) by following the bilinear methodof Hirota [75]. The multi-phase solutions (periodic multi-solitons) have been constructed by Satsuma–Ishimori [126] at first. We point out the work of Amick–Toland [4] on the characterization of 1-solitonsolutions which can also be revisited by theorem 1 and proposition 4.1.4. In Dobrokhotov–Krichever [30],the multi-phase solutions are constructed by finite zone integration and they have also established aninversion formula for multi-phase solutions. Compared to their work, we give a geometric description ofthe inverse spectral transform by proving the real bi-analyticity and the symplectomorphism property ofthe action–angle map. Furthermore, the inverse spectral formula
Πu(x) = iQ′u(x)
Qu(x), Qu(x) = det(x−G|Hpp(Lu)) = det(x−M(u)), ∀x ∈ R. (4.1.19)
4.2. THE LAX OPERATOR 91
provides a spectral connection between the Lax operator Lu and the infinitesimal generator G. The ideaof introducing generating function Hλ has also been used for the quantum BO equation in Nazarov–Sklyanin [111]. Their method has also been developed by Moll [109] for the classical BO equation. Theasymptotic stability of soliton solutions and of solutions starting with sums of widely separated solitonprofiles is obtained by Kenig–Martel [85].
Concerning the investigation of integrability for the BO equation on R besides the discovery of Lax pairformulation, we mention the pioneering work of Ablowitz–Fokas [1], Coifman–Wickerhauser [28], Kaup–Matsuno [84] and Wu [146, 147] for the inverse scattering transform. In the space-periodic regime, theBO equation on the torus T admits global Birkhoff coordinates on L2
r,0(T) := v ∈ L2(T,R) :∫T v = 0
in Gerard–Kappeler [54]. We refer to Gerard–Kappeler–Topalov [55] to see that the Birkhoff coordinatesof the BO equation on the torus can be extended to a larger Sobolev space Hs
r,0(T) := v ∈ Hs(T,R) :∫T v = 0, for every − 1
2 < s < 0. We point out that both Korteweg–de Vries equation on T (seeKappeler–Poschel [81]) and the defocusing cubic Schrodinger equation on T (see Grebert–Kappeler [65])admit global Birkhoff coordinates. The theory of finite-dimensional Hamiltonian system is transferredto the BO, KdV and dNLS equation on T through the submanifolds of corresponding finite-gap poten-tials, which are introduced to solve the periodic KdV initial problem. We refer to Matveev [105] for details.
Moreover, the cubic Szego equation both on T (see Gerard–Grellier [47, 49, 51, 52]) and on R (see Po-covnicu [119, 120]) admit global (generalized) action–angle coordinates on all finite-rank generic rationalfunction manifolds, denoted respectively byM(N)Tgen andM(N)Rgen. Moreover, the cubic Szego equationboth on T and on R have inverse spectral formulas which permit the Szego flows to be expressed explicitlyin terms of time-variables and initial data without using action–angle coordinates. The shift semigroup(S(η)∗)η≥0 and its infinitesimal generator G are also used in Pocovnicu [120] to establish the integrabilityof the cubic Szego equation on the line.
The BO equation admits an infinite hierarchy of conservation laws controlling every Hs-norm (seeAblowitz–Fokas [1], Coifman–Wickerhauser [28] in the case 2s ∈ N and Talbut [136] in the case− 1
2 < s < 0and conservation law controlling Besov norms etc.), so does the KdV equation and the NLS equation (seeKillip–Visan–Zhang [90], Koch–Tataru [91], Faddeev–Takhtajan [33], Gerard [46] and Sun [131] etc.)
Throughout this chapter, the main results of each section are stated at the beginning. Theirproofs are left inside the corresponding subsections.
4.2 The Lax operator
This section is dedicated to studying the Lax operator Lu in the Lax pair formulation of the BOequation (4.1.1), discovered by Nakamura [110] and Bock–Kruskal [12]. Then we describe the locationand revisit the simplicity of eigenvalues of Lu. At last, we introduce a generating functional Hλ whichencodes the entire BO hierarchy. The equation ∂tu = ∂x∇uHλ(u) also enjoys a Lax pair structure withthe same Lax operator Lu.
Definition 4.2.1. Given u ∈ L2(R,R), its associated Lax operator Lu is an unbounded operator on L2+,
given by Lu := D− Tu, where D : h ∈ H1+ 7→ −i∂xh ∈ L2
+ and Tu denotes the Toeplitz operator of symbolu, defined by Tu : h ∈ H1
+ 7→ Π(uh) ∈ L2+, where the Szego projector Π : L2(R)→ L2
+ is given by (4.1.12).If u ∈ H1(R,R) in addition, we define Bu := i(T|D|u − T 2
u) ∈ B(H1+, L
2+).
92 CHAPITRE 4. INTEGRABILITY OF THE BO EQUATION ON THE LINE
Both D and Tu are densely defined symmetric operators on L2+ and ‖Tu(h)‖L2 ≤ ‖u‖L2‖h‖L∞ , for every
h ∈ H1+ and u ∈ L2(R,R). Moreover, the Fourier–Plancherel transform implies that D is a self-adjoint
operator on L2+, whose domain of definition is H1
+.
Proposition 4.2.2. If u ∈ L2(R,R), then Lu is an unbounded self-adjoint operator on L2+, whose
domain of definition is D(Lu) = H1+. Moreover, Lu is bounded from below. The essential spectrum of Lu
is σess(Lu) = σess(D) = [0,+∞) and its pure point spectrum satisfies σpp(Lu) ⊂ [−C2
4 ‖u‖2L2 ,+∞), where
C = inff∈H1+\0
‖|D|14 f‖L2
‖f‖L4denotes the Sobolev constant.
Thanks to an identity firstly found by Wu [146] in the negative eigenvalue case, we show the simplicityof the pure point spectrum σpp(Lu), if u ∈ L2(R, (1 + x2)dx) is real-valued.
Proposition 4.2.3. Assume that u ∈ L2(R;R) and x 7→ xu(x) ∈ L2(R). For every λ ∈ R and ϕ ∈Ker(λ− Lu), we have uϕ ∈ C1(R)
⋂H1(R) and the following identity holds,∣∣∣ ∫
Ruϕ∣∣∣2 = −2πλ
∫R|ϕ|2. (4.2.1)
Thus σpp(Lu) ⊂ (−∞, 0) and for every λ ∈ σpp(Lu), we have
Ker(λ− Lu) ⊂ ϕ ∈ H1+ : ϕ|R+
∈ C1(R+)⋂H1(R+) and ξ 7→ ξ[ϕ(ξ) + ∂ξϕ(ξ)] ∈ L2(R+). (4.2.2)
Corollary 4.2.4. Assume that u ∈ L2(R;R) and x 7→ xu(x) ∈ L2(R). Then every eigenvalue of Lu is
simple. If u ∈ L∞(R) in addition, then σpp(Lu) is a finite subset of [−C2‖u‖2
L2
4 , 0).
Demonstration. Fix λ ∈ σpp(Lu) and set Vλ = Ker(λ−Lu), then dimC(Vλ) ≥ 1. We define a linear formA : Vλ → C such that
A(ϕ) :=
∫Ruϕ
Then identity (4.2.1) yields that Ker(A) = 0. Thus V ∼= V/Ker(A) ∼= Im(A) → C. So we havedimC(Vλ) = 1. When u ∈ L∞(R) in addition, the finiteness of σpp(Lu)
⋂(−∞, 0) is given by Theorem
1.2 of Wu [146].
We recall some known results of global well-posedness of the BO equation on the line.
Proposition 4.2.5. For every s ≥ 0, the Frechet space C(R, Hs(R)) is endowed with the topologyof uniform convergence on every compact subset of R. There exists a unique continuous mapping u0 ∈Hs(R) 7→ u ∈ C(R, Hs(R)) such that u solves the BO equation (4.1.1) with initial datum u(0) = u0.
Demonstration. See Tao [137], Burq–Planchon [21], Ionescu–Kenig [80], Molinet–Pilod [108], Ifrim–Tataru[79] etc.
Proposition 4.2.6. For every n ∈ N, if u0 ∈ Hn2 (R,R), let u : t ∈ R 7→ u(t) ∈ H
n2 (R,R) solves
equation (4.1.1) with initial datum u(0) = u0, then C(‖u0‖H n2
) := supt∈R ‖u(t)‖Hn2< +∞.
Demonstration. See Ablowitz–Fokas [1], Coifman–Wickerhauser [28].
When u ∈ H2(R,R), the Toeplitz operators T|D|u and Tu are bounded both on L2+ and on H1
+. So Bu isa bounded skew-adjoint operator both on L2
+ and on H1+.
4.2. THE LAX OPERATOR 93
Proposition 4.2.7. Let u : t ∈ R 7→ u(t) ∈ H2(R,R) denote the unique solution of equation (4.1.1),then
∂tLu(t) = [Bu(t), Lu(t)] ∈ B(H1+, L
2+), ∀t ∈ R. (4.2.3)
Let U : t 7→ U(t) ∈ B(L2+) := B(L2
+, L2+) denote the unique solution of the following equation
U ′(t) = Bu(t)U(t), U(0) = IdL2+, (4.2.4)
if u : t ∈ R 7→ u(t) ∈ H2(R,R) denote the unique solution of equation (4.1.1). The system (4.2.4) isglobally well-posed in B(L2
+), thanks to proposition 4.2.6, the following estimate
‖Bu(h)‖L2 . (‖u‖H2 + ‖u‖2H1)‖h‖L2 , ∀h ∈ L2+, ∀u ∈ H2(R,R).
and a classical Cauchy theorem (see for instance lemma 7.2 of Sun [131]). Since B∗u = −Bu, the operatorU(t) is unitary for every t ∈ R. Thus, the Lax pair formulation (4.2.3) of the BO equation (4.1.1) isequivalent to the unitary equivalence between Lu(t) and Lu(0),
Lu(t) = U(t)Lu(0)U(t)∗ ∈ B(H1+, L
2+). (4.2.5)
On the one hand, the spectrum of Lu is invariant under the BO flow. In particular, we have σpp(Lu(t)) =σpp(Lu(0)). On the other hand, there exists a sequence of conservation laws controlling every Sobolev
norms Hn2 (R), n ≥ 0. Furthermore, the Lax operator in the Lax pair formulation is not unique. If
f ∈ L∞(R) and p is a polynomial with complex coefficients, then
f(Lu(t)) = U(t)f(Lu(0))U(t)∗ ∈ B(L2+), p(Lu(t)) = U(t)p(Lu(0))U(t)∗ ∈ B(HN
+ , L2+), (4.2.6)
where N is the degree of the polynomial p.
Proposition 4.2.8. Given n ∈ N, let u : t ∈ R 7→ u(t) ∈ H n2 (R,R) denote the solution of equation
(4.1.1), we setEn(u) := 〈LnuΠu,Πu〉
H−n2 ,H
n2. (4.2.7)
Then En(u(t)) = En(u(0)), for every t ∈ R. In particular, E1 = E on H12 (R,R), where the energy
functional E is given by (4.1.6).
Definition 4.2.9. Given u ∈ L2(R,R) and λ ∈ C\σ(−Lu), the C-linear transformation λ + Lu isinvertible in B(H1
+, L2+) and the generating function is defined by Hλ(u) = 〈(Lu + λ)−1Πu,Πu〉L2 . The
subset X := (λ, u) ∈ R× L2(R,R) : 4λ > C2‖u‖2L2 is open in the R-Banach space R× L2(R,R), where
the Sobolev constant is given by C = inff∈H1+\0
‖|D|14 f‖L2
‖f‖L4and we have σ(Lu) ⊂ [−C
2‖u‖2L2
4 ,+∞) by
proposition 4.2.2.
The map (λ, u) ∈ X 7→ Hλ(u) = 〈(Lu + λ)−1Πu,Πu〉L2 ∈ R is real analytic.
Proposition 4.2.10. Let u : t ∈ R 7→ u(t) ∈ H∞(R,R) denote the solution of the BO equation (4.1.1)and we choose λ ∈ C\σ(−Lu(0)), then Hλ(u(t)) = Hλ(u(0)), for every t ∈ R.
Given (λ, u) ∈ X , there exists a neighbourhood of u in L2(R,R), denoted by Vu such that the restrictionHλ : v ∈ Vu 7→ Hλ(v) ∈ R is real analytic. The Frechet derivative of Hλ at u is computed as follows,
dHλ(u)(h) = 〈wλ,Πh〉L2 + 〈wλ,Πh〉L2 + 〈Thwλ, wλ〉L2 = 〈h,wλ + wλ + |wλ|2〉L2 , ∀h ∈ L2(R,R).
94 CHAPITRE 4. INTEGRABILITY OF THE BO EQUATION ON THE LINE
where wλ ∈ H1+ is given by wλ ≡ wλ(u) ≡ wλ(x, u) = [(Lu + λ)−1 Π]u(x), for every x ∈ R. Then
∇uHλ(u) = |wλ(u)|2 + wλ(u) + wλ(u). (4.2.8)
Given (λ, u0) ∈ X fixed, the pseudo-Hamiltonian equation associated to Hλ is defined by
∂tu = ∂x∇uHλ(u) = ∂x(|wλ(u)|2 + wλ(u) + wλ(u)
), u(0) = u0. (4.2.9)
There exists an open subset Vu0of L2(R,R) such that v ∈ Vu0
7→ ∂x(|wλ(v)|2 + wλ(v) + wλ(v)
)∈
L2(R,R) is real analytic and u0 ∈ Vu0. Hence (4.2.9) admits a local solution by Cauchy–Lipschitz theorem.
Remark 4.2.11. The word ’pseudo-Hamiltonian’ is used here because no symplectic form has been definedon L2(R,R) until now. In section 4.4, we show that ∂x∇f(u) is exactly the Hamiltonian vector field ofthe smooth function f : UN → R with respect to the symplectic form ω on the N -soliton manifold UNdefined in (4.4.2).
Proposition 4.2.12. Given (λ, u0) ∈ X fixed, there exists ε > 0 such that (λ, u(t)) ∈ X , for everyt ∈ (−ε, ε), where u : t ∈ (−ε,+ε) 7→ u(t) ∈ L2(R,R) denotes the local solution of (4.2.9) with initialdatum u(0) = u0. We have
∂tLu(t) = [Bλu(t), Lu(t)], where Bλv := i(Twλ(v)Twλ(v) + Twλ(v) + Twλ(v)), if (λ, v) ∈ X . (4.2.10)
i.e. (Lu, Bλu) is a Lax pair of equation (4.2.9).
Remark 4.2.13. The Toeplitz operators Twλ(v) and Twλ(v) are bounded both on L2+ and on H1
+, so is
the skew-adjoint operator Bλv , if (λ, v) ∈ X .
For every u ∈ H∞(R,R) and ε ∈ (0, 4C2‖u‖2
L2), we set Hε(u) := 1
εH 1ε(u) and Bε,u := 1
εB1εu . Recall that
En(u) = 〈LnuΠu,Πu〉L2 , we have the following Taylor expansion
Hε(u) =
M∑k=0
(−ε)nEn(u)− (−ε)M 〈(Lu + 1ε )−1Πu, LMu Πu〉L2 , ∀M ∈ N. (4.2.11)
Proposition 4.2.12 then leads to a Lax pair formulation for the equations corresponding to the conservationlaws in the BO hierarchy,
∂tLu = [dn
dεn
∣∣∣ε=0
Bε,u, Lu],
where now u evolves according to the pseudo-Hamiltonian flow of En = (−1)n dn
dεn
∣∣ε=0Hε. In the case
n = 1, we have E1 = E and Bu = ddε
∣∣ε=0
Bε,u.
This section is organized as follows. In subsection 4.2.1, we recall some basic facts concerning unitarilyequivalent self-adjoint operators on different Hilbert spaces. The subsection 4.2.2 is dedicated to theproofs of proposition 4.2.2 and 4.2.3. Proposition 4.2.8 and 4.2.10 that concern the conservation laws areproved in subsection 4.2.3. Proposition 4.2.7 and proposition 4.2.12 that indicate the Lax pair structuresare proved in subsection 4.2.4.
4.2. THE LAX OPERATOR 95
4.2.1 Unitary equivalence
Generally, if E1 and E2 are two Hilbert spaces, let A be a self-adjoint operator defined on D(A) ⊂ E1 andB be a self-adjoint operator defined on D(B) ⊂ E2. Both A and B have spectral decompositions
E1 = Hac(A)⊕
Hsc(A)⊕
Hpp(A), E2 = Hac(B)⊕
Hsc(B)⊕
Hpp(B). (4.2.12)
If A and B are unitarily equivalent i.e. there exists a unitary operator U : E1 → E2 such that
B = UAU∗, D(B) = UD(A), (4.2.13)
then we have the following identification result.
Proposition 4.2.14. The operators A and B have the same spectrum and UHxx(A) = Hxx(B), forevery xx ∈ ac, sc,pp. Moreover, for every bounded borel function f : R→ C, f(A) is a bounded operatoron E1, f(B) is a bounded operator on E2, we have f(B) = Uf(A)U∗.
Demonstration. If f is a bounded Borel function, ψ ∈ E1, consider the spectral measure of A associatedto the vector ψ ∈ E1, denoted by µAψ . Similarly, we denote by µBUψ the spectral measure of B associatedto the vector Uψ ∈ E2. Clearly, we have
supp(µAψ ) ⊂ σ(A) ⊂ R, supp(µBUψ) ⊂ σ(B) ⊂ R.
For every λ ∈ C\σ(A) = C\σ(B), formula (4.2.13) implies that U(λ − A)−1U∗ = (λ − B)−1. So theBorel–Cauchy transforms of these two spectral measures are the same.∫
R
dµAψ (ξ)
λ− ξ= 〈(λ−A)−1ψ,ψ〉E1 = 〈(λ− B)−1Uψ,Uψ〉E2 =
∫R
dµBUψ(ξ)
λ− ξ.
Both of these two spectral measures have finite total variations : µAψ (R) = µBUψ(R) = ‖ψ‖2E1 . Sinceevery finite Borel measure is uniquely determined by its Borel–Cauchy transform (see Theorem 3.21 ofTeschl [139] page 108), we have µAψ = µBUψ. So the restriction U|Hxx(A) : Hxx(A) → Hxx(B) is a linearisomorphism, for every xx ∈ ac, sc,pp. Finally, we use the definition of the spectral measures to obtain
〈f(A)ψ,ψ〉E1 =
∫Rf(ξ)dµAψ (ξ) =
∫Rf(ξ)dµBUψ(ξ) = 〈f(B)Uψ,Uψ〉E2
We may assume that f is real-valued, so that f(A) is self-adjoint. The polarization identity implies that〈f(A)ψ, φ〉E1 = 〈f(B)Uψ,Uφ〉E2 , for every ψ, φ ∈ E1. So we obtain f(B) = Uf(A)U∗ in the case f isreal-valued bounded Borel function. In the general case, it suffices to use f = Ref + iImf .
4.2.2 Spectral analysis I
In this subsection, we study the essential spectrum and discrete spectrum of the Lax operator Lu byproving proposition 4.2.2 and 4.2.3. The spectral analysis of Lu such that u is a multi-soliton in definition4.1.1, will be continued in subsection 4.4.2.
Proof of proposition 4.2.2. For every h ∈ L2+, let µD
h denote the spectral measure of D associated to h,then
〈f(D)h, h〉L2 =
∫ +∞
0
f(ξ)|h(ξ)|2
2πdξ =⇒ dµD
h (ξ) =1[0,+∞)(ξ)|h(ξ)|2
2πdξ.
96 CHAPITRE 4. INTEGRABILITY OF THE BO EQUATION ON THE LINE
Thus we have σ(D) = σess(D) = σac(D) = [0,+∞). If u ∈ L2(R,R), we claim that Pu := Tu (D + i)−1
is a Hilbert–Schmidt operator on L2+.
Recall that R∗+ = (0,+∞). In fact, let F : h ∈ L2+ 7→ h√
2π∈ L2(R∗+) denotes the renormalized Fourier–
Plancherel transform, then Au := F Pu F−1 is an operator on L2(R∗+). Then we have
Aug(ξ) =
∫ +∞
0
Ku(ξ, η)g(η)dη, Ku(ξ, η) :=u(ξ − η)
2π(η + i), ∀ξ, η ∈ R∗+.
Hence its Hilbert–Schmidt norm ‖Au‖HS(L2(R∗+)) ≤ ‖K‖L2(R∗+×R∗+) ≤‖u‖L2
2 . Since Pu is unitarily equiva-
lent to Au, we have ‖Pu‖2HS(L2+)
=∑λ∈σ(Pu) λ
2 =∑λ∈σ(Au) λ
2 = ‖Au‖2HS(L2(R∗+)) ≤‖u‖2
L2
4 .
Then the symmetric operator Tu is relatively compact with respect to D and Weyl’s essential spectrumtheorem (Theorem XIII.14 of Reed–Simon [122]) yields that σess(Lu) = σess(D) and Lu is self-adjoint withD(Lu) = D(D) = H1
+. An alternative proof of the self-adjointness of Lu can be given by Kato–Rellichtheorem (Theorem X.12 of Reed–Simon [121]) and the following estimate, for every f ∈ H1
+,
2π‖f‖L∞ ≤ ‖f‖L1 ≤ ‖f‖L2
√A+ ‖∂xf‖L2
√A−1 ≤ 2
(‖f‖L2‖∂xf‖L2
) 12
, A =
√‖∂xf‖L2
‖f‖L2.
So ‖Tu(f)‖L2 ≤ ‖u‖L2‖f‖L∞ ≤ 2π‖∂xf‖L2 +
‖u‖2L2
4 ‖f‖L2 .
Moreover, |〈Tuf, f〉L2 | = |∫R u|f |
2| ≤ ‖u‖L2‖f‖2L4 ≤ C‖u‖L2‖f‖L2‖|D| 12 f‖L2 holds by Sobolev embed-
ding ‖f‖L4 ≤ C‖|D| 14 f‖L2 , for every f ∈ H1+. Then Lu is bounded from below, precisely
〈Luf, f〉L2 = ‖|D| 12 f‖2L2 − 〈Tuf, f〉L2 ≥ −C2‖u‖2
L2‖f‖2L2
4 .
When λ < −C2‖u‖2
L2
4 , the map Lu − λ : H1+ → L2
+ is injective. Hence σpp(Lu) ⊂ [−C2
4 ‖u‖2L2 ,+∞).
Before the proof of proposition 4.2.3, we recall a lemma concerning the regularity of convolutions.
Lemma 4.2.15. For every p ∈ (1,+∞) and m,n ∈ N, we have
Wm,p(R) ∗Wn, pp−1 (R) → Cm+n(R)
⋂Wm+n,+∞(R). (4.2.14)
For every f ∈Wm,p(R) ∗Wn, pp−1 (R), we have lim|x|→+∞ ∂αx f(x) = 0, for every α = 0, 1, · · · ,m+ n.
Demonstration. In the case m = n = 0, it suffices use Holder’s inequality and the density argument ofthe Schwartz class S (R) ⊂ Wm,p(R). In the case m = 0 and n = 1, recall that a continuous functionwhose weak-derivative is continuous is of class C1 and 〈f, ϕ〉D(R)′,D(R) = f ∗ ϕ(0), we use the densityargument of the test function class D(R) ⊂ Lp(R). We conclude by induction on n ≥ 1 and m ∈ N.
Remark 4.2.16. Identity (4.2.1) was firstly found by Wu [146] in the case λ < 0. We show that (4.2.1)still holds in the case λ ≥ 0. Hence the operator Lu has no eigenvalues in [0,+∞).
4.2. THE LAX OPERATOR 97
Proof of proposition 4.2.3. We choose u ∈ L2(R; (1+x2)dx) such that u(R) ⊂ R, λ ∈ R and ϕ ∈ D(Lu) =H1
+ such that Lu(ϕ) = λϕ. Applying the Fourier–Plancherel transform, we obtain
uϕ(ξ)1ξ≥0 = (ξ − λ)ϕ(ξ) =: gλ(ξ). (4.2.15)
Since u ∈ H1(R) and ϕ ∈ L2(R), their convolution uϕ = 12π u ∗ ϕ ∈ C
1(R)⋂C0(R), where C0(R) denotes
the uniform closure of Cc(R) with respect to the L∞(R)-norm, by lemma 4.2.15. Recall R+ = [0,+∞).
We claim that if λ < 0, then ϕ ∈ C1(R+);
if λ ≥ 0, then ϕ ∈ C(R+)⋂C1(R+\λ).
In fact, if λ ≥ 0, we have gλ(λ) = 0. Otherwise, λ would be a singular point of ϕ that prevents ϕ frombeing a L2 function on R+, because ξ → 1
ξ−λ /∈ L2(R+). By using the fact gλ ∈ C1(R+) (gλ is right
differentiable at ξ = 0 and the derivative g′λ is right continuous at ξ = 0), we have
ϕ(ξ) =gλ(ξ)− gλ(λ)
ξ − λ→
g′λ(λ), if λ > 0;
g′λ(0+), if λ = 0;
when ξ → λ. So ϕ ∈ C(R+) and limξ→+∞ ϕ(ξ) = 0. Then we derive formula (4.2.15) with respect to ξ toget the following
− ixu ∗ ϕ(ξ) = g′λ(ξ) = (uϕ)′(ξ) = ϕ(ξ) + (ξ − λ)(ϕ)′(ξ), ∀ξ ∈ [0,+∞)\λ. (4.2.16)
Thus we have
d
dξ[(ξ − λ)|ϕ(ξ)|2] = |ϕ(ξ)|2 + 2Re[((ξ − λ)(ϕ)′(ξ))ϕ(ξ)] = 2Re[(uϕ)′(ξ)ϕ(ξ)]− |ϕ(ξ)|2. (4.2.17)
When λ < 0, it suffices to integrate equation (4.2.17) on [0,+∞) and use the Plancherel formula∫ +∞
0
(uϕ)′(ξ)ϕ(ξ)dξ = −2πi
∫Rxu(x)|ϕ(x)|2dx.
We also use the fact (ξ − λ)|ϕ(ξ)|2 = uϕ(ξ)ϕ(ξ)→ 0, as ξ → +∞. Thus,
λ|ϕ(0)|2 =
∫ +∞
0
d
dξ[(ξ − λ)|ϕ(ξ)|2]dξ = 4πIm
∫Rxu(x)|ϕ(x)|2dx−
∫ +∞
0
|ϕ(ξ)|2dξ = −2π‖ϕ‖2L2(R).
When λ > 0, there may be some problem of derivability of ϕ at ξ = λ. We replace the integral∫ +∞
0by
two integrals∫ λ−ε
0and
∫ +∞λ+ε
, for some ε ∈ (0, λ). Set
I(ε) :=λ|ϕ(0)|2 − ε|ϕ(λ− ε)|2 − ε|ϕ(λ+ ε)|2
=2Re
(∫ +∞
0
(uϕ)′(ξ)ϕ(ξ)dξ −∫ λ+ε
λ−ε(uϕ)′(ξ)ϕ(ξ)dξ
)−∫ +∞
0
|ϕ(ξ)|2dξ +
∫ λ+ε
λ−ε|ϕ(ξ)|2dξ
Thanks to the continuity of ϕ on R+, we have λ|ϕ(0)|2 = limε→0+ I(ε) = −2π‖ϕ‖2L2(R).
When λ = 0, we use the same idea and integrate (4.2.17) over interval [ε,+∞), for some ε > 0. Then
98 CHAPITRE 4. INTEGRABILITY OF THE BO EQUATION ON THE LINE
J (ε) := −ε|ϕ(ε)|2 = 2Re
∫ +∞
ε
(uϕ)′(ξ)ϕ(ξ)dξ −∫ +∞
ε
|ϕ(ξ)|2dξ → 0,
as ε→ 0. So we always have
− 2π‖ϕ‖2L2(R) = λ|ϕ(0)|2, if ϕ ∈ Ker(λ− Lu). (4.2.18)
As a consequence Lu has only negative eigenvalues, if the real-valued function u ∈ L2(R, (1 + x2)dx).Finally we use uϕ(0) = −λϕ(0) to get identity (4.2.1). If λ ∈ σpp(Lu) and ϕ ∈ Ker(λ−Lu)\0, we wantto prove that
ξ 7→ (1 + |ξ|)∂ξϕ(ξ) ∈ L2(0,+∞). (4.2.19)
In fact, since ϕ ∈ H1+ → L∞(R) and u ∈ L2(R, (1+x2)dx), we have uϕ = u∗ϕ
2π ∈ H1(R). Formula (4.2.15)
yields that ξ 7→ (|λ| + ξ)ϕ(ξ) ∈ L2(R) and we have ϕ ∈ L1(R). The hypothesis u ∈ L2(R, x2dx) impliesthat the convolution term xu ∗ ϕ ∈ L2(R). Since λ < 0, we obtain (4.2.19) by using formula (4.2.16).
4.2.3 Conservation laws
Proposition 4.2.8 and 4.2.10 are proved in this subsection. We begin with the following proposition.
Proposition 4.2.17. If u : t ∈ R 7→ u(t) ∈ H2(R,R) denotes the unique solution of the BO equation(4.1.1), then we have
∂tΠu(t) = Bu(t)(Πu(t)) + iL2u(t)(Πu(t)) ∈ L2
+. (4.2.20)
Demonstration. For every u ∈ H2(R,R), Bu is a bounded operator on both L2+ and H1
+, Πu ∈ D(Lu) =
H1+. We have u(−ξ) = u(ξ), u = Πu+Πu and |D|u = DΠu−DΠu. Since DΠu ∈ L2
−, we have Π(ΠuDΠu) =
Π(uDΠu). Thus the following two formulas hold,
Bu(Πu) = i(T|D|u − T 2u)(Πu) = i(Πu)(DΠu)− iΠ(uDΠu)− iT 2
u(Πu) = Πu∂xΠu−Π(u∂xΠu)− iT 2u(Πu),
iL2u(Πu) = iD2Πu− iTu(DΠu)− iD Tu(Πu) + iT 2
u(Πu) = −i∂2xΠu− Tu(∂xΠu)− ∂x[Tu(Πu)] + iT 2
u(Πu).
Then we add them together to get the following
Bu(Πu) + iL2u(Πu) = −i∂2
xΠu− 2Π[Πu∂xΠu+ Πu∂xΠu+ Πu∂xΠu]
Finally we replace u by u(t), where u : t ∈ R 7→ u(t) ∈ H2(R,R) solves equation (4.1.1) to obtain(4.2.20).
Proof of proposition 4.2.8. It suffices to prove (4.2.7) in the case u0 ∈ H∞(R,R). Then we use the densityargument and the continuity of the flow map
u0 ∈ Hs(R) 7→ u ∈ C([−T, T ];Hs(R)) with T > 0, s ≥ 0,
in proposition 4.2.5. We choose u = u(t) ∈ H∞(R,R) =⋂s≥0H
s(R,R), so the functions LnuΠu, ∂tΠuand ∂t(L
nu)Πu = [Bu, L
nu]Πu are in H∞(R,C). Thus
∂tEn(u) = 2Re〈LnuΠu, ∂tΠu〉L2 + 〈∂t(Lnu)Πu,Πu〉L2 .
4.2. THE LAX OPERATOR 99
Since Bu + iL2u is skew-adjoint, we use formula (4.2.20) to get the following
2Re〈LnuΠu, ∂tΠu〉L2 = 〈[Lnu, Bu + iL2u]Πu,Πu〉L2 = 〈[Lnu, Bu]Πu,Πu〉L2 .
Since (Lnu, Bu) is also a Lax pair of the Benjamin–Ono equation (4.1.1), we have
∂tEn(u) = 〈([Lnu, Bu] + ∂t(Lnu))Πu,Πu〉L2 = 0.
In the case n = 1, we assume that u ∈ H1(R,R). Since u = Πu+Πu, |D|u = DΠu−DΠu and∫R(Πu)3 = 0,
we have 〈|D|u, u〉L2 = 2〈DΠu,Πu〉L2 and∫R u
3 = 3∫R(Πu+ Πu)|Πu|2 = 3
∫R u|Πu|
2. In the general case
u ∈ H 12 (R,R), we use the density argument.
Proof of proposition 4.2.10. Let u : t ∈ R 7→ u(t) ∈ H∞(R,R) solve equation (4.1.1). Since σ(−Lu(t)) =σ(−Lu(0)) by proposition 4.2.14, the operator λ+ Lu(t) ∈ B(H1
+, L2+) is invertible and we have
∂tHλ(u) = 2Re〈(Lu + λ)−1Πu, ∂tΠu〉L2 − 〈(Lu + λ)−1∂tLu(Lu + λ)−1Πu,Πu〉L2 . (4.2.21)
Formula (4.2.20) yields that
2Re〈(Lu + λ)−1Πu, ∂tΠu〉L2 = 〈[(Lu + λ)−1, Bu + iL2u]Πu,Πu〉L2 = 〈[(Lu + λ)−1, Bu]Πu,Πu〉L2 ,
〈[(Lu + λ)−1, Bu]Πu,Πu〉L2 = 〈(Lu + λ)−1[Bu, Lu + λ](Lu + λ)−1Πu,Πu〉L2 .
Then ∂tLu = [Bu, Lu] yields that ∂tHλ(u(t)) = 0.
4.2.4 Lax pair formulation
In this subsection, we prove proposition 4.2.12 and 4.2.7. The Hankel operators whose symbols are inL2(R)
⋃L∞(R) will be used to calculate the commutators of Toeplitz operators. We notice that the Hankel
operators are C-anti-linear and the Toeplitz operators are C-linear. For every symbol v ∈ L2(R)⋃L∞(R),
we define its associated Hankel operator to be Hv(h) = Thv = Π(vh), for every h ∈ H1+. If v ∈ L∞(R),
then Hv : L2+ → L2
+ is a bounded operator. If v ∈ L2(R), then Hv may be an unbounded operator onL2
+ whose domain of definition contains H1+. For every b ∈ H1(R), we have ‖Tb(h)‖H1 + ‖Hb(h)‖H1 .
‖b‖H1‖h‖H1 , for every h ∈ H1+, so both Tb and Hb are bounded on L2
+ and on H1+.
Lemma 4.2.18. For every v, w ∈ L2+
⋂L∞(R) and u ∈ L2(R), we have
[Tv, Tw] = −Hv Hw ∈ B(L2+). (4.2.22)
If w ∈ H1+ in addition, then we have Tu(w) ∈ L2
+ and
HTuw = Tw HΠu +Hw Tu = Tu Hw +HΠu Tw ∈ B(H1+, L
2+). (4.2.23)
Demonstration. For every v, w ∈ L2+
⋂L∞(R) and h ∈ L2
+, we have wh = Π(wh) + Π(wh) ∈ L2+. Thus,
[Tv, Tw]h = Π(vΠ(wh)− wΠ(vh)) = Π(vwh− vΠ(wh)− vwh) = −Π(vΠ(wh)) = −Hv Hw(h) ∈ L2+.
Given u ∈ L2(R) and w ∈ H1+, for every h ∈ H1
+, we have wh = Π(wh) + Π(wh) ∈ H1(R) and
Hw(h), Tw(h) ∈ H1+. So Π(uΠ(wh)) = Π(Π(wh)Πu) = HΠu Tw(h) ∈ L2
+ and we have
HTuw(h) = Π(Π(uw)h) = Π(uwh) = Π(uΠ(wh) + uΠ(wh)) = (Tu Hw +HΠu Tw)(h) ∈ L2+.
100 CHAPITRE 4. INTEGRABILITY OF THE BO EQUATION ON THE LINE
Similarly, h ∈ H1+ yields that uh = Π(uh) + Π(uh) ∈ L2(R) and Π(uh) = Π(hΠu) = HΠu(h) ∈ L2
+. Thus,
HTuw(h) = Π(wuh) = Π(wΠ(uh) + wΠ(uh)) = (Tw HΠu +Hw Tu)(h) ∈ L2+.
Lemma 4.2.19. Given (λ, u) ∈ X given in definition 4.2.9, set wλ(u) = (Lu + λ)−1 Π(u) ∈ H1
+, then
[D− Tu, Twλ(u)Twλ(u) + Twλ(u) + Twλ(u)] = TD[|wλ(u)|2+wλ(u)+wλ(u)] ∈ B(H1+, L
2+). (4.2.24)
Demonstration. We use abbreviation wλ := wλ(u) ∈ H1+, then wλ ∈ H1
−. If f+, g+ ∈ H1+ and f−, g− ∈
H1−, then we have [Tf+ , Tg+ ] = [Tf− , Tg− ] = 0, because for every h ∈ L2
+, we have
Tf+ [Tg+(h)] = f+g+h = Tg+ [Tf+(h)], ∀h ∈ L2+.
and Tf− [Tg−(h)] = Π(f−Π(g−h)) = Π(f−g−h) = Π(g−Π(f−h)) = Tg− [Tf−(h)]. Since Πu ∈ L2+ and
Πu ∈ L2−, we use Leibnitz’s rule and formula (4.2.22) to obtain that
[D− Tu, Twλ + Twλ ] =TDwλ + TDwλ − [Tu, Twλ ]− [Tu, Twλ ]
=TDwλ + TDwλ − [TΠu, Twλ ]− [TΠu, Twλ ]
=TDwλ + TDwλ −HwλHΠu +HΠuHwλ .
(4.2.25)
Similarly, formula (4.2.22) implies that
[Tu, TwλTwλ ] =[Tu, Twλ ]Twλ + Twλ [Tu, Twλ ]
=[TΠu, Twλ ]Twλ + Twλ [TΠu, Twλ ]
=HwλHΠuTwλ − TwλHΠuHwλ .
(4.2.26)
For every h ∈ H1+, since wλ,Dwλ ∈ L2
−, we have
[D, TwλTwλ ]h =[D, Twλ ]Twλh+ Twλ [D, Twλ ]h
=TDwλ(Twλh) + Twλ(TDwλh)
=Π[DwλΠ(wλh) + wλΠ(Dwλh)] = Π[(wλDwλ + wλDwλ)h] ∈ L2+.
So [D, TwλTwλ ] = TD|wλ|2 ∈ B(H1+, L
2+). We use formula (4.2.22) and Leibnitz’s Rule to obtain that
[D, TwλTwλ ] = [D, TwλTwλ ]− [D, H2wλ
] = TD|wλ|2 −HDwλHwλ +HwλHDwλ (4.2.27)
Recall that wλ = (λ+ Lu)−1Πu, then we have
Dwλ = Tu(wλ)− λwλ + Πu. (4.2.28)
The formula (4.2.23) and (4.2.28) yield that
HDwλ − TwλHΠu = HTuwλ − λHwλ +HΠu − TwλHΠu = HwλTu − λHwλ +HΠu (4.2.29)
and
HDwλ −HΠuTwλ = HTuwλ − λHwλ +HΠu −HΠuTwλ = TuHwλ − λHwλ +HΠu. (4.2.30)
4.2. THE LAX OPERATOR 101
We use formulas (4.2.26), (4.2.27), (4.2.29) and (4.2.30) to get the following formula
[D− Tu, TwλTwλ ]
=TD|wλ|2 − (HDwλ − TwλHΠu)Hwλ +Hwλ(HDwλ −HΠuTwλ)
=TD|wλ|2 − (HwλTuHwλ − λH2wλ
+HΠuHwλ) + (HwλTuHwλ − λH2wλ
+HwλHΠu)
=TD|wλ|2 −HΠuHwλ +HwλHΠu
(4.2.31)
At last, we combine formulas (4.2.25) and (4.2.31) to obtain formula (4.2.24).
End of the proof of proposition 4.2.12. Since L : u ∈ L2(R,R) 7→ Lu = D− Tu ∈ B(H1+, L
2+) is R-affine,
for every u : t 7→ u(t) ∈ L2(R,R) solves equation (4.2.9), we have
d
dt(L u)(t) = −T∂tu(t) = −iTD(wλ(u(t))wλ(u(t))+wλ(u(t))+wλ(u(t))).
Thus the Lax equation (4.2.10) is equivalent to identity (4.2.24) in lemma 4.2.19.
The proof of proposition 4.2.7 can be found in Gerard–Kappeler [54], Wu [146] etc. In order to make thispaper self contained, we recall it here.
Proof of proposition 4.2.7. Since the Lax map L : u ∈ H2(R,R) 7→ D− Tu ∈ B(H1+, L
2+) is R-affine,
d
dt(L u)(t) = −T∂tu(t) = −TH∂2
xu(t)−∂x(u(t)2).
It suffices to prove [Bu, Lu] + TH∂2xu−∂x(u2) = 0 for every u ∈ H2(R,R).
In fact, u is real-valued, we have u(−ξ) = u(ξ), u = Πu + Πu and |D|u = DΠu − DΠu. Since both Tuand Bu are bounded operators L2
+ → L2+ and bounded operators H1
+ → H1+ , their Lie Bracket [Bu, Lu]
is given by
[Bu, Lu]f =−Π(f∂x|D|u) + iΠ[uΠ(f |D|u)− |D|uΠ(uf)] + Π[∂xuΠ(uf) + uΠ(f∂xu)]
=−Π(fH∂2xu) + I1 + I2 ∈ L2
+,(4.2.32)
for every f ∈ H1+, where the terms I1 and I2 are given by
I1 :=iΠ[uΠ(f |D|u)− |D|uΠ(uf)]
=Π[fΠu∂xΠu+ fΠu∂xΠu]−ΠuΠ(f∂xΠu)−Π(fΠu)∂xΠu+ Π[Π(fΠu)∂xΠu−ΠuΠ(f∂xΠu)],
I2 :=Π[∂xuΠ(uf) + uΠ(f∂xu)] = Π(fΠu)∂xΠu+ ΠuΠ(f∂xΠu) + Π(ΠuΠ(f∂xΠu))
+ 2fΠu∂xΠu+ Π[fΠu∂xΠu+ fΠu∂xΠu+ Π(fΠu)∂xΠu].
If h1 ∈ H1− and h2 ∈ L2
−, then h1h2 ∈ L2−. Since ∂xΠu ∈ L2
−, we have Π[Π(fΠu)∂xΠu] = Π[fΠu∂xΠu].Thus
I1 + I2 = 2fΠu∂xΠu+ 2Π[fΠu∂xΠu+ fΠu∂xΠu+ Π(fΠu)∂xΠu] = Π[f∂x(u2)] ∈ H1+. (4.2.33)
Formulas (4.2.32) and (4.2.33) yield that [Bu, Lu]f = Π[f(∂x(u2)− H∂2xu)]. Thus equation (4.2.3) holds
along the evolution of equation (4.1.1).
102 CHAPITRE 4. INTEGRABILITY OF THE BO EQUATION ON THE LINE
Remark 4.2.20. As indicated in Gerard–Kappeler [54], there are many choices of the operator Bu. Wecan replace Bu by any operator of the form Bu + Pu such that Pu is a skew-adjoint operator commutingwith Lu. For instance, we set Cu := Bu + iL2
u and we obtain Cu = iD2 − 2iDTu + 2iTDΠu. So (Lu, Cu)is also a Lax pair of the BO equation (4.1.1). The advantage of the operator Bu = i(T|D|u − T 2
u) is thatBu : L2
+ → L2+ is bounded if u is sufficiently regular. For instance, u ∈ H2(R,R).
4.3 The action of the shift semigroup
In this section, we introduce the semigroup of shift operators (S(η)∗)η≥0 acting on the Hardy space L2+
and classify all finite-dimensional translation-invariant subspaces of L2+.
For every η ≥ 0, we define the operator S(η) : L2+ → L2
+ such that S(η)f = eηf , where eη(x) = eiηx. Itsadjoint is given by S(η)∗ = Te−η . We have
S(η)∗ Lu S(η) = Lu + ηIdL2+, ∀η ≥ 0.
Since ‖S(η)∗‖B(L2+) = ‖S(η)‖B(L2
+) = 1, (S(η)∗)η≥0 is a contraction semi-group. Let −iG be its infinite-
simal generator, i.e. Gf = i ddη
∣∣∣η=0+
S(η)∗f ∈ L2+, ∀f ∈ D(G), where
D(G) :=f ∈ L2+ : f|R+
∈ H1(0,+∞), (4.3.1)
because limε→0 ‖ψ−τεψε −∂xψ‖L2(0,+∞) = 0, where τεψ(x) = ψ(x−ε) and ψ ∈ H1(0,+∞). Every functionf ∈ D(G) has bounded Holder continuous Fourier transform by Morrey’s inequality and Sobolev extension
operator yields the existence of f(0+) := limξ→0+ f(ξ). The operator G is densely defined and closed.The Fourier transform of Gf is given by
Gf(ξ) = i∂ξ f(ξ), ∀f ∈ D(G), ∀ξ > 0. (4.3.2)
In accordance with the Hille–Yosida theorem, we have
(−∞, 0) ⊂ ρ(iG), ‖(G− λi)−1‖B(L2+) ≤ λ−1, ∀λ > 0. (4.3.3)
Lemma 4.3.1. For every b ∈ L2(R)⋂L∞(R), we have Tb(D(G)) ⊂ D(G) and the following identity
[G,Tb]ϕ =iϕ(0+)
2πΠb (4.3.4)
holds for every ϕ ∈ D(G).
Demonstration. For every η > 0 and ϕ ∈ D(G), both S(η)∗ and Tb are bounded operators, so we have
([S(η)∗
η , Tb]ϕ)∧
(ξ) =b ∗ ϕ(ξ + η)− b ∗ [1R+
(τ−ηϕ)](ξ)
2πη=
1
2πη
∫ ξ+η
ξ
b(ζ)ϕ(ξ + η − ζ)dζ, ∀ξ > 0,
4.3. THE ACTION OF THE SHIFT SEMIGROUP 103
where τ−ηϕ(x) = ϕ(x+ η), for every x ∈ R. Then we change the variable ζ = ξ + tη, for 0 ≤ t ≤ 1,([S(η)∗−Id
L2+
η , Tb]ϕ
)∧(ξ) =
1
2π
∫ 1
0
b(ξ + tη)ϕ((1− t)η)dt = aη b(ξ) + φη(ξ), ∀ξ > 0, (4.3.5)
where aη := 12π
∫ 1
0ϕ((1− t)η)dt ∈ C and φη ∈ L2
+ such that
φη(ξ) :=1
2π
∫ 1
0
[b(ξ + tη)− b(ξ)]ϕ((1− t)η)dt, ∀ξ > 0.
Since ϕ|R+ ∈ H1(0,+∞), ϕ is bounded and limη→0+ ϕ(η) = ϕ(0+), Lebesgue’s dominated convergence
theorem yields that limη→0+ aη = ϕ(0+)2π . Since b ∈ L2(R), we have limε→0 ‖τεb − b‖L2 = 0. By using
Cauchy–Schwarz inequality and Fubini’s theorem, we have
‖φη‖2L2 . ‖ϕ‖2L∞∫ 1
0
∫ +∞
0
|b(ξ + tη)− b(ξ)|2dξdt = ‖ϕ‖2L∞∫ 1
0
‖τ−tη b− b‖2L2dt→ 0,
when η → 0+, by Lebesgue’s dominated convergence theorem. Thus (4.3.5) implies that
[S(η)∗−Id
L2+
η , Tb]ϕ = aηΠb+ φη →ϕ(0+)
2πΠb, in L2
+, when η → 0+.
Since ϕ ∈ D(G) and Tb is bounded, we have 1ηTb[(S(η)∗ − IdL2
+)ϕ]→ (TbG)ϕ in L2
+, consequently
1η (S(η)∗ − IdL2
+)(Tbϕ)→ (TbG)ϕ+
ϕ(0+)
2πin L2
+, when η → 0+.
So Tbϕ ∈ D(G) and (4.3.4) holds.
The following scalar representation theorem of Lax [96] allows to classify all translation-invariant sub-spaces of the Hardy space L2
+, which plays the same role as Beurling’s theorem in the case of Hardy spaceon the circle (see Theorem 17.21 of Rudin [124]).
Theorem 4.3.2 (Beurling–Lax). Every nonempty closed subspace of L2+ that is invariant under the
semigroup of shift operators (S(η))η≥0 is of the form ΘL2+, where Θ is a holomorphic function in the
upper-half plane C+ = z ∈ C : Imz > 0. We have |Θ(z)| ≤ 1, for all z ∈ C+ and |Θ(x)| = 1, ∀x ∈ R.Moreover, Θ is uniquely determined up to multiplication by a complex constant of absolute value 1.
The following lemma classifies all finite-dimensional subspaces that are invariant under the semi-group(S(η)∗)η≥0, which is a weak version of theorem 4.3.2.
Lemma 4.3.3. Let M be a subspace of L2+ of finite dimension N = dimCM ≥ 1 and G(M) ⊂ M .
Then there exists a unique monic polynomial Q ∈ CN [X] such that Q−1(0) ⊂ C− and M =C≤N−1[X]
Q ,
where C≤N−1[X] denotes all the polynomials whose degrees are at most N − 1. Q is the characteristicpolynomial of the operator G|M .
104 CHAPITRE 4. INTEGRABILITY OF THE BO EQUATION ON THE LINE
Demonstration. We set M = f ∈ L2(0,+∞) : f ∈ M, then dimC M = N . Since Gf = i∂ξ f on R\0,the restriction G|M is unitarily equivalent to i∂ξ|M by the renormalized Fourier–Plancherel transforma-
tion. So the characteristic polynomial Q ∈ CN [X] of i∂ξ|M is well defined, let β1, β2, · · · , βn ⊂ Cdenote the distinct roots of Q and mj denote the multiplicity of βj , we have
∑nj=1mj = N and
Q(z) = det(z − i∂ξ|M ) =
n∏j=1
(z − βj)mj = zd +
N−1∑k=0
ckzk, ck ∈ C.
The Cayley–Hamilton theorem implies that Q(i∂ξ) = 0 on the subspace M . If ψ ∈ M ⊂ L2(0,+∞), thenψ is a weak-solution of the following differential equation
i−NQ(−D)ψ = ∂Nξ ψ +
N−1∑k=0
ik−Nck∂kξψ = 0 on (0,+∞), ψ ≡ 0 on (−∞, 0). (4.3.6)
The differential operator Q(−D) is elliptic is on the open interval (0,+∞) in the following sense : thesymbol of the principal part of Q(−D), denoted by aQ : (x, ξ) ∈ (0,+∞)× R 7→ (−ξ)N , does not vanishexcept for ξ = 0. Theorem 8.12 of Rudin [125] yields that ψ is a smooth function. The solution space
Sol(4.3.6) = SpanCfj,l0≤l≤mj−1,1≤j≤n, fj,l(ξ) = ξle−iβjξ1R+. (4.3.7)
has complex dimension∑nj=1mj = N so we have Sol(4.3.6) = M ⊂ L2
+ and Imβj = Re(iβj) > 0 and
Q−1(0) ⊂ C−. At last, we have M = SpanCfj,l0≤l≤mj−1,1≤j≤n =C≤N−1[X]
Q , where
fj,l(x) =l!
2π[(−i)(x− βj)]l+1, ∀x ∈ R. (4.3.8)
The uniqueness is obtained by identifying all the roots.
4.4 The manifold of multi-solitons
This section is dedicated to a geometric description of the multi-soliton subsets in definition 4.1.1. Wegive at first a polynomial characterization then a spectral characterization for the real analytic symplecticmanifold of N -solitons in order to prove the global well-posedness of the BO equation with N -solitonsolutions (4.1.6).
Recall that every N -soliton has the form u(x) =∑Nj=1Rη−1
j(x−xj) =
∑Nj=1
2ηj(x−xj)2+η2
jwith xj ∈ R and
ηj > 0, then we have the following polynomial characterization of the N -solitons.
Proposition 4.4.1. The N -soliton subset UN ⊂ H∞(R,R)⋂L2(R, x2dx) and UN
⋂UM = ∅, for every
M 6= N . Moreover, each of the following three properties implies the others :
(a). u ∈ UN .(b). There exists a unique monic polynomial Qu ∈ CN [X] whose roots are contained in the lower half-
plane C− such that Πu = iQ′uQu
.
(c). There exists Q ∈ CN [X] such that Q−1(0) ⊂ C− and Πu = iQ′
Q .
4.4. THE MANIFOLD OF MULTI-SOLITONS 105
Demonstration. We only prove the uniqueness in (a)⇒ (b). If Πu = iQ′uQu
= iP′
P , then we have(PQu
)′≡ 0
on R. Since P and Qu are monic polynomials, we have P = Qu. The other assertions are consequencesof u = Πu+ Πu.
Definition 4.4.2. For every u ∈ UN , the unique monic polynomial Qu ∈ CN [X] given by proposition4.4.1 is called the characteristic polynomial of u. Its roots are denoted by zj = xj − iηj ∈ C−, for1 ≤ j ≤ N (not necessarily all distinct). The unordered N -uplet cl(z1, z2, · · · , zN ) ∈ CN−/SN is calledthe translation–scaling parameters of u, where CN−/SN denotes the orbit space of the action (4.6.3) ofsymmetric group SN on CN− .
The real analytic structure of UN is given in the next proposition.
Proposition 4.4.3. Equipped with the subspace topology of L2(R,R), the subset UN is a connected, realanalytic, embedded submanifold of the R-Hilbert space L2(R,R) and dimR UN = 2N . For every u ∈ UN ,its translation–scaling parameters are denoted by cl(x1 − iη1, x2 − iη2, · · · , xN − iηN ) for some xj ∈ Rand ηj > 0, then the tangent space to UN at u is given by
Tu(UN ) =
N⊕j=1
(Rfuj⊕
Rguj ), where fuj (x) =2[(x−xj)2−η2
j ]
[(x−xj)2+η2j ]2, guj (x) =
4ηj(x−xj)[(x−xj)2+η2
j ]2. (4.4.1)
Every tangent space Tu(UN ) is contained in the auxiliary space T defined by (4.1.5) in which the global2-covector ω ∈ Λ2(T ∗) is well defined. Recall that the nondegenerate 2-form ω on UN is given by
ωu(h1, h2) = ω(h1, h2) =i
2π
∫R
h1(ξ)h2(ξ)
ξdξ, ∀h1, h2 ∈ Tu(UN ). (4.4.2)
It provides the symplectic structure of the manifold UN .
Proposition 4.4.4. The nondegenerate real analytic 2-form ω is closed on UN . Endowed with thesymplectic form ω, the real analytic manifold (UN , ω) is a symplectic manifold.
For every smooth real-valued function f : UN → R, let Xf ∈ X(UN ) denote its Hamiltonian vector field,defined as follows : for every u ∈ UN and h ∈ Tu(UN ),
df(u)(h) = 〈h,∇uf(u)〉L2 =i
2π
∫R
h(ξ)
ξiξ(∇uf(u))∧(ξ)dξ = ωu(h,Xf (u)).
Then we haveXf (u) = ∂x∇uf(u) ∈ Tu(UN ), ∀u ∈ UN . (4.4.3)
Remark 4.4.5. There are several ways to prove the simple connectedness of UN . Firstly, it is irrelevantto the proof of proposition 4.5.16. In subsection 4.5.4, we show that the real analytic manifold UN isdiffeomorphic to some open convex subset of R2N , hence UN is homotopy equivalent to a one-point space.On the other hand, the simple connectedness of the Kahler manifold Π(UN ) can be directly obtained fromits construction (see proposition 4.6.5).
Then, we return back to spectral analysis in order to establish a spectral characterization of the manifold
UN . For every monic polynomial Q ∈ CN [X] with roots in C−, we set Θ = ΘQ := QQ ∈ Hol(C+), where
Q(x) :=
N−1∑j=0
ajxj + xN , if Q(x) =
N−1∑j=0
ajxj + xN .
106 CHAPITRE 4. INTEGRABILITY OF THE BO EQUATION ON THE LINE
Then Θ is an inner function on the upper half-plane C+, because |Θ| ≤ 1 on C+ and |Θ| = 1 on R.Recall the shift operator S(η) : L2
+ → L2+ defined in section 4.3, we have S(η)[Θh] = Θ[S(η)h], for every
h ∈ L2+, so ΘL2
+ is a closed subspace of L2+ that is invariant by the semigroup (S(η))η≥0 (see also the
Beurling–Lax theorem 4.3.2 of the complete classification of the translation-invariant subspaces of theHardy space L2
+). We define KΘ to be the orthogonal complement of ΘL2+, thus
L2+ = ΘL2
+
⊕KΘ, S(η)∗(KΘ) ⊂ KΘ and G(D(G)
⋂KΘ) ⊂ KΘ. (4.4.4)
where the infinitesimal generator G is defined in (4.3.2). Recall that the C-vector space C≤N−1[X] consists
of all polynomials with complex coefficients of degree at most N − 1. SoC≤N−1[X]
Q is an N -dimensional
subspace of L2+.
The Lax map L : u ∈ L2(R,R) 7→ Lu = D − Tu ∈ B(H1+, L
2+) is R-affine. Defined on D(Lu) = H1
+, theunbounded self-adjoint operator Lu has the following spectral decomposition
L2+ = Hac(Lu)
⊕Hsc(Lu)
⊕Hpp(Lu). (4.4.5)
The following proposition gives an identification of these subspaces in the spectral decomposition (4.4.5).
Proposition 4.4.6. If u ∈ UN , then Lu has exactly N simple negative eigenvalues. Let Qu denote the
characteristic polynomial of the N -soliton u given in definition 4.4.2 and Θu := ΘQu = QuQu
denote theassociated inner function. Then we have the following identification,
Hac(Lu) = ΘuL2+, Hsc(Lu) = 0, Hpp(Lu) = KΘu =
C≤N−1[X]
Qu. (4.4.6)
For every u ∈ UN , we have the following spectral decomposition of Lu :
σ(Lu) = σac(Lu)⋃σsc(Lu)
⋃σpp(Lu), where σac(Lu) = [0,+∞), σsc(Lu) = ∅ (4.4.7)
and σpp(Lu) = λu1 , λu2 , · · · , λuN consists of all eigenvalues of Lu. Proposition 4.2.2 yields that Lu is
bounded from below and −C2
4 ‖u‖2L2 ≤ λu1 < · · · < λuN < 0, where C = inff∈H1
+\0‖|D|
14 f‖L2
‖f‖L4denotes the
Sobolev constant. Hence the min-max principle (Theorem XIII.1 of Reed–Simon [122]) yields that
λun = supdimC F=n−1
I(F,Lu), I(F,Lu) = inf〈Luh, h〉L2 : h ∈ H1+
⋂F⊥, ‖h‖L2 = 1 (4.4.8)
where, the above supremum, F describes all subspaces of L2+ of complex dimension n, 1 ≤ n ≤ N . When
n ≥ N + 1, supdimC F=n I(F,Lu) = inf σess(Lu) = 0. Proposition 4.2.3 and corollary 4.2.4 yield that thereexist eigenfunctions ϕj : u ∈ UN 7→ ϕuj ∈Hpp(Lu) such that
Ker(λuj − Lu) = Cϕuj , ‖ϕuj ‖L2 = 1, 〈ϕuj , u〉L2 =
∫Ruϕuj =
√2π|λuj |, (4.4.9)
for every j = 1, 2, · · · , N . Then ϕu1 , ϕu2 , · · · , ϕuN is an orthonormal basis of the subspace Hpp(Lu). Wehave the following result.
Proposition 4.4.7. For every j = 1, 2, · · · , N , the j th eigenvalue λj : u ∈ UN 7→ λuj ∈ R is realanalytic.
4.4. THE MANIFOLD OF MULTI-SOLITONS 107
We refer to proposition 4.4.14 and formula (4.4.4) to see that the subspace Hpp(Lu) ⊂ D(G) is invariantby G. The matrix representation of G|Hpp(Lu) with respect to the orthonormal basis ϕu1 , ϕu2 , · · · , ϕuN isgiven in proposition 4.5.4. Then the following theorem gives the spectral characterization for N -solitons.
Theorem 4.4.8. A function u ∈ UN if and only if u ∈ L2(R, (1+x2)dx) is real-valued, dimC Hpp(Lu) =N and Πu ∈Hpp(Lu). Moreover, we have the following inversion formula
Πu(x) = id
dx det(x−G|Hpp(Lu))
det(x−G|Hpp(Lu)), ∀x ∈ R. (4.4.10)
Then Qu in definition 4.4.2 is the characteristic polynomial of G|Hpp(Lu). The translation–scaling para-meters of u can be identified as the spectrum of G|Hpp(Lu). Finally the invariance of UN under the BOflow is obtained by its spectral characterization, so we have the global well-posedness of the BO equationin the N -soliton manifold (4.1.6).
Proposition 4.4.9. If u0 ∈ UN , we denote by u : t ∈ R 7→ u(t) ∈ H∞(R,R) the solution of the BOequation (4.1.1) with initial datum u(0) = u0. Then u(t) ∈ UN , for every t ∈ R.
This section is organized as follows. The real analytic structure and the symplectic structure are given insubsection 4.4.1. Then the spectral decomposition of the Lax operator Lu and the real analyticity of itseigenvalues are given in subsection 4.4.2, for every u ∈ UN . The characterization theorem 4.4.8 is provedin subsection 4.4.3. Finally, we show the stability of UN under the BO flow in subsection 4.4.4.
4.4.1 Differential structure
The construction of real analytic structure and symplectic structure of UN is divided into three steps.Firstly, we describe the complex structure of Π(UN ). Then the Hermitian metric H for the complex ma-nifold Π(UN ) is introduced in (4.4.15) and we establish a real analytic diffeomorphism between UN andΠ(UN ). The third step is to prove dω = 0 on UN . Since ω = −Π∗(ImH), (Π(UN ),H) is a Kahler manifold.
Step I. The Viete map V : (β1, β2, · · · , βN ) ∈ CN 7→ (a0, a1, · · · , aN−1) ∈ CN is defined as follows
N∏j=1
(X − βj) =
N−1∑k=0
akXk +XN . (4.4.11)
Both addition and multiplication of two complex numbers are open continuous maps C2 → C, the Vietemap V : CN → CN is an open quotient map. So V(CN− ) is an open connected subset of CN (see also
proposition 4.6.5). With the subspace topology and the Hermitian form HCN (X,Y ) = 〈X,Y 〉CN = XTY ,the subset (V(CN− ),HCN ) is a connected Kahler manifold of complex dimension N .
Lemma 4.4.10. Equipped with the subspace topology of L2+, the subset Π(UN ) is a connected topological
manifold of complex dimension N and it has a unique complex analytic structure making it into anembedded submanifold of the C-Hilbert space L2
+. For every u ∈ UN , its translation–scaling parametersare denoted by cl(x1 − iη1, x2 − iη2, · · · , xN − iηN ), for some xj ∈ R and ηj > 0, then the tangent spaceto Π(UN ) at Πu is given by
TΠu(Π(UN )) =
N⊕j=1
Chuj , where huj (x) =1
(x− xj + ηji)2. (4.4.12)
108 CHAPITRE 4. INTEGRABILITY OF THE BO EQUATION ON THE LINE
Demonstration. We define ΓN : a = (a0, a1, · · · , aN−1) ∈ V(CN− ) 7→ Πu = iQ′
Q ∈ Π(UN ) ⊂ L2+ such that
Q(X) =
N−1∑k=0
akXk +XN .
The surjectivity of ΓN is given by the definition of UN . Since the monic polynomial Q is uniquelydetermined by u ∈ UN , the map ΓN is injective. For every h = (h0, h1, · · · , hN−1) ∈ CN , we have
dΓN (a0, a1, · · · , aN−1)h = iQH ′ −Q′H
Q2, where H(X) =
N−1∑k=0
hkXk.
If dΓN (a0, a1, · · · , aN−1)h = 0, then (HQ )′ ≡ 0. Since degH ≤ degQ − 1, we have H = 0. Thus
ΓN : V(CN− )→ L2+ is a complex analytic immersion. We claim that ΓN is a topological embedding.
In fact we set a(n) = (a(n)0 , a
(n)1 , · · · , a(n)
N−1) ∈ V(CN− ) such that
∂xQnQn
→ ∂xQ
Qin L2
+, as n→ +∞, where Qn(x) =
N−1∑j=0
a(n)j xj + xN , ∀x ∈ R.
Since a(n) ∈ V(CN− ), we have a(n)0 = Qn(0) 6= 0. For every x ∈ R, we have
Qn(x)
Qn(0)= exp(
∫ x
0
∂yQn(y)
Qn(y)dy)→ exp(
∫ x
0
∂yQ(y)
Q(y)dy) =
Q(x)
Q(0), as n→ +∞. (4.4.13)
Every coefficient of QnQn(0) converges to the corresponding coefficient of Q(x)
Q(0) . Since Qn, Q are monic, we
have limn→+∞1
Qn(0) = 1Q(0) and limn→+∞ a(n) = a. Then Γ−1
N : Π(UN ) ⊂ L2+ → V(CN− ) is continuous.
Since ΓN is a complex analytic embedding, with the subspace topology of L2+, there exists a unique
complex analytic structure making Π(UN ) = ΓN V(CN− ) into an embedded complex analytic submanifold
of L2+. The map ΓN : V(CN− ) → Π(UN ) is biholomorphic. Set u(x) =
∑Nj=1
2ηj(x−xj)2+η2
jfor some xj =
xj(u) ∈ R and ηj = ηj(u) > 0. Then every h ∈ TΠu(Π(UN )) is identified as the velocity of the smoothcurve c : t ∈ (−1, 1)→ Π(UN ) such that c(0) = Πu at t = 0. If we choose
c(t, x) =
N∑j=1
i
x− xj(t) + ηj(t)iwhere xj(t) ∈ R, ηj(t) > 0.
Then we have xj(0) = xj , ηj(0) = ηj and
h(x) = ∂t∣∣t=0
c(t, x) =
N∑j=1
η′j(0) + ix′j(0)
(x− xj + ηji)2. (4.4.14)
We have huj = Πfuj = −iΠguj and (huj )∧(ξ) = −2π1ξ≥0ξe−(ixj(u)+ηj(u))ξ. For every h ∈ TΠu(Π(UN )), we
have ξ 7→ ξ−1h(ξ) ∈ L2(R) (see also Hardy’s inequality (4.4.18)).
Step II. Given u ∈ UN , the Hermitian metric HΠu is defined as follows
HΠu(h1, h2) =
∫ +∞
0
h1(ξ)h2(ξ)
πξdξ, ∀h1, h2 ∈ TΠu(Π(UN )). (4.4.15)
4.4. THE MANIFOLD OF MULTI-SOLITONS 109
The sesquilinear form HΠu is positive definite because HΠu(h, h) =∫ +∞
0|h(ξ)|2πξ dξ > 0, if h 6= 0. Hence
the smooth symmetric covariant 2-tensor field ReH is positive definite on Π(UN ), so (Π(UN ),ReH) is aRiemannian manifold of real dimension 2N .
We consider the R-linear isomorphism between the Hilbert spaces
Π : u ∈ L2(R,R) 7→ Πu ∈ L2+, f ∈ L2
+ 7→ 2Ref ∈ L2(R,R).
Then Π 2Re = IdL2+
and 2Re Π = IdL2(R,R) and ‖u‖L2 =√
2‖Πu‖L2 . Then UN = 2Re Π(UN ) is a
real analytic manifold of real dimension 2N . Furthermore we have fuj = 2Rehuj , guj = 2iRehuj and
2Re : TΠu(Π(UN ))→ Tu(UN ) (4.4.16)
is an R-linear isomorphism. Since H is Hermitian, the 2-form ω = −Π∗(ImH) is nondegenerate on UN .
Step III. We set E := L2(R,R)⋂L2(R, x2dx), Ec := u ∈ E :
∫R u = c, for every c ∈ R. Then
UN ⊂ E2πN , Tu(UN ) ⊂ T := E0, ∀u ∈ UN .
The nondegenerate 2-form ω can be extended to a 2-covector of the subspace T . Recall that
ω(h1, h2) =i
2π
∫R
h1(ξ)h2(ξ)
ξdξ, ∀h1, h2 ∈ T . (4.4.17)
If h ∈ T , then we have h(0) = 0 and h ∈ H1(R). Hence the Hardy’s inequality (see Brezis [16], Bahouri–Chemin–Danchin [5] etc.) yields that∫
R
|h(ξ)|2
|ξ|2dξ ≤ 4‖∂ξh‖2L2 =⇒ ξ 7→ h(ξ)
ξ∈ L2(R), (4.4.18)
so the 2-covector ω ∈ Λ2(T ∗) is well defined and ωu(h1, h2) = ω(h1, h2). For every smooth vector fieldX ∈ X(UN ), let Xyω ∈ Ω1(UN ) denote the interior multiplication by X, i.e. (Xyω)(Y ) = ω(X,Y ), forevery Y ∈ X(UN ). We shall prove that dω = 0 on UN by using Cartan’s formula :
LXω = Xy(dω) + d(Xyω). (4.4.19)
Proof of proposition 4.4.4. For any smooth vector field X ∈ X(UN ), let φ denote the smooth maximalflow of X. If t is sufficiently close to 0, then φt : u ∈ UN 7→ φ(t, u) ∈ UN is a local diffeomorphism by thefundamental theorem on flows (see Theorem 9.12 of Lee [98]). For every u ∈ UN , h1, h2 ∈ Tu(UN ), wecompute the Lie derivative of ω with respect to X,
(LXω)u(h1, h2) = limt→0
ωφt(u)(dφt(u)h1,dφt(u)h2)− ωu(h1, h2)
t
= limt→0
ω
(dφt(u)h1 − h1
t,dφt(u)h2
)+ limt→0
ω
(h1,
dφt(u)h2 − h2
t
).
Since limt→0dφt(u)hj−hj
t = dX(u)hj ∈ Tu(UN ), for every j = 1, 2, we have
(LXω)u(h1, h2) = ω(dX(u)h1, h2) + ω(h1,dX(u)h2) = (h1ω(X,h2)) (u)− (h2ω(X,h1)) (u).
110 CHAPITRE 4. INTEGRABILITY OF THE BO EQUATION ON THE LINE
We choose (V, xi) a smooth local chart for UN such that u ∈ V and the tangent vector hk has the
coordinate expression hk =∑2Nj=1 h
(j)k
∂∂xj
∣∣u, for some h
(j)k ∈ R, j = 1, 2 · · · , 2N and k = 1, 2. The tangent
vector hk can be identified as some locally constant vector field Yk ∈ X(UN ) defined by
Yk : v ∈ V 7→2N∑j=1
h(j)k
∂
∂xj
∣∣∣v∈ Tv(UN ), Yk : u 7→ (Yk)u = hk, k = 1, 2.
Then the vector field [Y1, Y2] vanishes in the open subset V . The exterior derivative of the 1-form β = Xyωis computed as dβ(Y1, Y2) = Y1 (β(Y2))− Y2 (β(Y1)) + β([Y1, Y2]). Thus
d(Xyω)u(h1, h2) = h1ωu(Xu, h2)− h2ωu(Xu, h1) + ωu(Xu, [Y1, Y2]u) = (LXω)u(h1, h2).
Then Cartan’s formula (4.4.19) yields that Xy(dω) = 0. Since X ∈ X(UN ) is arbitrary, we have dω = 0.As a consequence, the real analytic 2-form ω : u ∈ UN 7→ ω ∈ Λ2(T ∗) is a symplectic form.
Since ImH = (−2Re)∗ω, where −2Re : Π(UN ) → UN is a real analytic diffeomorphism, the associated2-form ImH is closed. So (Π(UN ),H) is a Kahler manifold. The simple connectedness of Π(UN ) is provedin subsection 4.6.1.
4.4.2 Spectral analysis II
We continue to study the spectrum of the Lax operator Lu introduced in definition 4.2.1. The generalcases u ∈ L2(R,R) and u ∈ L2(R, (1 + x2)dx) have been studied in subsection 4.2.2. We restrict ourstudy to the case u ∈ UN in this subsection. Let Q = Qu denote the characteristic polynomial of u and
Θ := QQ , KΘ = (ΘL2
+)⊥. Since Lu is an unbounded self-adjoint operator of L2+, we have the following
L2+ = ΘL2
+
⊕KΘ = Hac(Lu)
⊕Hsc(Lu)
⊕Hpp(Lu).
We shall at first identify those subspaces by proving proposition 4.4.6 and formula (4.4.7). Then we turnto study the real analyticity of each eigenvalue λj : u ∈ UN 7→ λuj ∈ R.
Proof of proposition 4.4.6. The first step is to prove KΘ =C≤N−1[X]
Q . In fact, for every h ∈ L2+ and
f = PQ ∈
C≤N−1[X]
Q , for some P ∈ C≤N−1[X], we have
〈f,Θh〉L2 =
∫R
P (x)Θ(x)h(x)
Q(x)dx =
∫R
P (x)h(x)
Q(x)dx = 〈P
Q, h〉L2 .
Since Q(x) =∏Nj=1(x− αj) with Im(αj) > 0, the meromorphic function P
Qhas poles in C+, so P
Q∈ L2
−.
Thus 〈f,Θh〉L2 = 〈PQ, h〉L2 = 0. Thus
C≤N−1[X]
Q ⊂ (ΘL2+)⊥ = KΘ.
Conversely, if f ∈ KΘ, then 〈Θ−1f, h〉L2 = 〈f,Θh〉L2 = 0, for every h ∈ L2+. Thus g := Q
Qf ∈ L2
−. It
suffices to prove that P := Qf = Qg ∈ C[X]. In fact,
Qf = Q(i∂ξ)f and supp(f) ⊂ [0,+∞) =⇒ supp(Qf) ⊂ [0,+∞).
4.4. THE MANIFOLD OF MULTI-SOLITONS 111
Similarly, supp((Qg)∧) ⊂ (−∞, 0]. Thus supp(P ) ⊂ 0 and P is a polynomial. Since f = PQ ∈ L
2(R),
we have degP ≤ N − 1. So KΘ ⊂C≤N−1[X]
Q .
The second step is to prove Lu(ΘL2+) ⊂ ΘL2
+. Precisely, we have
Lu(Θh) = ΘDh, ∀h ∈ L2+. (4.4.20)
SinceC≤N−1[X]
Q ⊂ L2+, Θ = Q
Q and DΘΘ = DQ
Q− DQ
Q = iQ′
Q − iQ′
Q= Πu+ Πu = u on R, we have
Lu(Θh) = (D− Tu)(Θh) = ΘDh+ h(
DΘ− iQ′
Q Θ + iQ′
Q
)= ΘDh+ hΘ
(DΘΘ − i
Q′
Q + Q′
Q
)= ΘDh.
Recall that Lu = L∗u, so we have Lu(KΘ) ⊂ KΘ. Since dimCKΘ = N , corollary 4.2.4 yields that theHermitian matrix Lu|KΘ
has exactly N distinct eigenvalues. Hence KΘ ⊂Hpp(Lu).
On the other hand, we set UΘ : L2+ → ΘL2
+ such that UΘh = Θh. Thus ‖UΘ‖B(L2+,ΘL
2+) = 1 and
U−1Θ = U∗Θ : g ∈ ΘL2
+ 7→ Θ−1g ∈ L2+.
So UΘ : L2+ → ΘL2
+ is a unitary operator. UΘ(H1+) = ΘH1
+ = H1+
⋂ΘL2
+. Formula (4.4.20) yields that
U∗ΘLu|ΘL2+UΘ = D, UΘ[D(D)] = ΘH1
+ = H1+
⋂ΘL2
+ = D(Lu|ΘL2+
).
For every bounded Borel function f : R → C, we have f(Lu)UΘ = UΘf(D) by proposition 4.2.14. Wedenote by µψ = µLuψ the spectral measure of Lu associated to ψ ∈ L2
+, then ∀h ∈ L2+, we have∫
Rf(ξ)dµΘh(ξ) = 〈f(Lu)UΘh, UΘh〉L2 = 〈Θf(D)h,Θh〉L2 = 〈f(D)h, h〉L2 =
1
2π
∫ +∞
0
f(ξ)|h(ξ)|2dξ.
So dµΘh(ξ) =1R+|h(ξ)|2
2π dξ. The spectral measure µΘh is absolutely continuous with respect to the Le-besgue measure on R. Thus ΘL2
+ ⊂Hac(Lu) ⊂Hcont(Lu) = (Hpp(Lu))⊥ ⊂ ΘL2+ and (4.4.6) is obtained.
We have supp(µΘh) ⊂ [0,+∞), for every h ∈ L2+. ∀ξ ≥ 0, there exists h ∈ L2
+ such that h(ξ) 6= 0. So wehave σess(Lu) = σcont(Lu) = σac(Lu) = [0,+∞).
Before proving the real analyticity of each eigenvalue, we show its continuity at first.
Lemma 4.4.11. For every j = 1, 2, · · · , N , the j th eigenvalue λj : u ∈ UN 7→ λuj ∈ R is Lipschitzcontinuous on every compact subset of UN .
Demonstration. For every f ∈ H1(R), the Sobolev embedding ‖f‖L4 ≤ C‖|D| 14 f‖L2 yields that ∀u, v ∈UN , ∣∣〈Luh, h〉L2 − 〈Lvh, h〉L2
∣∣ ≤ ‖u− v‖L2‖h‖2L4 ≤ C‖u− v‖L2‖|D| 12h‖L2‖h‖L2 , ∀h ∈ H1+. (4.4.21)
Given j = 1, 2, · · · , N and a subspace F ⊂ L2+ with complex dimension j − 1, we choose
h ∈ F⊥⋂ j⊕
k=1
Ker(λuk − Lu) ⊂ H1+, ‖h‖L2 = 1, h =
j∑k=1
hkϕuk .
112 CHAPITRE 4. INTEGRABILITY OF THE BO EQUATION ON THE LINE
Then 〈Luh, h〉L2 =∑jk=1 |hk|2λuk ≤ λuj < 0, because λuk < λuk+1. We have the following estimate
‖D| 12h‖2L2 = 〈Dh, h〉L2 = 〈Luh, h〉L2 + 〈uh, h〉L2 ≤ λuj +‖u‖L2‖h‖2L4 ≤ C‖u‖L2‖|D| 12h‖L2‖h‖L2 . (4.4.22)
So estimates (4.4.21) and (4.4.22) yield that 〈Lvh, h〉L2 ≤ λuj + C2‖u‖L2‖u− v‖L2 . Since F is arbitrary,the max–min formula (4.4.8) implies that
|λuj − λvj | ≤ C2(‖u‖L2 + ‖v‖L2)‖u− v‖L2 .
Every compact subset K ⊂ UN is bounded in L2(R,R). Hence u ∈ K 7→ λuj ∈ R is Lipschitz continuous.
Proof of proposition 4.4.7. For every u ∈ UN , the Lax operator Lu has N negative simple eigenvalues,denoted by λu1 < λu2 < · · · < λuN < 0. Let Pju denotes the Riesz projector of the eigenvalue λuj and
D(z, ε) = η ∈ C : |η − z| < ε, C (z, ε) = ∂D(z, ε) = η ∈ C : |η − z| = ε, ∀z ∈ C, ε > 0.
Then there exists ε0 > 0 such that the family of closed discs D(λuj , ε0)1≤j≤N⋃D(0, ε0) is mutually
disjoint and for every j, k = 1, 2 · · · , N and any closed path Γuj (piecewise C1 closed curve) in D(λuj , ε0)with respect to which the eigenvalue λuj has winding number 1, we have
Pju =1
2πi
∮Γuj
(ζ − Lu)−1dζ, Pju Pju = Pju, Pjuϕuk = 1j=kϕuk . (4.4.23)
by Theorem XII.5 of Reed–Simon [122]. We choose Γuj to be the counterclockwise-oriented circle C (λuj , ε)
in (4.4.23) for some ε ∈ (0, ε0). We claim that ImPju = Ker(λuj − Lu) = Cϕuj .
It suffices to show that Pju|Hac(Lu) = 0. In fact the operator Pju = gλuj (Lu) is self-adjoint by Theorem
VIII.6 of Reed–Simon [123], where the real-valued bounded Borel function gλ : R→ R is given by
gλ(x) :=1
2πi
∮C (λ,ε)
(ζ − x)−1dζ = 1(λ−ε,λ+ε)(x), a.e. on R,
for every λ ∈ R. Since Pju(Hpp(Lu)) ⊂ Cϕuj ⊂Hpp(Lu), we have Pju(Hac(Lu)) ⊂Hac(Lu). Let µψ = µLuψdenote the spectral measure of Lu associated to the function ψ ∈Hac(Lu), whose support is included in[0,+∞) by formula (4.4.7), we have
〈Pjuψ,ψ〉L2 =1
2πi
∮C (λuj ,ε)
〈(ζ − Lu)−1ψ,ψ〉L2dζ =1
2πi
∫ +∞
0
(∮C (λuj ,ε)
(ζ − ξ)−1dζ
)dµψ(ξ) = 0.
Set ψ = Pjuψ ∈Hac(Lu), then ‖ψ‖2L2 = 〈Pjuψ, ψ〉L2 = 0. So the claim is obtained.
For every fixed j = 1, 2, · · ·N , we have λuj = Tr(Lu Pju). Since every eigenvalue λk : v ∈ UN 7→ λvk ∈ R iscontinuous, there exists an open subset V ⊂ UN containing u such that supv∈V sup1≤k≤N |λvk − λuk | <
ε03 .
We set ε = 2ε03 , then λvj ∈ D(λuj , ε)\D(λuk , ε0), for every v ∈ V and k 6= j. For example, in the next picture,
the dashed circles denote respectively C (λuj , ε0) and C (λuk , ε0) ; the smaller circles denote respectivelyC (λuj , ε) and C (λuk , ε) with j < k. The segments inside small circles denote the possible positions of λvjand λvk.
4.4. THE MANIFOLD OF MULTI-SOLITONS 113
λuj
λvj λvk
λuk
0
Then σ(Lv)⋂D(λuj , ε0) = λvj and C (λuj , ε) is a closed path in D(λuj , ε0) with respect to which λvj has
winding number 1. Thus,
Pjv =1
2πi
∮C (λuj ,ε)
(ζ − Lv)−1dζ, λvj = Tr(Lv Pjv), ∀v ∈ V. (4.4.24)
Since v ∈ V 7→ Lv ∈ B(H1+, L
2+) is R-affine and i : A ∈ BI(H1
+, L2+) 7→ A−1 ∈ B(L2
+, H1+) is complex
analytic, where BI(H1+, L
2+) ⊂ B(H1
+, L2+) denotes the open subset of all bijective bounded C-linear
transformations H1+ → L2
+, we have the real analyticity of the following map
(ζ, v) ∈(D(λuj ,
3
4ε0)\D(λuj ,
1
2ε0)
)× V 7→ (ζ − Lv)−1 ∈ B(L2
+, H1+). (4.4.25)
Hence the maps Pj : v ∈ V 7→ Pjv ∈ B(L2+, H
1+) and λj : v ∈ V 7→ Tr(Lv Pjv) ∈ R are both real analytic
by composing (4.4.24) and (4.4.25).
Recall that Hpp(Lu) =C≤N−1[X]
Qu, where Qu denotes the characteristic polynomial of u ∈ UN whose zeros
are contained in C−, so Hpp(Lu) ⊂ D(G) is given by (4.3.7). We have the following consequence.
Corollary 4.4.12. For every j = 1, 2, · · · , N , the map fj : u ∈ UN 7→ 〈Gϕuj , ϕuj 〉L2 ∈ C is real analytic.
Demonstration. For every u, v ∈ UN , we have Pjvϕuj = 〈ϕuj , ϕvj 〉L2ϕvj . Since the Riesz projector Pj : v ∈UN 7→ Pjv ∈ B(L2
+, H1+) is real analytic in the proof of proposition 4.4.7 and ‖Pjuϕuj ‖L2 = 1, there exists
a neighbourhood of u, denoted by V, such that ‖Pjvϕuj ‖L2 > 12 for every v ∈ V and Pj : v ∈ V 7→ Pjv ∈
B(L2+, H
1+) can be expressed by power series. Then
ϕvj =Pjvϕuj
〈ϕuj , ϕvj 〉L2
, fj(v) =〈G Pjv(ϕuj ),Pjv(ϕuj )〉L2
‖Pjv(ϕuj )‖2L2
.
Hence the restriction fj : v ∈ V 7→ ‖Pjv(ϕuj )‖−2L2 〈G Pjv(ϕuj ),Pjv(ϕuj )〉L2 ∈ C is real analytic.
114 CHAPITRE 4. INTEGRABILITY OF THE BO EQUATION ON THE LINE
4.4.3 Characterization theorem
The characterization theorem 4.4.8 is proved in this subsection. The direct sense is given by proposition4.4.1 and proposition 4.4.6. Before proving the converse sense of theorem 4.4.8, we need the followinglemmas to prove the invariance of Hpp(Lu) under G, if u ∈ L2(R, (1+x2)dx) is real-valued, Πu ∈Hpp(Lu)and dimC Hpp(Lu) = N ≥ 1. The following lemma gives another version of formula of commutators (seealso lemma 4.3.1).
Lemma 4.4.13. For u ∈ L2(R, (1 + x2)dx), ϕ ∈ Ker(λ − Lu) for some λ ∈ σpp(Lu), then we haveϕ, Tuϕ,Luϕ ∈ D(G) and
[G,Tu]ϕ =iϕ(0+)
2πΠu, [G,Lu]ϕ = iϕ− iϕ(0+)
2πΠu. (4.4.26)
where Θ = Θu = QuQu
with Qu the characteristic polynomial of u.
Demonstration. In proposition 4.2.3, we have shown that uϕ ∈ H1(R), so (Tuϕ)∧ = uϕ1R+∈ H1(0,+∞)
and Tuϕ ∈ D(G). We recall the regularity of eigenfunctions (4.2.2)
Ker(λ− Lu) ⊂ ϕ ∈ H1+ : ϕ|R+
∈ C1(R+)⋂H1(R+) and ξ 7→ ξ[ϕ(ξ) + ∂ξϕ(ξ)] ∈ L2(R+). (4.4.27)
So Gϕ ∈ H1+ = D(Lu) = D(Tu). Moreover, we have ϕ is right-continuous at ξ = 0+ and ϕ ∈ C1(0,+∞).
The weak-derivative of ϕ is denoted by ∂wξ ϕ, δ0 denotes the Dirac measure with support 0, then
∂wξ ϕ = 1R∗+d
dξϕ+ ϕ(0+)δ0, ∂ξ(u ∗ ϕ) = ∂wξ (u ∗ ϕ) = u ∗ ∂wξ ϕ (4.4.28)
by lemma 4.2.15. Since ϕ = 1R∗+ ϕ a.e. in R and u ∈ H1(R), we have u ∗ Gϕ(ξ) = u ∗ [1R∗+Gϕ](ξ), for
every ξ > 0 and ([G,Tu]ϕ)∧(ξ) = i2π∂ξ(u ∗ ϕ)(ξ) − i
2π u ∗ [1R∗+ddξ f ](ξ) = i
2π ϕ(0+)u(ξ). Together with
(4.5.9), the first formula of (4.4.26) is obtained. Since Lu = D − Tu, we claim that Dϕ ∈ D(G). In fact,
∂ξ(Dϕ)∧(ξ) = ϕ(ξ) + ξ∂ξϕ(ξ), ∀ξ > 0. Thus (4.4.27) implies that Dϕ ∈ H1(0,+∞). Then
([G,D]ϕ)∧
(ξ) = i∂ξ(ξϕ)(ξ)− ξ · i∂ξϕ(ξ) = iϕ(ξ), ∀ξ > 0. (4.4.29)
So we have [∂x, G] = IdL2+
. The second formula of (4.4.26) holds.
Proposition 4.4.14. If u ∈ L2(R, (1 + x2)dx) is real-valued, dimC Hpp(Lu) = N ≥ 1 and Πu ∈Hpp(Lu), then we have Hpp(Lu) ⊂ D(G) and G(Hpp(Lu)) ⊂Hpp(Lu).
Demonstration. There exists an orthonormal basis of Hpp(Lu), denoted by ψ1, ψ2, · · · , ψN, such that
Luψj = λjψj , where σpp(Lu) = λ1, λ2, · · · , λN ⊂ (−∞, 0), λj < λj+1.
Since (4.4.27) implies that Hpp(Lu) ⊂ G−1(H1+)⋂
D(G), formula (4.4.26) gives that
fj := [Lu, G]ψj = −iψj +iψj(0
+)
2πΠu ∈Hpp(Lu), ∀j = 1, 2, · · · , N.
So we have 〈fj , ψj〉L2 = 〈Gψj , Luψj〉L2 − 〈GLuψj , ψj〉L2 = λ(〈Gψj , ψj〉L2 − 〈Gψj , ψj〉L2) = 0.
4.4. THE MANIFOLD OF MULTI-SOLITONS 115
For every j = 1, 2, · · · , N , we set gj :=∑
1≤k≤N,k 6=j〈fj ,ψk〉L2
λk−λj ψk. Since fj =∑
1≤k≤N,k 6=j〈fj , ψk〉L2ψk,
we have (Lu − λj)gj = fj = (Lu − λj)Gψj . Then Gψj − gj ∈ Ker(Lu − λj) = Cψj and
Gψj ∈ gj + Cψj ⊂Hpp(Lu).
We conclude by Hpp(Lu) = SpanCψ1, ψ2, · · · , ψN. (see also formulas (4.4.4) and (4.4.6))
Now, we perform the proof of converse sense of theorem 4.4.8 give the explicit formula of Qu.
End of the proof of theorem 4.4.8. ⇐ : Proposition 4.4.14 yields that G(Hpp(Lu)) ⊂ Hpp(Lu). Let Q
denote the characteristic polynomial of the operator G|Hpp(Lu), then we have Hpp(Lu) =C≤N−1[X]
Q by
lemma 4.3.3. So Πu = P0
Q , for some P0 ∈ C[X] such that deg P0 ≤ N − 1. It remains to show that
P0 = iQ′. Since Hpp(Lu) is invariant under Lu, for every P ∈ C≤N−1[X], we have
Lu(P
Q) = (D− TP0
Q− TP0
Q
)(P
Q) =
DP
Q−Π(
P0P
QQ) +
(iQ′ − P0)P
Q2∈ C≤N−1[X]
Q.
Partial-fraction decomposition implies that Π(P0PQQ
) ∈ C≤N−1[X]
Q . So (iQ′−P0)PQ ∈ C≤N−1[X] for every
P ∈ C≤N−1[X]. Choose P = 1, since deg(iQ′ − P0) ≤ N − 1, we have P0 = iQ′, so u ∈ UN . SinceQ ∈ CN [X] is monic and Q−1(0) ⊂ C−, we have Qu(x) = Q(x) = det(x−G|Hpp(Lu)).
4.4.4 The stability under the Benjamin–Ono flow
Finally we prove proposition 4.4.9 in this subsection. Two lemmas will be proved at first in order toobtain the invariance of the property x 7→ xu(x) ∈ L2(R) under the BO flow.
Lemma 4.4.15. If u0 ∈ H2(R,R)⋂L2(R, x2dx), let u = u(t, x) solves the BO equation (4.1.1) with
initial datum u(0) = u0, then u(t) ∈ L2(R, x2dx), for every t ∈ R.
Remark 4.4.16. This result can be strengthened by replacing the assumption u0 ∈ H2(R,R) by a weaker
assumption u0 ∈ H32 +(R,R) =
⋃s> 3
2Hs(R,R), because one can construct the conservation law of BO
equation controlling the Hs-norm for every s > − 12 by using the method of perturbation of determinants.
We refer to Talbut [136] to see details and Killip–Visan–Zhang [90] for the KdV and the NLS cases(see also Koch–Tataru [91]). Lemma 4.4.15 is a special case of the result in Fonseca–Ponce [37] andFonseca–Linares–Ponce [38].
It suffices to use lemma 4.4.15 to prove proposition 4.4.9. Before proving lemma 4.4.15, we need somecommutator estimates used in Gerard–Lenzmann–Pocovnicu–Raphael [56], we recall it here.
Lemma 4.4.17. For a general locally Lipschitz function χ : R → R such that ∂xχ, ∂3xχ, ∂
5xχ ∈ L1(R),
then we have the following commutator estimates
‖[|D|, χ]g‖L2 + ‖[∂x, χ]g‖L2 . (‖∂xχ‖L1‖∂3xχ‖L1)
12 ‖g‖L2 , ∀g ∈ L2(R),
‖|D|[∂x, χ]g‖L2 . (‖∂xχ‖L1‖∂3xχ‖L1)
12 ‖∂xg‖L2 + (‖∂xχ‖L1‖∂5
xχ‖L1)12 ‖g‖L2 , ∀g ∈ H1(R).
(4.4.30)
116 CHAPITRE 4. INTEGRABILITY OF THE BO EQUATION ON THE LINE
Demonstration. We use∣∣|ξ| − |η|∣∣ ≤ |ξ − η| to estimate the Fourier modes of [|D|, χ]g.
2π∣∣∣ ([|D|, χ]g)
∧(ξ)∣∣∣ ≤ ∫
η∈R
∣∣|ξ| − |η|∣∣|χ(ξ − η)||g(η)|dη ≤∫η∈R|ξ − η||χ(ξ − η)||g(η)|dη = |∂xχ| ∗ |g|(ξ).
Then Young’s convolution inequality yields that ‖[|D|, χ]g‖L2 . ‖∂xχ| ∗ |g|‖L2 . ‖∂xχ‖L1‖g‖L2 . In order
to estimate ‖∂xχ‖L1 , we divide the integral as two parts. Wet set R1 = ‖∂xχ‖− 1
2
L1 ‖∂3xχ‖
12
L1 , so
‖∂xχ‖L1 ≤ ‖∂xχ‖L∞∫|ξ|≤R1
dξ +
∫|ξ|>R1
‖∂3xχ‖L∞|ξ|2
dξ . ‖∂xχ‖L1R1 +‖∂3xχ‖L1
R1= (‖∂xχ‖L1‖∂3
xχ‖L1)12 .
Similarly, we have ‖[∂x, χ]g‖L2 . ‖∂xχ‖L1‖g‖L2 . (‖∂xχ‖L1‖∂3xχ‖L1)
12 . Thus (4.4.30) is obtained.
2π∣∣∣ (|D|[∂x, χ]g)
∧(ξ)∣∣∣ ≤|ξ|∫
η∈R|ξ − η||χ(ξ − η)||g(η)|dη
≤∫η∈R|ξ − η|2
∣∣|χ(ξ − η)||g(η)|dη +
∫η∈R|ξ − η||χ(ξ − η)||η||g(η)|dη
=|∂2xχ| ∗ |g|(ξ) + |∂xχ| ∗ |∂xg|(ξ)
So we have ‖|D|[∂x, χ]g‖L2 . ‖|∂2xχ| ∗ |g|‖L2 +‖|∂xχ| ∗ |∂xg|‖L2 . ‖∂2
xχ‖L1‖g‖L2 +‖∂xχ‖L1‖∂xg‖L2 . Then
we use the same idea to estimate ‖∂2xχ‖L1 , we set R2 := ‖∂xχ‖
− 14
L1 ‖∂5xχ‖
14
L1 . Thus,
‖∂2xχ‖L1 ≤ ‖∂xχ‖L∞
∫|ξ|≤R1
|ξ|dξ+
∫|ξ|>R1
‖∂5xχ‖L∞|ξ|3
dξ . ‖∂xχ‖L1R22 +‖∂5xχ‖L1
R22
= (‖∂xχ‖L1‖∂5xχ‖L1)
12 .
Finally, we add them together to get the second estimate in (4.4.30).
Now we prove the invariance of the property x 7→ xu(x) ∈ L2(R) is invariant under the BO flow.
Proof of lemma 4.4.15. We choose a cut-off function χ ∈ C∞c (R) such that χ decreases in [0,+∞), χ iseven and
0 ≤ χ ≤ 1, χ ≡ 1 on [−1, 1], supp(χ) ⊂ [−2, 2]. (4.4.31)
If u0 ∈ H2(R)⋂L2(R, x2dx), we claim that there exists a constant C = C(‖u(0)‖H1) such that
I(R, t) :=
∫Rχ2( xR )|x|2|u(t, x)|2dx ≤ Ce|t|(
∫R|x|2|u(0, x)|2dx+ 1), ∀t ∈ R, ∀R > 1, (4.4.32)
if u solves the BO equation ∂tu = H∂2xu− ∂x(u2) = |D|∂xu− 2u∂xu.
In fact, we define ρ(x) := xχ(x). For every R > 0, we set ρR(x) := Rρ( xR ) = xχ( xR ). Thus
∂tI(R, t) = 2Re〈ρ2R∂tu(t), u(t)〉L2 = 2Re〈ρ2
R|D|∂xu(t)− 2ρ2Ru(t)∂xu(t), u(t)〉L2 = J1(u(t)) + J2(u(t)),
where for every u ∈ H2(R), we define
J1(u) := −4Re〈ρ2Ru∂xu, u〉L2 =⇒ |J1(u)| ≤ 4‖∂xu‖L∞‖ρRu‖2L2 . ‖u‖H2‖ρRu‖2L2 (4.4.33)
4.4. THE MANIFOLD OF MULTI-SOLITONS 117
andJ2(u) := 2Re〈ρ2
R|D|∂xu, u〉L2 = 〈[ρ2R, |D|∂x]u, u〉L2 ,
because |D|∂x = −(|D|∂x)∗ is an unbounded skew-adjoint operator on L2(R), whose domain of definitionis H2(R), u 7→ ρRu is a bounded self-adjoint operator on Hs(R), for every s ≥ 0. Since
[ρ2R, |D|∂x] = ρR[ρR, |D|∂x] + [ρR, |D|∂x]ρR, [ρR, |D|∂x] = [ρR, |D|∂x]∗ = [ρR, |D|]∂x + |D|[ρR, ∂x],
we have
J2(u) =〈ρR[ρR, |D|∂x]u+ [ρR, |D|∂x]ρRu, u〉L2
=2Re〈[ρR, |D|∂x]u, ρRu〉L2
=2Re〈[ρR, |D|]∂xu, ρRu〉L2 + 2Re〈|D|[ρR, ∂x]u, ρRu〉L2 .
(4.4.34)
Since ‖∂xρR‖L1 = R‖∂xρ‖L1 , ‖∂3xρR‖L1 = R−1‖∂xρ‖L1 and ‖∂5
xρR‖L1 = R−3‖∂xρ‖L1 , the commutatorestimates (4.4.30) yield that if u ∈ H2(R), then
|J2(u)| ≤2‖ρRu‖2L2 + ‖[ρR, |D|]∂xu‖2L2 + ‖|D|[ρR, ∂x]u‖2L2
.‖ρRu‖2L2 + ‖∂xρR‖L1‖∂3xρR‖L1‖∂xu‖2L2 + ‖∂xρR‖L1‖∂5
xρR‖L1‖u‖2L2
.‖ρRu‖2L2 + ‖∂xρ‖L1‖∂3xρ‖L1‖∂xu‖2L2 +R−2‖∂xρ‖L1‖∂5
xρ‖L1‖u‖2L2
.‖ρRu‖2L2 + ‖u‖2H1
(4.4.35)
for every R ≥ 1. Proposition 4.2.6 and 4.2.8 yield that there exists a conservation law of (4.1.1) controllingH2-norm of the solution. Let u : t ∈ R 7→ u(t) ∈ H2(R) denote the solution of the BO equation (4.1.1).Then supt∈R ‖u(t)‖H2 .‖u0‖H2
1. Since I(R, t) = ‖ρRu(t)‖2L2 , estimates (4.4.33) and (4.4.35) imply that
|∂tI(R, t)| ≤ C(I(R, t) + 1), t ∈ R,
for some constant C = C(‖u0‖H2). Thus (4.4.32) is obtained by Gronwall’s inequality. Let R → +∞, weconclude by using Lebesgue’s monotone convergence theorem.
Since the generating function λ ∈ C\σ(−Lu) 7→ Hλ(u) ∈ C is the Borel–Cauchy transform of the spectralmeasure of Lu, the invariance of the N−soliton manifold UN under BO flow is obtained by using theinverse spectral transform.
End of the proof of proposition 4.4.9. If u0 ∈ UN ⊂ H∞(R,R)⋂L2(R, x2dx), let u = u(t, x) be the
unique solution of the BO equation (4.1.1) with initial datum u(0) = u0, then u(t) ∈ H∞(R,R)⋂L2(R, x2dx)
by proposition 4.2.5 and lemma 4.4.15. Recall the generating function Hλ : u ∈ L2(R,R)→ R defined as
Hλ(u) = 〈(λ+ Lu)−1Πu,Πu〉L2 =
∫R
dmu(ξ)
ξ + λ, mu := µLuΠu, ∀λ ∈ C\σ(−Lu), (4.4.36)
where µLuψ denotes the spectral measure of Lu associated to the function ψ ∈ L2+. So the holomorphic
function λ ∈ C\σ(−Lu) 7→ Hλu is the Borel–Cauchy transform of the positive Borel measure mu. Werecall that the total variation mu(R) = ‖Πu‖2L2 is a conservation law of the BO equation (4.1.1) byproposition 4.2.8 and formula (4.2.20). Every finite Borel measure is uniquely determined by its Borel–Cauchy transform (see Theorem 3.21 of Teschl [139] page 108), precisely for every a ≤ b real numbers,we use Stieltjes inversion formula to obtain that
1
2mu((a, b)) +
1
2mu([a, b]) = − 1
πlimε→0+
∫ b
a
ImHx+iε(u)dx.
118 CHAPITRE 4. INTEGRABILITY OF THE BO EQUATION ON THE LINE
For every t ∈ R, proposition 4.2.10 yields that Hλ[u(t)] = Hλ[u(0)], ∀λ ∈ C\σ(−Lu(0)) = C\σ(−Lu(t)).Since u(0) ∈ UN , we have Π[u(0)] ∈Hpp(Lu(0)) by proposition 4.4.6 and there exist c1, c2, · · · , cN ∈ R\0such that
µLu(t)
Π[u(t)] = mu(t) = mu(0) = µLu(0)
Π[u(0)] =
N∑j=1
cjδλu(0)j
.
The spectral measure µLu(t)
Π[u(t)] is purely point, so Π[u(t)] ∈Hpp(Lu(t)) for every t ∈ R. The Lax pair struc-
ture yields the unitary equivalence between Lu(t) and Lu(0). So dimC Hpp(Lu(t)) = dimC Hpp(Lu(0)) = Nis given by proposition 4.2.14. We conclude by theorem 4.4.8.
4.5 The generalized action–angle coordinates
In this section, we construct the (generalized) action–angle coordinates ΦN in theorem 1 of the BOequation (4.1.6) with solutions in the real analytic symplectic manifold (UN , ω) of real dimension 2Ngiven in proposition 4.4.3. The goal of this section is to establish the diffeomorphism property and thesymplectomorphism property of ΦN .
Recall that the BO equation with N -soliton solutions is identified as a globally well-posed Hamiltoniansystem reading as
∂tu(t) = XE(u(t)), u(t) ∈ UN , (4.5.1)
whose energy functional E(u) = 〈LuΠu,Πu〉L2 is well defined on UN and the Hamiltonian vector fieldXE : u ∈ UN 7→ XE(u) = ∂x(|D|u − u2) ∈ Tu(UN ) coincides with the definition (4.4.3). The Poissonbracket of two smooth functions f, g : UN → R is given by
f, g : u ∈ UN 7→ ωu(Xf (u), Xg(u)) = 〈∂x∇uf(u),∇ug(u)〉L2 ∈ R. (4.5.2)
Given u ∈ UN , proposition 4.4.6 yields that there exist λu1 < λu2 < · · · < λuN < 0 and ϕuj ∈ Ker(λuj −Lu) ⊂D(G) such that ‖ϕuj ‖L2 = 1 and 〈u, ϕuj 〉L2 =
√2π|λuj |, thanks to the spectral analysis in subsection 4.4.2.
Definition 4.5.1. For every j = 1, 2, · · · , N , the map Ij : u ∈ UN 7→ 2πλuj ∈ R is called the j th action.The map γj : u ∈ UN 7→ Re〈Gϕuj , ϕuj 〉L2 ∈ R is called the j th (generalized) angle.
Set ΩN := (r1, r2, · · · , rN ) ∈ RN : r1 < r2 < · · · < rN < 0 ⊂ RN , the canonical symplectic form on
R2N = (r1, r2, · · · , rN ;α1, α2, · · · , αN ) : ∀rj , αj ∈ R is given by ν =∑Nj=1 drj ∧ dαj . Endowed with
the subspace topology and the embedded real analytic structure of R2N , the submanifold (ΩN × RN , ν)is a symplectic manifold of real dimension 2N . The action–angle map is defined by
ΦN : u ∈ UN 7→ (I1(u), I2(u), · · · , IN (u); γ1(u), γ2(u), · · · , γN (u)) ∈ ΩN × RN . (4.5.3)
Theorem 1 is restated here.
Theorem 4.5.2. The map ΦN has following properties :
(a). The map ΦN : UN → ΩN × RN is a real analytic diffeomorphism.(b). The pullback of ν by ΦN is ω, i.e. Φ∗Nν = ω.
(c). We have E Φ−1N : (r1, r2, · · · , rN ;α1, α2, · · · , αN ) ∈ ΩN × RN 7→ − 1
2π
∑Nj=1 |rj |2 ∈ (−∞, 0).
4.5. THE GENERALIZED ACTION–ANGLE COORDINATES 119
Remark 4.5.3. The real analyticity of ΦN : UN → ΩN ×RN is given by proposition 4.4.7 and corollary4.4.12. The symplectomorphism property (b) is equivalent to the following Poisson bracket characterization(see proposition 4.5.24)
Ij , Ik = 0, Ij , γk = 1j=k, γj , γk = 0 on UN , ∀j, k = 1, 2, · · · , N. (4.5.4)
The family (XI1 , XI2 , · · · , XIN ;Xγ1, Xγ2
, · · · , XγN ) is linearly independent in X(UN ) and we have
dΦN (u) : XIk(u) 7→ ∂
∂αk
∣∣∣ΦN (u)
, dΦN (u) : Xγk(u) 7→ − ∂
∂rk
∣∣∣ΦN (u)
.
The assertion (c) is obtained by a direct calculus : Πu =∑Nj=1〈Πu, ϕuj 〉L2ϕuj , formula (4.4.9) yields that
E(u) = 〈Lu(Πu),Πu〉L2 =
N∑j=1
|〈Πu, ϕuj 〉L2 |2λuj = −N∑j=1
Ij(u)2
2π.
Thus theorem 4.5.2 introduces (generalized) action–angle coordinates of the BO equation (4.5.1) in thesense of (4.1.8), i.e. Ij , E(u) = 0 and γj , E(u) = 2λuj , for every u ∈ UN .
This section is organized as follows. The matrix associated to G|Hpp(Lu) is expressed in terms of ac-tions and angles in subsection 4.5.1. Then the injectivity of ΦN is given by inversion formulas in sub-section 4.5.2. In subsection 4.5.3, the Poisson brackets of actions and angles are used to show thelocal diffeomorphism property of ΦN . The surjectivity of ΦN is obtained by Hadamard’s global in-verse theorem in subsection 4.5.4. Finally, we use subsection 4.5.5 and subsection 4.5.6 to prove thatΦN : (UN , ω)→ (ΩN × RN , ν) preserves the symplectic structure.
4.5.1 The associated matrix
We continue to study the infinitesimal generator G defined in (4.3.2) when restricted to the invariantsubspace Hpp(Lu) with complex dimension N . Let M(u) = (Mkj(u))1≤k,j≤N denote the matrix associa-ted to the operator G|Hpp(Lu) with respect to the basis ϕu1 , ϕu2 , · · · , ϕuN. Then we state a general linearalgebra lemma that describes the location of eigenvalues of the matrix M(u).
Proposition 4.5.4. For every u ∈ UN , the coefficients of matrix M(u) = (Mkj(u))1≤k,j≤N are givenby
Mkj(u) = 〈Gϕuj , ϕuk〉L2 =
i
λuk−λuj
√|λuk ||λuj |
, if j 6= k,
γj(u)− i2|λuj |
, if j = k.(4.5.5)
Demonstration. Since Lu is a self-adjoint operator on L2+ and Hpp(Lu) ⊂ D(G), we have
(λuj − λuk)Mkj(u) = 〈GLuϕuj , ϕuk〉L2 − 〈Gϕuj , Luϕuk〉L2 = 〈[G,Lu]ϕuj , ϕuk〉L2 .
Since formulas (4.2.15) and (4.4.9) imply that −λuj ϕuj (0) = uϕuj (0) =√
2π|λuj |, we use (4.4.26) to obtain
(λuj − λuk)Mkj(u) = 〈iϕuj −i
2πϕuj (0+)Πu, ϕuk〉L2 = − i
2πϕuj (0+)uϕuk(0) = −i
√|λuk ||λuj |
.
120 CHAPITRE 4. INTEGRABILITY OF THE BO EQUATION ON THE LINE
In the case k = j, we use Plancherel formula and integration by parts to calculate
〈G∗f, g〉L2 = 〈f,Gg〉L2 = − i2π
∫ +∞
0
f(ξ)∂ξ g(ξ)dξ = i2π
[f(0+)g(0+) +
∫ +∞
0
∂ξ f(ξ)g(ξ)dξ
]Thus we have 〈G∗f, g〉L2 = 〈Gf, g〉L2 + i
2π f(0+)g(0+), for every f, g ∈Hpp(Lu). Then
ImMjj(u) =1
2i(〈Gϕuj , ϕuj 〉L2 − 〈G∗ϕuj , ϕuj 〉L2) = −
|ϕuj (0)|2
4π= − 1
2|λuj |.
We conclude by γj(u) = Refj(u) = 〈Gϕuj , ϕuj 〉L2 defined in corollary 4.4.12.
Then we state a linear algebra lemma that describe the location of spectrum of all matrices of the formdefined as (4.5.5).
Lemma 4.5.5. For every N ∈ N+, we choose N negative numbers λ1 < λ2 < · · · < λN < 0 and N realnumbers γ1, γ2, · · · , γN ∈ R. The matrix M = (Mkj)1≤k,j≤N ∈ CN×N is defined as
Mkj =
iλk−λj
√|λk||λj | , if k 6= j,
γj − i2|λj | , if k = j.
(4.5.6)
Then ImM = M−M∗2i is negative semi-definite and σpp(M) ⊂ C−. Furthermore, the map
(λ1, λ2, · · · , λN ; γ1, γ2, · · · , γN ) 7→ M = (Mkj)1≤k,j≤N
defined as (4.5.6) is real analytic on ΩN × RN .
Demonstration. The vector Vλ ∈ RN is defined as V Tλ := ((2|λ1|)−12 , (2|λ2|)−
12 , · · · , (2|λN |)−
12 ). So we
have
ImM =
(− 1
2√|λj ||λk|
)1≤k,j≤N
= −Vλ · V Tλ .
Recall that 〈X,Y 〉CN := XT · Y , thus 〈(ImM)X,X〉CN = −|〈X,Vλ〉CN |2 ≤ 0. So ImM is a negativesemi-definite matrix. If µ ∈ σpp(M) and V ∈ Ker(µ−M)\0, it suffices to show that Imµ < 0.
− |〈V, Vλ〉CN |2 = 〈(ImM)V, V 〉CN = Imµ‖V ‖2CN , where ‖V ‖2CN = 〈V, V 〉CN > 0. (4.5.7)
So we have Imµ ≤ 0. Assume that µ ∈ R, then formula (4.5.7) yields that V ⊥ Vλ. Moreover, we have(M−M∗)V = −2i〈V, Vλ〉CNVλ = 0. We set Dλ ∈ CN×N to be the diagonal matrix whose diagonal
elements are λ1, λ2, · · · , λN , i.e. Dλ =
λ1
λ2
. . .λN
. Then we have the following formula
[M, Dλ] = i(IN + 2DλVλVTλ ). (4.5.8)
So [M, Dλ]V = iV by (4.5.8). Recall that M∗V =MV = µV . Finally,
i‖V ‖2CN = 〈[M, Dλ]V, V 〉CN = 〈(M− µ)DλV, V 〉CN = 〈DλV, (M∗ − µ)V 〉CN = 0
contradicts the fact that V 6= 0. Consequently, we have µ ∈ C−.
4.5. THE GENERALIZED ACTION–ANGLE COORDINATES 121
Corollary 4.5.6. For every u ∈ UN , let M(u) = (Mkj(u))1≤k,j≤N ∈ CN×N denote the matrix defined
by formula (4.5.5), then ImM(u) = M(u)−M(u)∗
2i is negative semi-definite and σpp(M(u)) ⊂ C−.
Remark 4.5.7. The fact σpp(M(u)) ⊂ C− can also be given by using the inversion formula (4.4.10) andproposition 4.4.1. The characteristic polynomial Qu(x) = det(x−M(u)) has zeros in C−.
4.5.2 Inverse spectral formulas
The injectivity of ΦN is proved in this subsection by using inverse spectral formulas. The followinglemma describes the relation between the Fourier transform of an eigenfunction ϕ ∈ Hpp(Lu) and the
inner function associated to u defined by Θu = QuQu
with Qu(x) = det(x−M(u)).
Lemma 4.5.8. For every monic polynomial Q ∈ CN [X] such that Q−1(0) ⊂ C−, the associated inner
function is defined by Θ = QQ . The following identity holds for every ϕ ∈ C≤N−1[X]
Q ,
ϕ(ξ) = 〈S(ξ)∗ϕ, 1−Θ〉L2 . (4.5.9)
In particular, ϕ(0+) = 〈ϕ, 1−Θ〉L2 .
Demonstration. Since ϕ = PQ , for some P ∈ C≤N−1[X] and Q−1(0) ⊂ C−, recall that Q(x) =
∏nj=1(x−
zj)mj with Imzj < 0, z1, z2, · · · , zN are all distinct and
∑nj=1mj = N . Formulas (4.3.7) and (4.3.8) imply
that
fj,l(x) =l!
2π[(−i)(x− zj)]l+1=⇒ fj,l(ξ) = ξle−izjξ1R+
(ξ).
Since ϕ ∈ SpanCfj,l1≤j≤mj ,1≤j≤n, partial-fractional decomposition implies that ϕ ∈ C1(R∗+), and the
right limit ϕ(0+) = limξ→0+ ϕ(ξ) exists. Recall that Θ = QQ , so we have Θϕ = Q
QPQ = P
Q∈ L2
−. Since
Θ(x) = 1 + 2i∑Nj=1
Imzjx−zj +O( 1
x2 ), when x→ +∞, we have 1−Θ ∈ L2+. Then
ϕ(ξ) =
∫Rϕ(y)(1−Θ(y))e−iyξdy = 〈ϕ, S(ξ)(1−Θ)〉L2 = 〈S(ξ)∗ϕ, 1−Θ〉L2 , ∀ξ ≥ 0.
Proposition 4.5.9. For every u ∈ UN , we set Qu ∈ CN [X] to be the characteristic polynomial of u and
we define the associated inner function as Θu = QuQu
. Then the following inversion formula holds,
f(z) =1
2πi〈(G− z)−1f, 1−Θu〉L2 , f ∈Hpp(Lu), ∀z ∈ C+. (4.5.10)
Demonstration. If f ∈Hpp(Lu) =C≤N−1[X]
Qu, then formula (4.5.9) yields that
f(ξ) = 〈S(ξ)∗f, 1−Θu〉L2 = 〈e−iξGf, 1−Θu〉L2 .
Since ImG := G−G∗2i is a negative semi-definite operator on Hpp(Lu) by proposition 4.5.4 and lemma
4.5.5, the operator Re(i(z−G))|Hpp(Lu) = (ImG− Imz)|Hpp(Lu) is negative definite, for every z ∈ C+. So
f(z) =1
2π
∫ +∞
0
〈eiξ(z−G)f, 1−Θu〉L2dξ =1
2πi〈(G− z)−1f, 1−Θu〉L2 .
122 CHAPITRE 4. INTEGRABILITY OF THE BO EQUATION ON THE LINE
Recall that 〈Πu, ϕuj 〉L2 =√
2π|λuj | and 〈1−Θ, ϕuj 〉L2 =√
2π|λuj |
, for every j = 1, 2, · · · , N , by (4.2.15) and
(4.4.9). Since Πu ∈ Hol(z ∈ C : Imz > −ε), for some ε > 0, we have the following inversion formula
Πu(x) = 12πi 〈(G− x)−1Πu, 1−Θ〉L2 = −i〈(M(u)− x)−1X(u), Y (u)〉CN , ∀x ∈ R, (4.5.11)
where the two vectors X(u), Y (u) ∈ RN are defined as
X(u)T = (√|λu1 |,
√|λu2 |, · · · ,
√|λuN |), Y (u)T = (
√|λu1 |−1,
√|λu2 |−1, · · · ,
√|λuN |−1), (4.5.12)
and M(u) is the N × N matrix of the infinitesimal generator G associated to the orthonormal basisϕu1 , ϕu1 , · · · , ϕuN, defined in (4.5.4). A consequence of the inverse spectral formula (4.5.11) is the explicitformula of the BO flow with N -soliton solutions as described by formula (4.1.11).
Corollary 4.5.10. The map ΦN : UN → ΩN × RN is injective.
Demonstration. If ΦN (u) = ΦN (v) for some u, v ∈ UN , then λuj = λvj and γj(u) = γj(v), for every j. So
M(u) = M(v), X(u) = X(v), Y (u) = Y (v).
Then the inversion formula (4.5.11) gives that Πu = Πv. Thus, u = 2ReΠu = 2ReΠv = v.
At last we show the equivalence between the inversion formulas (4.4.10) and (4.5.11).
Revisiting formula (4.4.10). For every k, j = 1, 2, · · · , N , let Kukj(x) denote the (N − 1)× (N − 1) sub-
matrix obtained by deleting the k th column and j th row of the matrix M(u) − x, for every x ∈ R. Sothe inversion formula (4.5.11) and the Cramer’s rule imply that
iΠu(x) =∑
1≤k,j≤N
(−1)k+j det(Kukj(x))
det(M(u)− x)
√λukλuj
=
∑Nj=1 det(Ku
jj(x)) +R
det(M(u)− x), (4.5.13)
where R :=∑
1≤k 6=j≤N (−1)k+j det(Kukj(x))
√λukλuj
. The coefficients of the matrix M(u)− x satisfies that
(M(u)− x)kj = Mkj(u) = iλuk−λ
uj
√λukλuj, if 1 ≤ j 6= k ≤ N,
by formula (4.5.5). Using expansion by minors, we have
iR =∑
1≤k,j≤N
(−1)k+j(λuk − λuj )(M(u)− x)kj det(Kukj(x)) = (
N∑k=1
λuk −N∑j=1
λuj ) det(M(u)− x) = 0.
Finally, let Q denote the characteristic polynomial of the operator G|Hpp(Lu), so
Q(x) = det(x−G|Hpp(Lu)) = det(x−M(u)), Q′(x) = (−1)NN∑j=1
det(Kujj(x)).
4.5. THE GENERALIZED ACTION–ANGLE COORDINATES 123
4.5.3 Poisson brackets
In this subsection, the Poisson bracket defined in (4.5.2) is generalized in order to obtain the first twoformulas of (4.5.4). It can be defined between a smooth function from UN to an arbitrary Banach spaceand another smooth function from UN to R.
The N-soliton subset (UN , ω) is a real analytic symplectic manifold of real dimension 2N , where
ωu(h1, h2) =i
2π
∫R
h1(ξ)h2(ξ)
ξdξ, ∀h1, h2 ∈ Tu(UN ), ∀u ∈ UN .
For every smooth function f : UN → R, its Hamiltonian vector field Xf ∈ X(UN ) is given by (4.4.3).Recall that Xf (u) = ∂x∇uf(u) and df(u)(h) = ω(h,Xf (u)), ∀h ∈ Tu(UN ). For any Banach space E andany smooth map F : u ∈ UN 7→ F (u) ∈ E , we define the Poisson bracket of f and F as follows
f, F : u ∈ UN 7→ f, F(u) := dF (u)(Xf (u)) ∈ TF (u)(E) = E . (4.5.14)
If E = R, then the definition in formula (4.5.14) coincide with (4.5.2) and we recall it here,
f, F(u) = dF (u)(Xf (u)) = ωu(Xf (u), XF (u)). (4.5.15)
For every λ ∈ C\σ(−Lu), the generating function Hλ(u) = 〈(Lu + λ)−1Πu,Πu〉L2 is well defined. Since
Πu =∑Nj=1〈Πu, ϕuj 〉L2ϕuj , we have
Hλ(u) =
N∑j=1
|〈Πu, ϕuj 〉L2 |2
λ+ λuj= −
N∑j=1
2πλujλ+ λuj
. (4.5.16)
The analytical continuation allow to extend the generating function λ 7→ Hλ(u) to the domain C\σpp(−Lu),
and it has simple poles at every λ = −λuj . Proposition 4.2.2 yields that −C2
4 ‖u‖2L2 ≤ λu1 < · · · < λuN < 0,
where C = inff∈H1+\0
‖|D|14 f‖L2
‖f‖L4denotes the Sobolev constant. So we introduce
Y = (λ, u) ∈ R× UN : 4λ > C2‖u‖2L2 = X⋂
(R× UN ) , (4.5.17)
where X is given by definition 4.2.9. Then the subset Y is open in R×UN and the map H : (λ, u) ∈ Y 7→−∑Nj=1
2πλujλ+λuj
∈ R is real analytic by proposition 4.4.7. Recall that the Frechet derivative (4.2.8) is given
by
dHλ(u)(h) = 〈wλ,Πh〉L2 + 〈wλ,Πh〉L2 + 〈Thwλ, wλ〉L2 = 〈h,wλ + wλ + |wλ|2〉L2 , ∀h ∈ Tu(UN ).
where wλ ∈ H1+ is given by wλ ≡ wλ(u) ≡ wλ(x, u) = [(Lu + λ)−1 Π]u(x), for every x ∈ R. Thus
XHλ(u) = ∂x∇uHλ(u) = ∂x(|wλ(u)|2 + wλ(u) + wλ(u)), ∀(λ, u) ∈ Y. (4.5.18)
by (4.4.3). The Lax map L : u ∈ UN 7→ Lu = D − Tu ∈ B(H1+, L
2+) is R-affine, hence real analytic. The
following proposition restates the Lax pair structure of the Hamiltonian equation associated to Hλ. Eventhough the stability of UN under the Hamiltonian flow of Hλ remains as an open problem, the Poissonbracket defined in (4.5.14) provides an algebraic method to obtain the first two formulas of (4.5.4).
124 CHAPITRE 4. INTEGRABILITY OF THE BO EQUATION ON THE LINE
Proposition 4.5.11. Given (λ, u) ∈ Y defined by (4.5.17), we have Hλ, L(u) = [Buλ , Lu] and
Hλ, λj(u) = 0, Hλ, γj(u) = Re〈[G,Bλu ]ϕuj , ϕuj 〉L2 = − λ
(λ+ λuj )2, (4.5.19)
for every j = 1, 2, · · · , N , where Buλ = i(Twλ(u)Twλ(u) + Twλ(u) + Twλ(u)).
Demonstration. Since L : u ∈ L2(R,R) 7→ Lu = D− Tu ∈ B(H1+, L
2+), for every u ∈ L2
+, we have
dL(u)(h) = −Th, ∀h ∈ L2+.
If (λ, u) ∈ Y, then the C-linear transformation Lu + λ ∈ B(H1+, L
2+) is bijective. So formula (4.5.18)
yields that Hλ, L(u) = dL(u)(XHλ(u)) = −TD(|wλ(u)|2+wλ(u)+wλ(u)). Then identity (4.2.24) yields theLax equation for the Hamiltonian flow of the generating function Hλ, i.e.
Hλ, L(u) = [Buλ , Lu] ∈ B(H1+, L
2+). (4.5.20)
Consider the map Lϕj : u ∈ UN 7→ Luϕuj = λujϕ
uj ∈ H1
+, for every (λ, u) ∈ Y, we have
Hλ, L(u)ϕuj + Lu (Hλ, ϕj(u)) = λuj Hλ, ϕj(u) + Hλ, λj(u)ϕuj
with Hλ, ϕj(u) ∈ H1+ and Hλ, λj(u) ∈ R. Then (4.5.20) yields that
(λuj − Lu)(Bλuϕ
uj − Hλ, ϕj(u)
)= Hλ, λj(u)ϕuj .
Since ϕuj ∈ Ker(λuj − Lu) and ‖ϕuj ‖L2 = 1 by the definition in (4.4.9), we have
Hλ, λj(u) = 〈(λuj − Lu)(Bλuϕ
uj − Hλ, ϕj(u)
), ϕuj 〉L2 = 0.
Let N2 : ϕ ∈ L2 7→ ‖ϕ‖2L2 , then we have N2 ϕj ≡ 1 on UN . Then we have
0 = d(N2 ϕj)(u) = 2Re〈ϕuj , Hλ, λj(u)〉L2 . (4.5.21)
So there exists r ∈ R such that Bλuϕuj − Hλ, ϕj(u) = irϕuj because Ker(λuj − Lu) = Cϕuj by corollary
4.2.4 and formula (4.5.21). Recall that Bλu is skew-adjoint and γj = Re〈Gϕuj , ϕuj 〉L2 , we have
Hλ, γj(u) = Re(〈GHλ, ϕj(u), ϕuj 〉L2 + 〈Gϕuj , Hλ, ϕj(u)〉L2
)= Re〈[G,Bλu ]ϕuj , ϕ
uj 〉L2 .
Furthermore, for every (λ, u) ∈ Y, formula (4.3.4) implies that [G,Twλ(u)] = 0 and
[G,Bλu ]f = i[G,Twλ(u)](Twλ(u)(f) + f) = − 12π [(wλ(u)f)∧(0+) + f(0+)]wλ(u), ∀f ∈ D(G). (4.5.22)
Since (wλ(u)ϕuj )∧(0+) = 〈ϕuj , wλ(u)〉L2 = (λ+ λuj )−1〈u, ϕuj 〉L2and 〈u, ϕuj 〉L2
= −λuj ϕuj (0+), we replace f
by ϕuj in formula (4.5.22) to obtain the following
〈[G,Bλu ]ϕuj , ϕuj 〉L2 =
〈u, ϕuj 〉L2
2π(
1
λuj− 1
λ+ λuj)〈wλ(u), ϕuj 〉L2 = − λ
(λ+ λuj )2, ∀(λ, u) ∈ Y.
4.5. THE GENERALIZED ACTION–ANGLE COORDINATES 125
Remark 4.5.12. Recall that Hε = 1εH 1
εand Bε,u := 1
εB1εu for every (ε−1, u) ∈ Y. In general, the identity
En, γj(u) = Re〈[G, dn
dεn
∣∣ε=0
Bε,u]ϕuj , ϕuj 〉L2 , 1 ≤ j ≤ N
holds for every conservation law En = (−1)n dn
dεn
∣∣ε=0Hε in the BO hierarchy.
Corollary 4.5.13. For every j, k = 1, 2, · · · , N , we have
2πλj , γk(u) = 1j=k, λk, λj(u) = 0, ∀u ∈ UN . (4.5.23)
Demonstration. Given u ∈ UN , for every λ >C2‖u‖2
L2
4 then (λ, u) ∈ Y, then (4.5.16) and (4.5.19) implythat
− λ
(λ+ λuj )2= Hλ, γj(u) = 2π
N∑k=1
λ
λ+ λuk, γj(u) = −2πλ
N∑k=1
λk, γj(u)
(λ+ λuk)2,
and 0 = Hλ, λj(u) = 2πλ∑Nk=1
λk,λj(u)(λ+λuk )2 , for every j = 1, 2, · · · , N . The uniqueness of analytic
continuation yields that the following formula holds for every z ∈ C\R,
− z
(z + λuj )2= −2πz
N∑k=1
λk, γj(u)
(z + λuk)2,
N∑k=1
λk, λj(u)
(z + λuk)2= 0.
Recall that the actions Ij : u ∈ UN 7→ 2πλuj and the generalized angles γj : u ∈ UN 7→ Re〈Gϕuj , ϕuj 〉L2 areboth real analytic functions by proposition 4.4.7 and corollary 4.4.12.
Proposition 4.5.14. For every u ∈ UN , the family of differentials
dI1(u),dI2(u), · · · dIN (u); dγ1(u),dγ2(u), · · · dγN (u)
is linearly independent in the cotangent space T ∗u (UN ).
Demonstration. For every a1, a2, · · · , aN , b1, b2, · · · , bN ∈ R such that N∑j=1
ajdIj(u) + bjdγj(u)
(h) = 0, ∀h ∈ Tu(UN ). (4.5.24)
Formula of Poisson brackets (4.5.23) yields that for every j, k = 1, 2, · · · , N , we have
dIj(u)(XIk(u)) = Ik, Ij(u) = 0, dγj(u)(XIk(u)) = Ik, γj(u) = 1j=k
We replace h by XIk(u) in (4.5.24) to obtain that bk = 0, ∀k = 1, 2, · · · , N . Then set h = Xγk(u)
−ak =
N∑j=1
ajγk, Ij(u) =
N∑j=1
ajdIj(u)
(Xγk(u)) = 0, ∀k = 1, 2, · · · , N.
126 CHAPITRE 4. INTEGRABILITY OF THE BO EQUATION ON THE LINE
As a consequence, ΦN : UN → ΩN × RN is a local diffeomorphism. Moreover, since all the actions(Ij)1≤j≤N are in evolution by (4.5.23) and the differentials (dIj(u))1≤j≤N are linearly independent forevery u ∈ UN , for every r = (r1, r2, · · · , rN ) ∈ ΩN , the level set
Lr =
N⋂j=1
I−1j (rj), where r = (r1, r2, · · · , rN )
is a smooth Lagrangian submanifold of UN and Lr is invariant under the Hamiltonian flow of Ij , for everyj = 1, 2, · · · , N , by the Liouville–Arnold theorem (see Theorem 5.5.21 of Katok–Hasselblatt [83], seealso Fiorani–Giachetta–Sardanashvily [35] and Fiorani–Sardanashvily [36] for the non-compact invariantmanifold case).
4.5.4 The diffeomorphism property
This subsection is dedicated to proving the real bi-analyticity of ΦN : UN → ΩN × RN . It remains toshow the surjectivity. Its proof is based on Hadamard’s global inverse theorem 4.5.18.
Lemma 4.5.15. The map Φ : UN → ΩN × RN is proper.
Demonstration. If K is compact in ΩN × RN , we choose un ∈ Φ−1N (K), so
ΦN (un) = (2πλun1 , 2πλun2 , · · · , 2πλunN ; γ1(un), γ2(un), · · · , γN (un)) ∈ K, ∀n ∈ N.
We assume that there exists (2πλ1, 2πλ2, · · · , 2πλN ; γ1, γ2, · · · , γN ) ∈ K such that λunj → λj and
γj(un) → γj up to a subsequence. So (M(un))n∈N converges to some matrix M ∈ CN×N whose co-efficients are defined as follows
Mkj =
iλk−λj
√|λk||λj | , if k 6= j,
γj − i2|λj | , if k = j.
Lemma 4.5.5 yields that σpp(M) ⊂ C−. We set Q(x) := det(x−M) and u = iQ′
Q − iQ′
Q∈ UN . The Viete
map V is defined in (4.4.11) and V(CN− ) is open in CN . Then there exists
a(n) = (a(n)0 , a
(n)1 , · · · , a(n)
N−1), a = (a0, a1, · · · , aN−1) ∈ V(CN− )
such that Qn(x) = det(x−M(un)) =∑N−1j=0 a
(n)j xj + xN and Q(x) =
∑N−1j=0 ajx
j + xN . We have
limn→+∞
Qn(x) = Q(x), ∀x ∈ R =⇒ limn→+∞
a(n) = a
The continuity of the map ΓN : a = (a0, a1, · · · , aN−1) ∈ V(CN− ) 7→ Πu = iQ′
Q ∈ L2+ yields that
Πun = iQ′nQn
= ΓN (a(n))→ ΓN (a) = iQ′
Q= Πu in L2
+, as n→ +∞.
Since UN inherits the subspace topology of L2(R,R), we have (un)n∈N converges to u in UN . The continuityof the map ΦN shows that ΦN (u) = (2πλ1, 2πλ2, · · · , 2πλN ; γ1, γ2, · · · , γN ) ∈ K.
4.5. THE GENERALIZED ACTION–ANGLE COORDINATES 127
Proposition 4.5.16. The map ΦN : UN → ΩN × RN is bijective and both ΦN and its inverse Φ−1N are
real analytic.
Demonstration. The analyticity of ΦN is given by proposition 4.4.7 and corollary 4.4.12. The injectivityis given by corollary 4.5.10. Proposition 4.5.14 yields that ΦN : UN → ΩN ×RN is a local diffeomorphismby inverse function theorem for manifolds. So ΦN is an open map. Since every proper continuous map tolocally compact space is closed, ΦN is also a closed map by lemma 4.5.15. Since the target space ΩN×RNis connected, we have ΦN (UN ) = ΩN × RN and ΦN : UN → ΩN × RN is a real analytic diffeomorphism.
Remark 4.5.17. We establish the relation between ΦN : UN → ΩN × RN and ΓN : V(CN− ) → Π(UN )introduced in proposition 4.4.10. We set M : ΩN × RN → CN×N to be the matrix-valued real analyticfunction M(η1, η2, · · · , ηN ; θ1, θ2, · · · , θN ) = (Mkj)1≤k,j≤N with coefficients defined as
Mkj =
2πi
ηk−ηj
√ηkηj, if k 6= j,
θj + πiηj, if k = j.
Then, we set C : M ∈ CN×N 7→ (a0, a1, · · · , aN−1) ∈ CN such that
Q(x) :=
N−1∑j=0
ajxj + xN = det(x−M). (4.5.25)
Since (−1)n−jaj = Tr(Λn−jM) is the sum of all principle minors of M of size (N − j) × (N − j), forevery j = 1, 2, · · · , N , the map C is real analytic on CN×N and C M(ΩN × RN ) ⊂ V(CN− ) by lemma4.5.5, where V denotes the Viete map defined as (4.4.11). In lemma 4.4.10, we have shown that the map
ΓN : a = (a0, a1, · · · , aN−1) ∈ V(CN− ) 7→ Πu = iQ′
Q ∈ Π(UN ) is biholomorphic, where the polynomial Q
is defined as (4.5.25). We conclude by the following identity
Φ−1N = 2Re ΓN C M (4.5.26)
The smooth manifolds Π(UN ) and V(CN− ) are both diffeomorphic to the convex open subset ΩN × RN ,so they are simply connected (see also proposition 4.6.5). At last, we recall Hadamard’s global inversetheorem.
Theorem 4.5.18. Suppose X and Y are connected smooth manifolds, then every proper local dif-feomorphism F : X → Y is surjective. If Y is simply connected in addition, then every proper localdiffeomorphism F : X → Y is a diffeomorphism.
Demonstration. For the surjectivity, see Nijenhuis–Richardson [112] and the proof of proposition 4.5.16.If the target space is simply connected, see Gordon [63] for the injectivity.
Remark 4.5.19. Since the target space ΩN ×RN is convex, there is another way to show the injectivityof ΦN without using the inversion formulas in subsection 4.5.2. It suffices to use the simple connectednessof ΩN × RN and Hadamard’s global inverse theorem 4.5.18.
128 CHAPITRE 4. INTEGRABILITY OF THE BO EQUATION ON THE LINE
4.5.5 A Lagrangian submanifold
In general, the symplectomorphism property of ΦN is equivalent to its Poisson bracket characterization(4.5.4), which will be proved in proposition 4.5.24. The first two formulas of (4.5.4) given in corollary4.5.13, lead us to focusing on the study of a special Lagrangian submanifold of UN , denoted by
ΛN := u ∈ UN : γj(u) = 0, ∀j = 1, 2, · · · , N, (4.5.27)
where the generalized angles γj : u ∈ UN 7→ Re〈Gϕuj , ϕuj 〉L2 are defined in (4.5.1). A characterizationlemma of ΛN is given at first.
Lemma 4.5.20. For every u ∈ UN , then each of the following four properties implies the others :
(a). u ∈ ΛN .(b). For every x ∈ R, we have Πu(x) = Πu(−x).(c). u is an even function R→ R.(d). The Fourier transform u is real-valued.
Then every element u ∈ ΛN has translation–scaling parameter in (iR)N/SN i.e. u(x) =∑Nj=1
2ηjx2+η2
j, for
some ηj > 0.
Demonstration. (a)⇒ (b) : If u ∈ ΛN , then the matrix M(u) defined in (4.5.5) is an N ×N matrix withpurely imaginary coefficients. Recall the definition of X(u), Y (u) ∈ RN in (4.5.12) :
X(u)T = (√|λu1 |,
√|λu2 |, · · · ,
√|λuN |), Y (u)T = (
√|λu1 |−1,
√|λu2 |−1, · · · ,
√|λuN |−1).
The inversion formula (4.5.11) yields that
Πu(x) = i〈(M(u)− x)−1X(u), Y (u)〉CN = −i〈(M(u) + x)−1X(u), Y (u)〉CN = Πu(−x).
(b)⇒ (c) is given by the formula u = Πu+ Πu. (c)⇒ (d) is given by u(x) = u(x) = u(−x).
(d) ⇒ (a) : Choose λ ∈ σpp(Lu) = λu1 , λu2 , · · · , λuN and ϕ ∈ Ker(λ− Lu). Since both u and its Fourier
transform u are real-valued, we have [(ϕ)∨]∧(ξ) = ϕ(ξ), where (ϕ)∨(x) := ϕ(−x), ∀x, ξ ∈ R. Thus,
Tu((ϕ)∨) = (Tuϕ)∨ =⇒ (ϕ)∨ ∈ Ker(λ− Lu).
We choose the orthonormal basis ϕu1 , ϕu2 , , · · · , ϕuN in Hpp(Lu) as in formula (4.4.9). Proposition 4.2.4
yields that dimC Ker(λ− Lu) = 1. For every j = 1, 2, · · · , N , there exists θj ∈ R such that
(ϕuj )∨ = eiθjϕuj ⇐⇒ (ϕuj )∧(ξ) = eiθj (ϕuj )∧(ξ), ∀ξ ∈ R.
So we set φuj := exp(iθj2 )ϕuj , then its Fourier transform (φuj )∧ is a real-valued function. Recall the definition
of G in (4.3.2) and γj in (4.5.5), then we have
γj(u) = Re〈Gϕuj , ϕuj 〉L2(R) = Re〈Gφuj , φuj 〉L2(R) = − 1
2πIm〈∂ξ[(φuj )∧], (φuj )∧〉L2(0,+∞) = 0.
by using Plancherel formula.
4.5. THE GENERALIZED ACTION–ANGLE COORDINATES 129
Lemma 4.5.21. The level set ΛN is a real analytic Lagrangian submanifold of (UN , ω).
Demonstration. The map γ : u ∈ UN 7→ (γ1(u), γ2(u), · · · , γN (u)) ∈ RN is a real analytic submersionby proposition 4.5.14. So the level set ΛN is a properly embedded real analytic submanifold of UN anddimR ΛN = N . The classification of the tangent space Tu(UN ) is given by formula (4.4.1). If u(x) =∑Nj=1
2ηjx2+η2
j, for some ηj > 0, every tangent vector h ∈ ΛN is an even function by lemma 4.5.20. So h is
real valued and we have
Tu(ΛN ) =
N⊕j=1
Rfuj , where fuj (x) =2[x2−η2
j ]
[x2+η2j ]2. (4.5.28)
We have (fuj )∧(ξ) = −2π|ξ|e−ηj |ξ|. Then by definition of ω, we have
ωu(h1, h2) =i
2π
∫R
h1(ξ)h2(ξ)
ξdξ =
i
2π
∫R
h1(ξ)h2(ξ)
ξdξ ∈ iR, ∀h1, h2 ∈ Tu(ΛN ). (4.5.29)
Since the symplectic form ω is real-valued, we have ωu(h1, h2) = 0, for every h1, h2 ∈ Tu(ΛN ). SincedimR(ΛN ) = N = 1
2 dimR UN , ΛN is a Lagrangian submanifold of UN .
4.5.6 The symplectomorphism property
Finally, we prove the assertion (b) in theorem 4.5.2, i.e. the map ΦN : (UN , ω) → (ΩN × RN , ν) is
symplectic, where ω(h1, h2) := i2π
∫Rh1(ξ)h2(ξ)
ξ dξ, for every h1, h2 ∈ Tu(UN ) and
ΩN × RN = (r1, r2, · · · , rN ;α1, α2, · · · , αN ) ∈ R2N : r1 < r2 < · · · < rN < 0, ν =
N∑j=1
drj ∧ dαj .
We set ΨN = Φ−1N : ΩN × RN → UN , let Ψ∗Nω denote the pullback of the symplectic form ω by ΨN , i.e.
for every p = (r1, r2, · · · , rN ;α1, α2, · · · , αN ) ∈ ΩN × RN , set u = ΨN (p) ∈ UN ,
(Ψ∗Nω)p (V1, V2) = ωu(dΨN (p)(V1),dΨN (p)(V2)), (4.5.30)
for every V1, V2 ∈ Tp(ΩN × RN ). The goal is to prove that
ν := Ψ∗Nω − ν = 0. (4.5.31)
Recall that the coordinate vectors ∂∂r1
∣∣p, ∂∂r2
∣∣p, · · · , ∂
∂rN
∣∣p; ∂∂α1
∣∣p, ∂∂α2
∣∣p, · · · , ∂
∂αN
∣∣p
form a basis for the
tangent space Tp(ΩN × RN ). We have the following lemma.
Lemma 4.5.22. For every u ∈ UN , set p = ΦN (u) ∈ ΩN × RN . Then we have
dΦN (u)(XIk(u)) =∂
∂αk
∣∣∣p, ∀k = 1, 2, · · · , N. (4.5.32)
130 CHAPITRE 4. INTEGRABILITY OF THE BO EQUATION ON THE LINE
Demonstration. Fix u ∈ UN and p = ΦN (u), for every h ∈ Tu(UN ), we have dΦN (u)(h) ∈ Tp(ΩN ×RN ).For every smooth function f : p = (r1, r2, · · · , rN ;α1, α2, · · · , αN ) ∈ ΩN × RN 7→ f(p) ∈ R, then
(dΦN (u)(h)) f = d(f ΦN )(u)(h) =
N∑j=1
(dIj(u)(h)
∂f
∂rj
∣∣∣p
+ dγj(u)(h)∂f
∂αj
∣∣∣p
). (4.5.33)
For every k = 1, 2, · · · , N , we replace h by XIk(u), where XIk denotes the Hamiltonian vector field of thek th action Ik defined in (4.5.1), thus the Poisson bracket formulas (4.5.23) yield that
∂f
∂αk
∣∣∣p
=
N∑j=1
(Ik, Ij(u)
∂f
∂rj
∣∣∣p
+ Ik, γj(u)∂f
∂αj
∣∣∣p
)= (dΦN (u)(XIk(u))) f.
Lemma 4.5.23. For every 1 ≤ j < k ≤ N , there exists a smooth function cjk ∈ C∞(ΩN × RN ) suchthat
ν =∑
1≤j<k≤N
cjkdrj ∧ drk,∂cjk∂αl
∣∣∣p
= 0, ∀j, k, l = 1, 2, · · · , N, (4.5.34)
for every p = (r1, r2, · · · , rN ;α1, α2, · · · , αN ) ∈ ΩN × RN .
Demonstration. The proof is divided into three steps. The first step is to prove that for every p ∈ ΩN×RNand every V ∈ Tp(ΩN × RN ),
νp(∂
∂αl
∣∣∣p, V ) = 0, ∀l = 1, 2, · · · , N. (4.5.35)
In fact, let u = ΨN (p) ∈ UN and p = (r1, r2, · · · , rN ;α1, α2, · · · , αN ), so rl = rl(p) = Il ΨN (p). Then
(Ψ∗Nω)p(∂
∂αl
∣∣∣p, V ) = ωu(dΨN (p)
(∂
∂αl
∣∣∣p
),dΨN (p)(V )) = ωu(XIl(u),dΨN (p)(V ))
by (4.5.32). Thus (Ψ∗Nω)p(∂∂αl
∣∣∣p, V ) = −dIl(u)(dΨN (p)(V )) = −d(Il ΨN )(p)(V ). On the other hand,
νp(∂
∂αl
∣∣∣p, V ) =
N∑j=1
(drj ∧ dαj)
(∂
∂αl
∣∣∣p, V
)= −drl(p)(V )
Thus (4.5.35) is obtained by ν = Ψ∗Nω − ν.
Since ν is a smooth 2-form on ΩN × RN , we have
ν =∑
1≤j<k≤N
(ajkdαj ∧ dαk + bjkdrj ∧ dαk + cjkdrj ∧ drk),
for some smooth functions ajk, bjk, cjk ∈ C∞(ΩN × RN ), 1 ≤ j < k ≤ N . The second step is to prove
that ajk = bjk = 0 on ΩN × RN , for every 1 ≤ j < k ≤ N . In fact, we have drj ∧ drk( ∂∂αl
∣∣∣p, V ) = 0,
drj ∧ dαk(∂
∂αl
∣∣∣p, V ) = −1k=ldr
j(p)(V ) and dαj ∧ dαk(∂
∂αl
∣∣∣p, V ) = 1j=ldα
k(p)(V )− 1k=ldαj(p)(V ).
4.5. THE GENERALIZED ACTION–ANGLE COORDINATES 131
Then, let l ∈ 2, · · · , N be fixed, for every 1 ≤ j < k ≤ N , we have∑1≤l<k≤N
alkdαk(p)(V )−∑
1≤j<l≤N
(ajldαj(p)(V ) + bjldr
j(p)(V )) = νp(∂
∂αl
∣∣∣p, V ) = 0. (4.5.36)
Then we replace V by ∂∂rj
∣∣∣p
and ∂∂αj
∣∣∣p
respectively in formula (4.5.36), then ajl = bjl = 0, for every
j = 1, 2, · · · , l − 1.
It remains to show that cjk depends on r1, r2, · · · , rN , for every 1 ≤ j < k ≤ N . The symplectic form ω
is closed by proposition 4.4.4 and ν = dκ is exact, where κ =∑Nj=1 r
jdαj . So
dν = d(Ψ∗Nω)− dν = Ψ∗N (dω) = 0.
The exterior derivative of ν =∑
1≤j<k≤N cjkdrj ∧ drk is computed as following
0 =∑
1≤j<k≤N
N∑l=1
(∂cjk∂αl
dαl ∧ drj ∧ drk +∂cjk∂rl
drl ∧ drj ∧ drk).
Since the family drj ∧ drk ∧ dαl1≤j<k≤N,1≤l≤N⋃drj ∧ drk ∧ drl1≤j<k<l≤N is linearly independent
in Ω3(UN ), we have∂cjk∂αl
= 0, for every 1 ≤ j < k ≤ N and l = 1, 2, · · · , N .
Since the 2-form ν is independent of α1, α2, · · · , αN , it suffices to consider points p = (r, α) ∈ ΩN × RNwith α = 0. We shall prove that ν = 0 by introducing the following Lagrangian submanifold of ΩN ×RN ,
ΩN × 0RN = (r1, r2, · · · , rN ; 0, 0, · · · , 0) ∈ R2N : r1 < r2 < · · · < rN < 0.
End of the proof of formula (4.5.31). The submersion level set theorem implies that ΩN × 0RN is aproperly embedded N -dimensional submanifold of ΩN × RN . We have ΩN × 0RN = ΦN (ΛN ), whereΛN is the Lagrangian submanifold of (UN , ω) defined by (4.5.27). For every q ∈ ΩN × 0RN , set v =ΨN (q) ∈ ΛN , we claim at first that
Tq(ΩN × 0RN ) =
N⊕j=1
R∂
∂rj
∣∣∣q
= dΦN (v)(Tv(ΛN )). (4.5.37)
In fact, every tangent vector V ∈ Tq(ΩN × 0RN ) is the velocity at t = 0 of some smooth curveξ : t ∈ (−1, 1) 7→ ξ(t) = (ξ1(t), ξ2(t), · · · , ξN (t); 0, 0, · · · , 0) ∈ ΩN × 0RN such that ξ(0) = q, i.e.
V f =d
dt
∣∣∣t=0
(f ξ) =
N∑j=1
ξ′j(0)∂f
∂rj
∣∣∣q, ∀f ∈ C∞(ΩN × RN ). (4.5.38)
So the first equality of (4.5.37) is obtained. Then we set η(t) = ΨN ξ(t), ∀t ∈ (−1, 1). For everyg ∈ C∞(UN ), we replace f by g ΨN ∈ C∞(ΩN × RN ) in (4.5.38) to obtain that
dΨN (q)(V )g = V (g ΨN ) =d
dt
∣∣∣t=0
(g η) = η′(0)g.
132 CHAPITRE 4. INTEGRABILITY OF THE BO EQUATION ON THE LINE
Since η is a smooth curve in the Lagrangian section ΛN such that η(0) = v, we have dΨN (q)(V ) = η′(0) ∈Tv(ΛN ). So formula (4.5.37) holds. Since ν =
∑Nj=1 drj∧dαj , the submanifold ΩN×0RN is Lagrangian.
For every p = (r1, r2, · · · , rN ;α1, α2, · · · , αN ) ∈ ΩN × RN and every V1, V2 ∈ Tp(ΩN × RN ), where
Vm =
N∑j=1
(a
(m)j
∂
∂rj
∣∣∣p
+ b(m)j
∂
∂αj
∣∣∣p
), a
(m)j , b
(m)j ∈ R, m = 1, 2,
we choose q = (r1, r2, · · · , rN ; 0, 0, · · · , 0) ∈ ΩN × 0RN and W1,W2 ∈ Tq(ΩN × 0RN ), where
Wm =
N∑j=1
a(m)j
∂
∂rj
∣∣∣p, m = 1, 2.
We set v = ΨN (q) ∈ ΛN . We have proved that cjk(p) = cjk(q), then (4.5.34) yields that
νp(V1, V2) =∑
1≤j<k≤N
a(1)j a
(2)k cjk(p) = νq(W1,W2) = ωv(dΨN (v)(W1),dΨN (v)(W2)),
because νq(W1,W2) = 0. The identification (4.5.37) yields that hm := dΨN (v)(Wm) ∈ Tv(ΛN ), form = 1, 2. Consequently, we have νp(V1, V2) = ωv(h1, h2) = 0.
Formula (4.5.31) is equivalent to Φ∗Nν = ω, so ΦN : (UN , ω) → (ΩN × RN , ν) is a symplectomorphism.Finally, we recall a basic property in symplectic geometry : the three formulas in (4.5.4) are equivalentto the symplectomorphism property of ΦN .
Proposition 4.5.24. If ΦN : (UN , ω)→ (ΩN × RN , ν) is a diffeomorphism,
ΦN (u) = (I1(u), I2(u), · · · , IN (u); γ1(u), γ2(u), · · · , γN (u)), ∀u ∈ UN ,
for some smooth functions Ij , γj on UN , then each of the following three properties implies the others :
(a). ΦN : (UN , ω)→ (ΩN × RN , ν) is a symplectomorphism, i.e. Φ∗Nν = ω.(b). For every j, k = 1, 2, · · · , N , we have Ij , Ik = γj , γk = 0 and Ij , γk = 1j=k on UN .(c). For every k = 1, 2, · · · , N , we have
dΦN (u)(XIk(u)) =
∂
∂αk
∣∣∣ΦN (u)
, dΦN (u)(Xγk(u)) = − ∂
∂rk
∣∣∣ΦN (u)
, ∀u ∈ UN .
Demonstration. (a) ⇒ (b). For any smooth function f : ΩN × RN → R, its Hamiltonian vector field isgiven by
Xf (p) =
N∑j=1
∂f
∂rj(p)
∂
∂αj
∣∣∣p− ∂f
∂αj(p)
∂
∂rj
∣∣∣p, ∀p ∈ ΩN × RN . (4.5.39)
If Φ∗Nν = ω, then XfΦN (u) = dΨN (p) Xf (p), if p = ΦN (u), where ΨN = Φ−1N . The Poisson bracket of
two smooth functions f, g on ΩN × RN is given by
f, gν(p) = (Xfg)∣∣∣p
= νp(Xf (p), Xg(p)) =
N∑j=1
∂f
∂rj(p)
∂g
∂αj(p)− ∂f
∂αj(p)
∂g
∂rj(p). (4.5.40)
4.6. APPENDICES 133
Then f ΦN , g ΦN = f, gν ΦN on UN . It suffices to choose f, g ∈ Ij ΨN , γj ΨN1≤j≤N .
(b)⇒ (c). We do the same calculus as in lemma 4.5.22 to obtain that
dΦN (u)(XIk(u)) =
∂
∂αk
∣∣∣ΦN (u)
, dΦN (u)(Xγk(u)) = − ∂
∂rk
∣∣∣ΦN (u)
, ∀u ∈ UN . (4.5.41)
(c) ⇒ (a). The same calculus as in lemma 4.5.22 yields that (c) ⇒ (b). Formula (4.5.41) implies thatXI1
, XI2, · · · , XIN
;Xγ1, Xγ2
, · · · , XγN forms a basis in X(UN ). Since the 2-covectors (Φ∗Nν)u and ωu
coincide at every couple of elements of this basis, they are the same, so Φ∗Nν = ω.
4.6 Appendices
We establish several topological properties of the N -soliton manifold UN without using the action–anglemap ΦN : UN → ΩN × RN . The Viete map V : (β1, β2, · · · , βN ) ∈ CN 7→ (a0, a1, · · · , aN−1) ∈ CN isdefined by
N∏j=1
(X − βj) =
N−1∑k=0
akXk +XN . (4.6.1)
Proposition 4.6.1. Endowed with the Hermitian form H introduced in (4.4.15), (Π(UN ),H) is a simplyconnected Kahler manifold which is biholomorphically equivalent to V(CN− ).
Proposition 4.6.2. The N -soliton manifold UN is a universal covering manifold of the following N -gappotential manifold for the BO equation on the torus T := R/2πZ as described by Gerard–Kappeler [54],
UTN = v = h+ h ∈ L2(T,R) : h : y ∈ T 7→ −eiyQ
′(eiy)
Q(eiy)∈ C, Q ∈ C+
N [X], (4.6.2)
where C+N [X] consists of all monic polynomial Q ∈ C[X] of degree N , whose roots are contained in the
annulus A := z ∈ C : |z| > 1. The fundamental group of UTN is (Z,+).
Remark 4.6.3. The real analytic symplectic manifold UTN is mapped real bi-analytically onto CN−1×C∗
by the restriction of the Birkhoff map constructed in Gerard–Kappeler [54]. The union of all finite gappotentials
⋃N≥0 U
TN is dense in L2
r,0(T) = v ∈ L2(T,R) :∫T v = 0. However
⋃N≥1 UN is not dense in
L2(R, (1 + x2)dx). We refer to Coifman–Wickerhauser [28] to see solutions with sufficiently small initialdata and the case of non-existence of rapidly decreasing solitons.
The simple connectedness of UN is proved in subsection 4.6.1. Then we establish a real analytic coveringmap UN → UT
N in subsection 4.6.2.
4.6.1 The simple connectedness of UNThanks to the biholomorphical equivalence between the Kahler manifolds Π(UN ) and V(CN− ) establishedin lemma 4.4.10, it suffices to prove the simple connectedness of the subset V(CN− ), where V denotes theViete map defined by (4.6.1). Since every fiber of the Viete map is invariant under the permutation of
134 CHAPITRE 4. INTEGRABILITY OF THE BO EQUATION ON THE LINE
components, we introduce the following group action. Equipped with the discrete topology, the symmetricgroup SN acts continuously on CN by permuting the components of every vector :
σ : (β0, β1, · · · , βN−1) ∈ CN 7→ (βσ(0), βσ(1), · · · , βσ(N−1)) ∈ CN , ∀σ ∈ SN . (4.6.3)
A subset A ⊂ CN is said to be stable under SN if⋃σ∈SN
σ(A) = A. We recall the basic property of theViete map V and the action of symmetric group SN .
Lemma 4.6.4. The Viete map V : CN → CN is a both open and closed quotient map. For everyA ⊂ CN , A is stable under SN if and only if A is saturated with respect to V, the quotient space A/SNis homeomorphic to V(A).
We set ∆ := (β, β, · · · , β) ∈ CN : ∀β ∈ C. The goal of this subsection is to prove the following result.
Proposition 4.6.5. For every open simply connected subset A ⊂ CN , if A is stable under the symmetricgroup SN and A
⋂∆ 6= ∅, then V(A) is an open simply connected subset of CN .
Demonstration. Let A ⊂ CN be a nonempty open simply connected subset that is stable by SN . Thesubset B := V(A) is open, connected and locally simply connected, then it admits a universal coveringspace E and a covering map π : E → B. The triple (E, π,B) is identified as a fiber bundle over B whosemodel fiber F is discrete. The target is to show that F has cardinality 1.
Let (P, q, B) denote the fiber product (Husemoller [77]) of bundles (A,V, B) and (E, π,B), defined by
P = A×B E := (β, e) ∈ A× E : π(e) = V(β), q : (β, e) ∈P 7→ V(β) = π(e) ∈ B. (4.6.4)
The total space P is equipped with the subspace topology of the product space A× E and projectionsonto the first factor and onto the second factor are denoted respectively by
p : (β, e) ∈P 7→ β ∈ A, W : (β, e) ∈P 7→ e ∈ E. (4.6.5)
Both p and W are continuous functions on P and the following diagram commutes.
P E
A B
p q
W
V
π
We claim two properties concerning the projections p and W.
i. W : P → E is an open quotient map and p : P → A is a covering map whose model fiber is F.ii. Equipped with the discrete topology, the symmetric group SN acts continuously on P by permutingcomponents of the first factor
σ : (β, e) ∈P 7→ (σ(β), e) ∈P, ∀σ ∈ SN ,
where σ ∈ GLN (C) is defined by (4.6.3). Hence the quotient map W : P → E is closed.
Thanks to the simple connectedness of the base space A, the covering space P is the disjoint unionof its connected components (Ak)k∈F and the restriction of the covering map p|Ak
: Ak → A is a
4.6. APPENDICES 135
homeomorphism. Since P is locally path-connected, every component Ak is both open and closed, thenW|Ak
: Ak → E an open closed quotient map. So is the lift gk := W|Ak (p|Ak
)−1 : A → E. Notethat π gk = V and SN stabilizes every element of ∆. We choose β ∈ A
⋂∆ and b := V(β). Since the
fiber V−1(b) = β is a singleton, so is the fiber π−1(b). Hence |F| = 1 and the universal covering mapp : E → B is a homeomorphism. So B is simply connected.
Remark 4.6.6. Let F be a closed submanifold of a smooth connected manifold M without boundary offinite dimension. If dimRM−dimR F ≥ 3, then the inclusion map i : M\F →M induces an isomorphismbetween the fundamental groups i∗ : π1(M\F, x) → π1(M,x), for every x ∈ M\F (see Theoreme 2.3 inP.146 of Godbillon [62]). Note that the closed submanifold ∆ ⊂ CN has real dimension 2. When N ≥ 3,the condition A
⋂∆ 6= ∅ cannot be deduced by the other three conditions in the hypothesis of proposition
4.6.5 : A is open, simply connected and stable by SN .
As a consequence, V(CN− ) is open and simply connected because CN− is an open convex subset of CNwhich is stable under the symmetric group SN and ∆
⋂CN− = (z, z, · · · , z) ∈ CN : Imz < 0. Together
with lemma 4.4.10, we finish the proof of proposition 4.6.1.
4.6.2 Covering manifold
The Szego projector on L2(T,C) is given by ΠTv(x) =∑n≥0 vne
inx, for every v ∈ L2(T,C) such that
v(x) =∑n∈Z vne
inx with vn = 12π
∫ 2π
0v(x)e−inxdx. Equipped with the subspace topology of ΠT(L2(T,C))
and the Hermitian form
HT(v1, v2) = 〈D−1ΠTv1,ΠTv2〉L2(T) =
1
2π
∫ 2π
0
D−1ΠTv1(x)ΠTv2(x)dx,
the subset ΠT(UTN ) is a Kahler manifold, which is mapped biholomorphically onto V(A N ) with A =
C\D(0, 1) = z ∈ C : |z| > 1 in Gerard–Kappeler [54].
Proposition 4.6.7. There exists a covering map π : V(CN− )→ V(A N ).
Remark 4.6.8. Consider the cubic Szego equation on the torus (see Gerard–Grellier [47, 49, 51, 52])
i∂twT = ΠT(|wT|2wT), (t, x) ∈ R× T, (4.6.6)
and the cubic Szego equation on the line (see Pocovnicu [119, 120]), we set ΠR := Π in (4.1.12),
i∂twR = ΠR(|wR|2wR), (t, x) ∈ R× R. (4.6.7)
The manifold of N -solitons for the cubic Szego equation on the line is not simply connected. Let M(N)R
denote all rational functions of the form wR : x ∈ R 7→ P (x)Q(x) ∈ C where P ∈ C≤N−1[X] and Q ∈ CN [X]
is a monic polynomial such that Q−1(0) ⊂ C− and P,Q have no common factors. Then M(N)R is aKahler manifold of complex dimension 2N . So is the subset M(N)T consisting of all rational functions
of the form wT : x ∈ T 7→ P (eix)Q(eix) ∈ C where P ∈ C≤N−1[X] and Q ∈ CN [X] is a monic polynomial such
that Q−1(0) ⊂ A and P,Q have no common factors. Both of them have rank characterization of Hankeloperators by Kronecker-type theorem (see Lemma 8.12 in Chapter 1 of Peller [117], p. 54). So the manifoldM(N)R (resp. M(N)T) is invariant under the flow of equation (4.6.7) (resp. of equation (4.6.6)) and the(generalized) action–angle coordinates of equation (4.6.7) (resp. of equation (4.6.6)) are defined in some
136 CHAPITRE 4. INTEGRABILITY OF THE BO EQUATION ON THE LINE
open dense subset of M(N)R (resp. of M(N)T). Moreover, if N ≥ 2 then M(N)R is simply connectedby proposition 4.6.5 and remark 4.6.6. There exists a holomorphic covering map M(N)R → M(N)T byfollowing the construction in proposition 4.6.7. The manifold of N -solitons for the cubic Szego equationon the line is an open dense subset of M(N)R
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Titre: Comportement en grand temps et intégrabilité de certaines équations dispersivessur l'espace de Hardy
Mots clés: Équation de Schrödinger non linéaire, Projecteur de Szeg®, Équation de BenjaminOno, Espace de Hardy, Onde progressive, Coordonnées d'actionangle
Résumé: On s'intéresse dans cette thèse àtrois modèles d'équations hamiltoniennes dis-persives non linéaires: l'équation de Schrödingercubique défocalisante ltrée par le projecteur deSzeg® ΠT, qui enlève tous les modes de Fourierstrictement négatifs, sur le tore T := R/2πZ(NLSSzeg® cubique), l'équation de Schrödingerquintique focalisante ltrée par le projecteur deSzeg® ΠR sur la droite R (NLSSzeg® quin-tique) et l'équation de BenjaminOno (BO) surla droite. Ces trois modèles nous donnentl'occasion d'étudier les propriétés qualitatives de
certaines ondes progressives, le phénomène decroissance de normes de Sobolev, le phénomènede diusion non linéaire et certaines propriétésd'intégrabilité de systèmes dynamiques hamil-toniens. Le but de cette thèse est de compren-dre l'inuence des opérateurs non locaux ΠT etΠR sur des équations de type de Schrödingeret d'adapter les outils liés à l'espace de Hardysur le cercle et sur la droite. Dans le troisièmemodèle, la théorie de l'intégrabilité permet defaire le lien avec certains aspects algébriques etgéométriques.
Title: Long time behavior and integrability of some dispersive equations on the Hardy
space
Keywords: Non linear Schrödinger equation, Szeg® projector, BenjaminOno equation, Hardyspace, Traveling wave, Actionangle coordinates
Abstract: We are interested in three non lin-ear dispersive Hamiltonian equations: the de-focusing cubic Schrödinger equation ltered bythe Szeg® projector ΠT that cancels every neg-ative Fourier modes, leading to the cubic NLSSzeg® equation on the torus T := R/2πZ; thefocusing quintic Schrödinger equation, which isltered by the Szeg® projector ΠR, leading to thequintic NLSSzeg® equation on the line R; andthe BenjaminOno (BO) equation on the line.These three models allow us to study their qual-
itative properties of some traveling waves, thephenomenon of the growth of Sobolev norms,the phenomenon of non linear scattering andsome properties about the complete integrabil-ity of Hamiltonian dynamical systems. The goalof this thesis is to investigate the inuence of thenon local operators ΠT and ΠR on some one-dimensional Schrödinger-type equations and toadapt the tools of the Hardy space on the torusand on the line. In the third model, the integra-bility theory allows to establish the connectionwith some algebraic and geometric aspects.
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