Digital Object Identifier (DOI) 10.1007/s00220-004-1260-yCommun. Math. Phys. 257, 51–85 (2005) Communications in

MathematicalPhysics

Travelling Breathers with Exponentially Small Tailsin a Chain of Nonlinear Oscillators

Guillaume James, Yannick Sire

Mathematiques pour l’Industrie et la Physique, UMR CNRS 5640, and Departement GMM, InstitutNational des Sciences Appliquees, 135 avenue de Rangueil, 31077 Toulouse Cedex 4, France.E-mail: james@insa-toulouse.fr; sire@insa-toulouse.fr

Received: 6 April 2004 / Accepted: 10 July 2004Published online: 11 January 2005 – © Springer-Verlag 2005

Abstract: We study the existence of travelling breathers in Klein-Gordon chains, whichconsist of one-dimensional networks of nonlinear oscillators in an anharmonic on-sitepotential, linearly coupled to their nearest neighbors. Travelling breathers are spatiallylocalized solutions which appear time periodic in a referential in translation at constantvelocity. Approximate solutions of this type have been constructed in the form of mod-ulated plane waves, whose envelopes satisfy the nonlinear Schrodinger equation (M.Remoissenet, Phys. Rev. B 33, n.4, 2386 (1986), J. Giannoulis and A. Mielke, Nonlin-earity 17, p. 551–565 (2004)). In the case of travelling waves (where the phase velocityof the plane wave equals the group velocity of the wave packet), the existence of nearbyexact solutions has been proved by Iooss and Kirchgassner, who have obtained exactsolitary wave solutions superposed on an exponentially small oscillatory tail (G. Iooss,K. Kirchgassner, Commun. Math. Phys. 211, 439–464 (2000)). However, a rigorousexistence result has been lacking in the more general case when phase and group veloc-ities are different. This situation is examined in the present paper, in a case when thebreather period and the inverse of its velocity are commensurate. We show that the cen-ter manifold reduction method introduced by Iooss and Kirchgassner is still applicablewhen the problem is formulated in an appropriate way. This allows us to reduce the prob-lem locally to a finite dimensional reversible system of ordinary differential equations,whose principal part admits homoclinic solutions to quasi-periodic orbits under generalconditions on the potential. For an even potential, using the additional symmetry of thesystem, we obtain homoclinic orbits to small periodic ones for the full reduced sys-tem. For the oscillator chain, these orbits correspond to exact small amplitude travellingbreather solutions superposed on an exponentially small oscillatory tail. Their princi-pal part (excluding the tail) coincides at leading order with the nonlinear Schrodingerapproximation.

52 G. James, Y. Sire

1. Introduction

We consider a one-dimensional lattice of nonlinear oscillators described by the followingsystem (Klein-Gordon system):

d2xn

dτ 2 + V ′(xn) = γ (xn+1 + xn−1 − 2xn), n ∈ Z, (1)

where xn is the displacement of thenth particle from an equilibrium position, the couplingconstant γ is strictly positive and the on-site potential V is analytic in a neighborhood ofx = 0 (with V ′(0) = 0, V ′′(0) > 0). This system describes a chain of particles linearlycoupled to their first neighbors, in the local anharmonic potential V .

In this paper, we consider solutions of (1) satisfying

xn(τ ) = xn−p(τ − T ), (2)

for a fixed T ∈ R and p ≥ 1. The case when p = 1 in (2) corresponds to travellingwaves. Solutions satisfying (2) for p �= 1 consist of pulsating travelling waves, whichare exactly translated by p sites after a fixed propagation time T and are allowed tooscillate as they propagate on the lattice. In particular, solutions of (1) having the formxn(τ ) = x(n− c τ, τ ) (x being T -periodic in its second argument) satisfy (2) under thecondition c = p/T . A different situation arises when c and 1/T are incommensurate,since the solution is not exactly translated on the lattice after time T but is modified by aspatial shift. Solutions of type (2) having the additional property of spatial localization(xn(τ ) → 0 as n → ±∞) are known as exact travelling breathers (with commensuratevelocity and frequency) and have been studied numerically in different systems.

Approximate travelling breather solutions propagating on the lattice at a non constantvelocity c have also drawn a lot of attention. They have been numerically observed invarious one-dimensional nonlinear lattices such as Fermi-Pasta-Ulam lattices [43, 8, 37,13], Klein-Gordon chains [9, 6] and the discrete nonlinear Schrodinger (DNLS) equa-tion [12]. The two last models exhibit similar features in some regimes where the DNLSequation can be derived from the Klein-Gordon system using appropriate scalings [35].Other references are available in the review paper [15].

One way of generating approximate travelling breathers consists of “kicking” staticbreathers consisting of spatially localized and time periodic oscillations (see the basicpapers [44, 30, 15, 25, 5] for more details on these solutions). Static breathers are put intomotion by perturbation in the direction of a pinning mode [6]. The possible existenceof an energy barrier that the breather has to overcome in order to become mobile hasdrawn a lot of attention, see e.g. [9, 6, 13, 26] and the review paper [39].

It is a more delicate task to examine the existence of exact travelling breathers usingnumerical computations. Indeed, these solutions might not exist without being super-posed on a small nonvanishing oscillatory tail which violates the property of spatiallocalization. This phenomenon is likely to occur since the existence of a nonvanish-ing oscillatory tail has been previously observed in some parameter regimes for soli-tary waves (spatially localized travelling waves) in Klein-Gordon chains [6]. Numericalresults indicate similar phenomena for the propagation of kinks [10, 38, 4]. Fine analysisof numerical convergence problems also suggests that different nonlinear lattices do notsupport exact solitary waves or travelling breathers in certain parameter regimes [42, 3].

Nevertheless, several formal analytical methods have been used to obtain travellingbreather solutions. On the one hand, approximate travelling breathers can be formallyobtained via effective Hamiltonians, which approximately describe the motion of thebreather center on the lattice, at a nonconstant velocity [31, 26]. On the other hand,

Travelling Breathers in Klein-Gordon Chains 53

multi-scale expansions provide evolution equations for the envelopes of well-preparedinitial conditions corresponding to modulated plane waves. This approach has been usedby Remoissenet for Klein-Gordon lattices [36] and yields the nonlinear Schrodinger(NLS) equation as a modulation equation. For good parameter values, the NLS equationadmits solitons corresponding (at least formally) to travelling breather solutions of theoriginal system, which propagate at a constant velocity (the group velocity of the wavepacket). At the order of the NLS approximation, the linear dispersion is exactly balancedby the effect of nonlinear terms. The same approach has been used by Tsurui for theFermi-Pasta-Ulam lattice [45]. For the Klein-Gordon system (and generalizations withanharmonic coupling), the validity of the nonlinear Schrodinger equation on large butfinite time intervals has been proved recently by Giannoulis and Mielke [19].

It is a challenging problem to determine if these approximate solutions could consti-tute the principal part of exact travelling breather solutions of the Klein-Gordon system.This would imply that linear dispersion is balanced by nonlinear terms at any order inthe above mentioned multi-scale expansion.

This problem has been solved by Iooss and Kirchgassner in the case of travellingwaves [22], where the phase velocity of the plane wave equals the group velocity of thewave packet. Travelling wave solutions of (1) (with p = 1 in (2)) are determined by thescalar advance-delay differential equation

d2x1

dτ 2 + V ′(x1) = γ (x1(τ − T )− 2x1 + x1(τ + T )). (3)

Iooss and Kirchgassner have studied small amplitude solutions of (3) in different param-eter regimes and have obtained in particular “nanopterons” consisting of a solitary wavesuperposed on an exponentially small oscillatory tail. The leading order part of thesesolutions (excluding their tail) coincides with approximate solutions obtained via theNLS equation.

However, the more general case when phase and group velocities are different hasremained open until now. More generally, different situations have been observed forthe existence of exact travelling breathers in various simpler models. On the one hand,exact travelling breathers can be explicitly computed in the integrable Ablowitz-Ladiklattice [1], and other examples of nonlinear lattices supporting exact travelling breatherscan be obtained using an inverse method [14]. On the other hand, travelling breathersolutions of the Ablowitz-Ladik lattice are not robust under various non-Hamiltonianreversible perturbations as shown in [7].

The aim of our study is to clarify the existence question of exact travelling breathersolutions in the Klein-Gordon lattice (1), in a case when the breather period and theinverse of its velocity are commensurate (we develop the results announced in [40]). Forfixed p ≥ 2, problem (1)–(2) reduces to the p-dimensional system of advance-delaydifferential equations

d2

dτ 2

x1...

xn...

xp

+

V ′(x1)...

V ′(xn)...

V ′(xp)

= γ

x2(τ )− 2x1(τ )+ xp(τ + T )...

xn+1(τ )− 2xn(τ )+ xn−1(τ )...

x1(τ − T )− 2xp(τ)+ xp−1(τ )

. (4)

For the sake of simplicity we restrict ourselves to the case p = 2 in (4). The generalcase p ≥ 2 is analyzed in a work in progress. The latter is technically more difficult butthe approach used in our paper works as well.

54 G. James, Y. Sire

We analyze small amplitude solutions of (4) (with p = 2) using the method devel-oped by Iooss and Kirchgassner [22] in the context of travelling waves (see [20] foran application of this method to Fermi-Pasta-Ulam lattices). The method is based on areduction to a center manifold in the infinite dimensional case as described in references[27, 33, 46]. System (4) is rewritten as a reversible evolution problem in a suitable func-tional space, and considered for parameter values (T , γ ) near a critical curve where theimaginary part of the spectrum consists of a pair of double eigenvalues and two pairsof simple ones. Close to this curve, the pair of double eigenvalues splits in two pairsof eigenvalues with opposite nonzero real parts, which opens the possibility of findinghomoclinic solutions to 0.

Near these parameter values, the center manifold theorem reduces the problem locallyto a reversible 8-dimensional system of differential equations. Thanks to an appropriatechoice of variables, the reduction procedure is similar to the case analyzed by Ioossand Kirchgassner [22]. However, the simplest homoclinic bifurcation yields in our casea higher-dimensional reduced system, with a supplementary pair of simple imaginaryeigenvalues.

The reduced system is put in a normal form which is integrable up to higher orderterms. In some regions of the parameter space, the truncated normal form admits revers-ible homoclinic orbits to 0, which bifurcate from the trivial state and correspond toapproximate solutions of (4). These approximate solutions coincide with spatially local-ized modulated plane waves obtained via the NLS equation. However, by analogy withresults of Lombardi [28] we conjecture that these solutions do not generically persistwhen higher order terms are taken into account in the normal form. To make a moreprecise statement fix V (x) = 1

2x2 + αx3 + βx4. We expect that a reversible solution of

the reduced equation homoclinic to 0 and close to a small amplitude homoclinic orbit ofthe truncated normal form might only exist if (T , γ, α, β) is chosen on a discrete collec-tion of codimension-m submanifolds of R

4 (m > 0). The codimension depends on thenumber of pairs of purely imaginary eigenvalues (i.e. the number of resonant phonons)in our parameter regime and symmetry assumptions. In our case (with two pairs of purelyimaginary eigenvalues, in addition to hyperbolic ones), we expect m = 2 when homo-clinic orbits to 0 correspond to travelling breather solutions of (1)–(2) (with p = 2), andm = 1 when homoclinic orbits to 0 correspond to solitary waves (homoclinic orbits to0 possess an additional symmetry in that case).

For general parameter values, instead of homoclinic orbits to 0 one can expectthe existence of reversible homoclinic orbits to exponentially small 2−dimensionaltori, originating from the two additional pairs of simple purely imaginary eigenvalues.These solutions should constitute the principal part of exact travelling breather solu-tions of (1) superposed on a small quasi-periodic oscillatory tail. However, in orderto obtain exact solutions one has to prove the persistence of the corresponding ho-moclinic orbits as higher order terms are taken into account in the normal form. Thisstep is non-trivial and would require to generalize results of Lombardi [28] availablewhen one pair of simple imaginary eigenvalues is removed. The most intricate partof the problem is to obtain a sharp (exponentially small) estimate of the minimal tailsize of solutions. Another promising approach for obtaining such estimates is devel-oped in the recent work of Iooss and Lombardi [23] on polynomial normal formswith exponentially small remainder for analytic vector fields. However the applica-tion of their theory to our situation would require several nontrivial extensions (to the(iω0)

2iω1iω2 resonance and to systems with an additional infinite-dimensional hyper-bolic part).

Travelling Breathers in Klein-Gordon Chains 55

In this paper we prove the persistence of some homoclinic solutions in the case whenthe on-site potential V is even. Indeed, due to the additional invariance xn → −xn onecan find solutions of (1)–(2) (with p = 2) satisfying xn(τ ) = −xn−1(τ − T

2 ). Thesesolutions correspond to solutions of the normal form system possessing a particularsymmetry. For the normal form restricted to the associated (6-dimensional) invariantsubspace, results of Lombardi [28] are applicable since the linear part does not possessan extra pair of simple purely imaginary eigenvalues (the bifurcation corresponds to apair of double eigenvalues and a pair of simple ones). As a result the full normal formadmits homoclinic orbits to small periodic ones for near-critical parameter values (T , γ ).These solutions correspond to exact travelling breather solutions of (1) superposed ona small periodic oscillatory tail, which can be made exponentially small with respect tothe central oscillation size. The minimal tail size should be generically nonzero for agiven value of (T , γ ), but might vanish on a discrete collection of curves in the (T , γ )parameter plane. As a consequence, in a given system (1) (with fixed coupling constantγ and symmetric on-site potential V ), exact travelling breather solutions decaying to 0at infinity (and satisfying (2) for p = 2) might exist in the small amplitude regime, forisolated values of the breather velocity 2/T .

We insist on the fact that our study is local, and analytical results for large amplitudesolutions would be of interest. Results of this type exist for solitary waves or kinks inseveral one-dimensional nonlinear lattices (see [18, 17, 32, 41, 16]) but the problem isstill open for large amplitude travelling breather solutions.

The paper is organized as follows. In Sect. 2 we formulate (1)-(2) as an evolutionproblem in an infinite-dimensional Banach space. Sections 3 and 4 are devoted to thelinearized problem (spectral study, optimal regularity result) and the reduction to a cen-ter manifold. In Sect. 5 we study the reduced equation and describe its small amplitudehomoclinic solutions when higher-order terms are neglected. These terms are taken intoaccount in the even-potential case. Section 6 describes the corresponding leading-ordertravelling breather solutions of the Klein-Gordon system, and exact solutions (with smalloscillatory tails) in the case of even potentials.

2. Formulation of the Problem

In this section, we formulate the initial problem (1)–(2) in an appropriate way.The case p = 2 in (2) leads to the following system:

d2

dτ 2

[x1x2

]+[V ′(x1)

V ′(x2)

]= γ

[x2(τ )− 2x1(τ )+ x2(τ + T )

x1(τ )− 2x2(τ )+ x1(τ − T )

]. (5)

Note that travelling wave solutions of (1) satisfying xn(τ ) = xn−1(τ − T/2) are par-ticular solutions of (2) with p = 2. Consequently, the solutions considered in our caseinclude those found by Iooss and Kirchgassner [22].

We shall analyze small amplitude solutions of (5) using the center manifold reductionmethod introduced by Iooss and Kirchgassner [22] in the context of reversible advance-delay differential equations. For this purpose, one has to make a convenient choice ofvariables which allows us to recover some essential estimates in their reduction process(optimal regularity result).

We rescale (5) using t = τT

and consider the new variable (u1(t), u2(t)) = (x1(τ ),

x2(τ + T2 )). This yields

56 G. James, Y. Sire

xn(τ ) = u1(τ

T− n− 1

2) if n is odd,

xn(τ ) = u2(τ

T− n− 1

2) if n is even. (6)

With this change of variables, we have

d2

dt2

[u1u2

]+ T 2

[V ′(u1)

V ′(u2)

]= γ T 2

[u2(t − 1

2 )− 2u1(t)+ u2(t + 12 )

u1(t + 12 )− 2u2(t)+ u1(t − 1

2 )

]. (7)

Note that solutions of (7) with u1 = u2 correspond to travelling wave solutions of (1)satisfying xn(τ ) = xn−1(τ − T

2 ).As in [22] we set U = (u1, u2, u1, u2, X1(t, v),X2(t, v))

T , where v ∈ [−1/2, 1/2]andX1(t, v) = u1(t+v),X2(t, v) = u2(t+v). We define the following trace operators:

δ1/2Xi(t, v) = Xi(t, 1/2), (8)

δ−1/2Xi(t, v) = Xi(t,−1/2). (9)

Furthermore, we assume V analytic in a neighborhood of 0, with the following Taylorexpansion at x = 0:

V (x) = 1

2x2 − a

3x3 − b

4x4 + h.o.t. (10)

We can write the system (7) as an evolution problem

dU

dt= LU + F(U) (11)

with L given by

L =

0 0 1 0 0 00 0 0 1 0 0α1 0 0 0 0 α2(δ1/2 + δ−1/2)

0 α1 0 0 α2(δ−1/2 + δ1/2) 00 0 0 0 ∂v 00 0 0 0 0 ∂v

, (12)

α2 = T 2γ and α1 = −T 2(1 + 2γ ). The nonlinear operator F is given by

F(U) = T 2(0, 0, f (u1), f (u2), 0, 0)T (13)

and

f (u) = au2 + bu3 + h.o.t. (14)

We now write (11) in appropriate function spaces. For this purpose we introduce theBanach spaces

H = R4 × (C0[−1/2, 1/2])2, (15)

D ={U ∈ R

4 × (C1[−1/2, 1/2])2/X1(0) = u1, X2(0) = u2

}. (16)

Travelling Breathers in Klein-Gordon Chains 57

The operator L maps D into H continuously, F : D → D is Ck−1 with F(U) =O(‖U‖2

D).

We observe that the symmetry R on H defined by

R(u1, u2, ξ1, ξ2, X1(v),X2(v))T = (u1, u2,−ξ1,−ξ2, X1(−v),X2(−v))T

satisfies (L+F) ◦R = −R(L+F). Therefore, if U is a solution of (11) then RU(−t)is also a solution, i.e. the system (11) is reversible under R. This property is due tothe invariance t → −t of (7). A solution U of (11) is said to be reversible under R ifRU(−t) = U(t) for all t ∈ R. Reversible solutions under R correspond to solutions of(1)–(2) satisfying x−n(−τ − T ) = xn(τ ).

In addition, note that the permutational symmetry

S(u1, u2, ξ1, ξ2, X1, X2)T = (u2, u1, ξ2, ξ1, X2, X1)

T (17)

commutes with L+ F . As we observed previously, travelling wave solutions (i.e. solu-tions of (1) satisfying xn(τ ) = xn−1(τ − T/2)) appear as fixed points of S.

This additional invariance implies that R1 = R S = S R is also a reversibility sym-metry for Eq. (11). Reversible solutions under R1 correspond to solutions of (1)–(2)satisfying x−2n(−τ − T/2) = x2n+1(τ ).

The problem (11) is ill-posed as an initial value problem in D. Nevertheless, it is pos-sible to construct bounded solutions for all t ∈ R. Using the method developed in [22],we are able to reduce (11) locally to a finite dimensional system of ordinary differentialequations. The dimension of this reduced system depends on the bifurcation parametersγ and T (we shall fix T > 0 since Eq. (11) is even in T ). In the next section, we describethe spectrum of L in various parameter regions.

3. Spectral Problem

The linear operator L is closed in H with domain D and has a compact resolvent. Itfollows that its spectrum consists of isolated eigenvalues σ with finite multiplicities.

Let us compute the eigenvalues of L. Solving LU = σ U with

U = (u1, u2, ξ1, ξ2, X1, X2)T

leads to the equationA(u1, u2)

T = 0,

where

A =(σ 2 + T 2(1 + 2γ ) −2T 2γ cosh(σ/2)−2T 2γ cosh(σ/2) σ 2 + T 2(1 + 2γ )

).

The dispersion relation detA = 0 reads

N(σ, T , γ ) := (σ 2 + T 2(1 + 2γ ))2 − 4(γ T 2)2 cosh2(σ/2) = 0. (18)

The spectrum of L is then given by the roots ofN(σ, T , γ ) = 0. Since L has real coeffi-cients and due to the reversibility, the spectrum is invariant under the reflection on thereal and the imaginary axis.

We need basic properties of the spectrum in order to apply the reduction method [22].As in reference [22], L is not bi-sectorial and the central part (σ = iq) of its spectrumis isolated from the hyperbolic part (σ �= iq). More precisely, the following result canbe obtained as in [22], p. 443.

58 G. James, Y. Sire

Lemma 3.1. For all (γ, T ) ∈ R2+, there exists p0 such that all eigenvalues σ = p+ iq

of L with p �= 0 satisfy |p| ≥ p0.

For the central part of the spectrum (σ = iq), the dispersion relation reads

(−q2 + T 2(1 + 2γ ))2 = 4(γ T 2)2 cos2(q/2). (19)

In what follows we study the solutions of (19). Since (19) is even in q, we restrictourselves to the case q ≥ 0.

3.1. Spectrum on the imaginary axis for γ T 2 < 4. The spectrum ofL on the imaginaryaxis has a particularly simple structure for γ T 2 < 4. From the previous relation wededuce two cases:

T 2(1 + 2γ )− q2 = ±2γ T 2 cos(q/2). (20)

Case + in (20). We consider the equation

T 2(1 + 2γ )− q2 = 2γ T 2 cos(q/2). (21)

This equation can be written

T 2 = q2 − 4γ T 2 sin2(q/4). (22)

We now consider (T 2, α2) as new parameters (recall α2 = γ T 2). Equation (22) reads

T 2 = fα2(q) = q2 − 4α2 sin2(q/4). (23)

If α2 < 4, fα2 : [0,+∞[→ R+ is a strictly increasing function of q and Eq. (23) yields

q = f−1α2(T 2). This proves the existence of a pair of simple eigenvalues σ = ±if−1

α2(T 2)

for γ T 2 < 4. The corresponding eigenvectors V, V read

V = (1, 1, iq, iq, eiqv, eiqv)T .

Note that R V = V and S V = V .

Case – in (20). We consider the equation

T 2(1 + 2γ )− q2 = −2γ T 2 cos(q/2). (24)

In this case, we have

T 2 = gα2(q) = q2 − 4α2 cos2(q/4).

If α2 < 4, gα2 : [0,+∞[→ R is a strictly increasing function of q and then q =g−1α2(T 2).

This proves the existence of another pair of simple eigenvalues σ = ±ig−1α2(T 2) for

γ T 2 < 4. The corresponding eigenvectors V, V read

V = (−1, 1,−iq, iq,−eiqv, eiqv)T .We observe that R V = V and S V = −V .

Note that if−1α2(T 2) = ig−1

α2(T 2) = i(2k + 1)π for T 2(1 + 2γ ) = (2k + 1)2π2

(k ∈ N). In this case, the two pairs of eigenvalues collide, yielding a pair of doublesemi-simple eigenvalues (with eigenvectors having different symmetries).

In what follows we extend the spectral study to the whole parameter space. In partic-ular we shall consider the occurrence of double and triple purely imaginary eigenvalues.

Travelling Breathers in Klein-Gordon Chains 59

3.2. Double and triple eigenvalues on the imaginary axis. For having (at least) doublepurely imaginary eigenvalues, we have to verify (19) and dN(iq,T ,γ )

dq= 0, i.e

2q(−q2 + T 2(1 + 2γ )) = (γ T 2)2 sin(q). (25)

Moreover, iq is a triple eigenvalue when q satisfies (19),(25) and the following equation

( d2N(iq,γ,T )

dq2 = 0):

−6q2 + 2T 2(1 + 2γ ) = (γ T 2)2 cos(q). (26)

The following lemma gives a description of the set of double and triple eigenvalues onthe imaginary axis, as a function of (γ, T ) ∈ R

2+. These results are sketched in Fig. 1.

Lemma 3.2. Consider the curve� parametrized by (T (q), γ (q))with q ∈ R+ and T , γ

defined by the system (19)–(25). This curve (which we call a bifurcation curve) is givenby:if q ∈ [4kπ, (2k + 1)2π ] (for an integer k ≥ 1),

T 2 = q2 − 4q tan(q/4), (27)

γ = 2q

T 2 sin(q/2), (28)

if q ∈ [(2k − 1)2π, 4kπ ] (k ≥ 1),

Σ1

Σ3

Σ3

Σ5

γ Τ2=4

γ

0

Σ4

Σ1 Σ2 Σ3 Σ4Σ0 Σ5.. .... ......Σ6 ....

x

π π π

ΤW Σ7

Σ7

Σ8

x

Σ7

....

x

x

Σ9

Σ9

. . x

Σ8

2 4

Σ0Σ0

Σ8

Σ8

Σ4

Σ5

. . x

x. . xΣ11

Σ11Σ11

Σ6

x

x. .

Σ12

Σ12

. *

*. Σ13

Σ6

Σ10

Σ10

Σ4

6κ=1 κ=2 κ=3κ=0

Σ1

Σ13

Σ2Σ10

Σ8

Σ11

Σ14

T

x Double eigenvalue

Simple eigenvalue

* Triple eigenvalue

.

Σ15

Σ12

Σ6

Σ6

Σ4

Σ4

Σ4

Σ5

Σ5

Σ16

TP

TP

Σ10

Σ11

Σ16Σ15Σ14 xx

x

..

. x. ..

(1+2γ)=(2κ+1)Τ2 2 π 2

Σ1Σ1

Σ8

x

x

. .

. . . .

Fig. 1. Bifurcation curves and purely imaginary eigenvalues of L (upper half complex plane). “TP”(respectively “TW”) stands for the curves corresponding mainly to pulsating travelling wave (respec-tively travelling wave) bifurcations. The bold line corresponds to the subset �

60 G. James, Y. Sire

T 2 = q2 + 4q

tan(q/4), (29)

γ = − 2q

T 2 sin(q/2). (30)

The range of q is determined by the condition T 2 > 0. We denote by �k the restriction of� to the interval q ∈ [2kπ, 2(k + 1)π ]. The curve � lies in the parameter region whereγ T 2 > 4.

For (T , γ ) ∈ � (except on a countable set of points ), the spectrum of L on theimaginary axis consists of a pair of double non-semi-simple eigenvalues ±i q and atleast two distinct pairs of simple eigenvalues.

The set of exceptional parameter values consists of the following types of points:

• Cusps on � correspond to the existence of a pair of triple eigenvalues ±iq (Jordanblock of index 3) satisfying tan(q/2) = q/2 and a pair of simple eigenvalues.• The point of tangent intersection between �k and the curve T 2(1 + 2γ ) = (2k +1)2π2 leads to the existence of a pair of triple eigenvalues (with a two-dimensionaleigenspace) and a pair of simple eigenvalues.• A point of transverse intersection between�m and a curveT 2(1+2γ ) = (2k+1)2π2

(k ∈ N) leads to the existence of two pairs of double eigenvalues (one being semi-simple and the other non-semi-simple), and at least one pair of simple eigenvaluesif m �= k.• Double points on� correspond to the existence of two pairs of double non semi-sim-ple eigenvalues, and pairs of simple eigenvalues, depending on the parameter region.

Proof. First, we divide (19) by (25) to obtain the following equation:

T 2(1 + 2γ ) = q2 + 4q

tan(q/2). (31)

Substituing the expression for T 2(1 + 2γ ) in (25), we obtain

γ = 2q

T 2| sin(q/2)| . (32)

We have to consider two cases : sin(q/2) > 0 and sin(q/2) < 0.Fixing γ = 2q

T 2 sin(q/2)in (31) yields

T 2 = q2 − 4q tan (q/4). (33)

In the same way, fixing γ = − 2qT 2 sin(q/2)

in (31) leads to

T 2 = q2 + 4q

tan(q/4). (34)

Furthermore, Eq. (32) shows that γ T 2 > 4.The spectrum of L on the imaginary axis as a function of γ, T is sketched in Fig. 1.

The spectrum outside � is obtained by continuity arguments.

Travelling Breathers in Klein-Gordon Chains 61

We note that for T 2(1 + 2γ ) = (2k+ 1)2π2, k ∈ N, q∗ = (2k+ 1)π is a solution of(20) for both cases + and −. Therefore, ±iq∗ = ±i(2k+1)π is a pair of at least doubleeigenvalues.

One can check that �k has a tangent intersection with the curve T 2(1 + 2γ ) =(2k+ 1)2π2 at the point (T , γ ) = (T (q∗), γ (q∗)). Moreover, Eq. (26) is satisfied at thispoint and consequently iq∗ is a triple eigenvalue of L (one can check that the associatedeigenspace is two-dimensional). The existence of another pair of simple eigenvaluesfollows by a continuity argument.

Moreover, one can show that �k has only one other (transverse) intersection with thecurve T 2(1 + 2γ ) = (2k + 1)2π2, at a point (T , γ ) = (T (q0), γ (q0)) with q0 �= q∗. Inthis case one has two pairs of double eigenvalues (iq∗ being semi-simple and iq0 non-semi-simple). Similar intersections between�m (m �= k) and T 2(1+2γ ) = (2k+1)2π2

lead to extra pairs of simple eigenvalues.Finally, for q �= (2k + 1)π Eqs. (31),(32) and (26) lead to

tan(q/2) = (q/2). (35)

In any fixed interval [2kπ, (2k + 1)π ] (k ≥ 1) this equation has a unique solution q(which determines γ, T uniquely). This solution corresponds to a triple eigenvalue iq(and one has a Jordan block of index 3). Such triple eigenvalues appear as cusp pointsof the bifurcation curve (( dT

dq) and ( dγ

dq) vanish at q = q). �

Remark. Since our bifurcating solutions include the travelling waves found by Ioossand Kirchgassner [22], it is interesting to compare our bifurcation diagram with the oneof reference [22].

More precisely, there exist travelling wave solutions of (1)–(2) (with p = 2) satisfy-ing

xn−1(τ − T

2) = xn(τ ). (36)

In order to establish a comparison of Lemma 3.2 with reference [22], we replace q by2q in the parametrization of �. This yields

γ = 4q

T 2| sin(q)| , (37)

and if q ∈ [2kπ, (2k + 1)π ]

T 2 = 4q2 − 8q tan(q/2), (38)

otherwise

T 2 = 4q2 + 8q

tan(q/2). (39)

Now replacing T by 2T in (38) yields exactly the parametrization of the bifurcationcurve given on p. 443 in [22]. Consequently, small amplitude solutions which bifurcatein the neighborhood of �2k include travelling wave solutions of reference [22]. Thesesolutions can be combined with an additional mode corresponding to an extra pair ofsimple eigenvalues on the imaginary axis.

62 G. James, Y. Sire

On the contrary, small amplitude solutions which bifurcate in the neighborhood of�2k+1 mainly consist (apart from spatially periodic travelling waves) of pulsating trav-elling waves not described in reference [22].

In what follows, we define � as the subset of � such that the central part of thespectrum is�0 = {±iq1,±iq2,±iq0}, where ±iq0 is a pair of non semi-simple doubleeigenvalues and ±iq1,±iq2 two pairs of simple ones (� corresponds to the bold line inFig. 1). One can check the following properties.

Lemma 3.3. Fix (T , γ ) ∈ � and let V0,V1,V2 be the eigenvectors associated to iq0,iq1, iq2 respectively. Denote by V0 the generalized eigenvector associated to iq0. Theeigenvectors can be chosen in the following way:

V1 = (−1, 1,−iq1, iq1,−eiq1v, eiq1v)T , V2 = (1, 1, iq2, iq2, eiq2v, eiq2v)T ,

V0 = (ε, 1, εiq0, iq0, εeiq0v, eiq0v)T , V0 = (0, 0, ε, 1, εveiq0v, veiq0v)T ,

where ε = −1 if q0 ∈ [(2k − 1)2π, 4kπ ] and ε = 1 if q0 ∈ [4kπ, (2k + 1)2π ] (k ≥ 1).Moreover these eigenvectors satisfy

RV0 = V0, RV1 = V1, RV2 = V2, RV0 = −V0,

SV0 = ε V0, SV1 = −V1, SV2 = V2, SV0 = εV0.

4. Optimal Regularity Problem and Reduction on a Center Manifold

In this section we fix (T , γ ) ∈ �, compute the spectral projection on the hyperbolicsubspace (invariant subspace underL corresponding to the hyperbolic spectral part) andprove an optimal regularity result for the associated inhomogeneous linearized equation.This result is a crucial assumption for applying center manifold reduction theory [46].Our proof closely follows the method given in [22].

We call P0, P1, P2 respectively the spectral projection on the 4-dimensional invariantsubspace associated to ±iq0, on the 2-dimensional subspace corresponding to ±iq1, onthe 2-dimensional subspace corresponding to ±iq2. We also define P = P0 + P1 + P2(spectral projection on the 8-dimensional central subspace) and use the notations Dh =(I − P)D, Hh = (I − P)H, Dc = PD, Uh = (I − P)U . The affine linearized systemon Hh reads

dUh

dt= LUh + Fh(t), (40)

where F(t) = (0, 0, f1(t), f2(t), 0, 0)T lies in the range of the nonlinear operator (13).We shall note Uh = (uh1, u

h2, ξ

h1 , ξ

h2 , X

h1 (v),X

h2 (v))

T .Our aim is to check the optimal regularity property of Eq. (40) (see [46], property

(ii) p.127). This property can be stated as follows. We introduce the following Banachspace, for a given Banach space Z and α ∈ R

+:

Eαj (Z) ={f ∈ Cj (R, Z) ‖f ‖j = max

0≤k≤jsupt∈R

e−α|t ||Dkf (t)| < ∞}. (41)

We need to check that system (40) admits a unique solutionUh inEα0 (Dh)⋂Eα1 (Hh) for

0 ≤ α < α0 (for some α0 > 0), the operatorKh : Eα0 (R2) → Eα0 (Dh), (f1, f2) → Uh

being bounded.As the linear operator L is not bi-sectorial, we do not have classical estimates on its

resolvent and have to compute Uh explicitly.

Travelling Breathers in Klein-Gordon Chains 63

4.1. Computation of the spectral projection on the hyperbolic subspace. The spectralprojection on the central subspace is defined by the Dunford integral

P = 1

2iπ

∫C

(σ I − L)−1dC, (42)

where C is a regular curve surrounding ±iq1,±iq2,±iq0. The spectral projection onthe hyperbolic subspace is Ph = I − P .

We shall use the following result for computing Ph.

Lemma 4.1. Let h(z) = f (z)g(z)

be a function of z ∈ C. Assume the function f (z) is entireand the function g(z) admits a double pole at z = z0. Then the residue of h at z = z0 isgiven by

Res(h, z0) = 2f ′(z0)g′′(z0)− 2

3f (z0)g′′′(z0)

g′′(z0)2. (43)

In the following lemma, we compute the spectral projection on the hyperbolic subspaceof a vector F lying in the range of the nonlinear operator (13).

Lemma 4.2. Let F ∈ D be a vector of the type F = (0, 0, f1, f2, 0, 0)T . Then theprojection of F on the hyperbolic subspace reads

Fh = (0, 0, k3f1 + k4f2, k5f1 + k6f2, k7(v)f1 + k8(v)f2, k9(v)f1 + k10(v)f2)T ,

(44)

where k3, k4, k5, k6 ∈ R and k7, k8, k9, k10 ∈ C∞([−1/2, 1/2]) depend on γ, T .

Proof. We first compute the resolvent of L. One has to solve (σ I − L)U = F , whichyields the system

ξ1 = σu1, ξ2 = σu2,

(σ 2 − α1)u1 − 2α2 cosh(σ/2)u2 =f1, (σ2 − α1)u2 − 2α2 cosh(σ/2)u1 =f2, (45)

X1(v) = u1eσv,X2(v) = u2e

σv,

with U = (u1, u2, ξ1, ξ2, X1(v),X2(v))T . We have then

(u1u2

)= 1

N(σ, γ, T )

((σ 2 − α1)f1 + 2α2 cosh(σ/2)f2(σ 2 − α1)f2 + 2α2 cosh(σ/2)f1

). (46)

Now we compute the spectral projection P1. Since σ = iq1 is a simple root of (18), onehas

Res(u1, iq1) = i(−(q21 + α1)f1 + 2α2 cos(q1/2)f2)

4q1(q21 + α1)+ 2α2

2 sin(q1),

Res(u2, iq1) = i(−(q21 + α1)f2 + 2α2 cos(q1/2)f1)

4q1(q21 + α1)+ 2α2

2 sin(q1).

Denoting (P1F)i the ith component of P1F , we get consequently

(P1F)1 = Res(u1, iq1)+ Res(u1,−iq1) = 0,

(P1F)2 = Res(u2, iq1)+ Res(u2,−iq1) = 0.

64 G. James, Y. Sire

In the same spirit

(P1F)3 = −2q1(−(q21 + α1)f1 + 2α2 cos(q1/2)f2)

4q1(q21 + α1)+ 2α2

2 sin(q1),

(P1F)4 = −2q1(−(q21 + α1)f2 + 2α2 cos(q1/2)f1)

4q1(q21 + α1)+ 2α2

2 sin(q1),

(P1F)5 = −2 sin(q1v)(−(q21 + α1)f1 + 2α2 cos(q1/2)f2)

4q1(q21 + α1)+ 2α2

2 sin(q1),

(P1F)6 = −2 sin(q1v)(−(q21 + α1)f2 + 2α2 cos(q1/2)f1)

4q1(q21 + α1)+ 2α2

2 sin(q1),

which completes the computation ofP1F . The computations are identical for the spectralprojection P2 associated to ±iq2. For computing the spectral projection P0 associ-ated to the double eigenvalues ±iq0, we use formula (43). These computations lead toEq. (44). �

4.2. Resolution of the affine equation for bounded functions of t . We first solve (40) inthe spaces Eαj with α = 0, i.e. we consider bounded functions of t (note that E0

j (H) =Cjb (H)). Fixing α = 0 will allow us to take the Fourier transform in time of the system

in the tempered distributional space S′(R).From (40), we directly deduce

Xh1 (t, v) = uh1(t + v)+∫ v

0(k7(s)f1(t + v − s)+ k8(s)f2(t + v − s))ds (47)

= uh1(t + v)+∫ t+v

t

(k7(t + v − s)f1(s)+ k8(t + v − s)f2(s))ds, (48)

Xh2 (t, v) = uh2(t + v)+∫ v

0(k9(s)f1(t + v − s)+ k10(s)f2(t + v − s))ds (49)

= uh2(t + v)+∫ t+v

t

(k9(t + v − s)f1(s)+ k10(t + v − s)f2(s))ds (50)

(this expression comes from the two last equations of the affine linear system and fromconditions X1(0, t) = u1(t),X2(0, t) = u2(t)).

From the previous equations and the fact that (ki)i=7..10 and their derivatives arebounded functions of v, we deduce that

‖Xh1‖E00 (C

1[−1/2,1/2]) ≤ ‖uh1‖E01+ C(‖f1‖E0

0+ ‖f2‖E0

0), (51)

‖Xh2‖E00 (C

1[−1/2,1/2]) ≤ ‖uh2‖E01+ C(‖f1‖E0

0+ ‖f2‖E0

0). (52)

We now have to estimate uh1, uh2, ξ

h1 , ξ

h2 . Taking the Fourier transform in time of the

system (40) in the tempered distributional space S′(R), we have

(ik − L)Uh = Fh. (53)

Travelling Breathers in Klein-Gordon Chains 65

We deduce

ξ h1 = ikuh1,

ξ h2 = ikuh2,

Xh1 = eikvuh1 + f1

∫ v

0eik(v−s)k7(s)ds + f2

∫ v

0eik(v−s)k8(s)ds,

Xh2 = eikvuh2 + f1

∫ v

0eik(v−s)k9(s)ds + f2

∫ v

0eik(v−s)k10(s)ds.

For uh1, uh2, we have

−(k2 + α1)uh1 − 2α2 cos(k/2)uh2 = ˆ(Fh)3 + C1(k)f1 + C2(k)f2, (54)

−(k2 + α1)uh2 − 2α2 cos(k/2)uh1 = ˆ(Fh)4 +D1(k)f1 +D2(k)f2,

where

ˆ(Fh)3 = k3f1 + k4f2,

ˆ(Fh)4 = k5f1 + k6f2,

andCi ,Di areC∞ functions of k, beingO(1/|k|) as k → ±∞. Solving the system (54)leads to

N(ik, γ, T )

(uh1

uh2

)=(h1h2

), (55)

where

h1 = −(k2 + α1)[(k3 + C1(k))f1 + (k4 + C2(k))f2] + 2α2 cos(k/2)[(k5 +D1(k))f1

+(k6 +D2(k))f2],

h2 = 2α2 cos(k/2)[(k3 + C1(k))f1 + (k4 + C2(k))f2] − (k2 + α1)[(k5 +D1(k))f1

+(k6 +D2(k))f2].

Equation (55) can be written

N(ik, γ, T )

(uh1 + H1f1 + H2f2

uh2 + G1f1 + G2f2

)=(

00

). (56)

As the operator (ik − Lh)−1 is analytic in a strip around the real axis, we deduce that

H1, H2, G1, G2 are analytic functions in this strip. Moreover, H1, H2, G1, G2 areO( 1k2 )

as k → ±∞ due to the fact that N(ik, γ, T ) = O(k4) and h1, h2 are O(k2) as k →±∞. Since N(iqj , γ, T ) = 0, N ′(iq0, γ, T ) = 0 and N ′(iq1, γ, T ), N ′(iq2, γ, T ),N ′′(iq0, γ, T ) do not vanish, Eq. (56) yields

uh1 + H1f1 + H2f2 = a+1 δiq1 + a−

1 δ−iq1 + a+2 δiq2 + a−

2 δ−iq2

+a+0 δiq0 + a−

0 δ−iq0 + b+0 δ

′iq0

+ b−0 δ

′−iq0

, (57)

66 G. James, Y. Sire

uh2 + G1f1 + G2f2 = c+1 δiq1 + c−1 δ−iq1 + c+2 δiq2 + c−2 δ−iq2

+c+0 δiq0 + c−0 δ−iq0 + d+0 δ

′iq0

+ d−0 δ

′−iq0

. (58)

Furthermore, k → (1+|k|2)1/2Hi and k → (1+|k|2)1/2Gi belong toL2(R). Therefore,using the inverse Fourier Transform and Lemma 3, p.448 of [22], there exist Gi,Hi ∈H 1δ (R) (i.e eδ|t |Hi ∈ H 1(R), eδ|t |Gi ∈ H 1(R), δ > 0 small enough) such that Gi, Hi

are the unique Fourier transforms of Gi,Hi . We have the following estimates

‖dH1

dt∗ f1‖C0

b= sup

t∈R

|∫

R

dH1

dt(t − s)f1(s)ds|

≤ C(δ)‖f1‖C0b‖H1‖H 1

δ (R). (59)

The same estimate is valid for dH2dt

∗ f2,dG1dt

∗ f1,dG2dt

∗ f2.

Now we make the solution of (40) explicit. We set Uh = (uh1, uh2, ξ

h1 , ξ

h2 , X

h1 , X

h2 )T

and

uh1 = −H1 ∗ f1 −H2 ∗ f2,

uh2 = −G1 ∗ f1 −G2 ∗ f2,

ξ h1 = duh1

dt, (60)

ξ h2 = duh2

dt,

Xh1 (t, v) = uh1(t + v)+∫ v

0(k7(s)f1(t + v − s)+ k8(s)f2(t + v − s))ds,

Xh2 (t, v) = uh2(t + v)+∫ v

0(k9(s)f1(t + v − s)+ k10(s)f2(t + v − s))ds.

By construction, uh satisfies (40) andP ˆUh = 0 (henceP Uh = 0) for (f1, f2) ∈ Eα0 (R2)

with α < 0 (fi are analytic functions in a strip around the real axis). Since the compu-

tations are formally the same for α = 0, we have P ˆUh = 0 for α = 0, hence P Uh = 0

for α = 0.Moreover, we have

‖Uh‖C0b (Dh)

⋂C1b (Hh)

≤ C(‖f1‖C0b (R)

+ ‖f2‖C0b (R)

) (61)

due to estimates (51), (52), (59) (with analogous estimates on H2, Gi). For α = 0, weobtain uh1, u

h2 by adding to uh1, u

h2 the inverse Fourier transforms of Dirac measures, i.e.

uh1 = uh1 + a+1 e

iq1t + a−1 e

−iq1t + a+2 e

iq2t + a−2 e

−iq2t

+(a+0 + itb+

0 )eiq0t + (a−

0 − itb−0 )e

−iq0t , (62)

uh2 = uh2 + c+1 eiq1t + c−1 e

−iq1t + c+2 eiq2t + c−2 e

−iq2t

+(c+0 + itd+0 )e

iq0t + (c−0 − itd−0 )e

−iq0t . (63)

Travelling Breathers in Klein-Gordon Chains 67

Since P Uh = 0, we have PUh = 0 if and only if

a±1 = a±

2 = c±1 = c±2 = b±0 = a±

0 = d±0 = 0. (64)

It follows that Uh = Uh. Finally, we have proved the following

Lemma 4.3. Assume F = (0, 0, f1, f2, 0, 0)T and f1, f2 ∈ C0b (R). Then the affine

linear system (40) has a unique bounded solution Uh ∈ C0b (Dh)

⋂C1b(Hh) and the

operator Kh : C0b (R

2) → C0b (Dh), (f1, f2) → Uh is bounded.

Remark. The first and second components of (44) vanish due to our choice of variables(u1, u2) in (7). This would not be the case using (x1, x2) and the proof of optimal reg-ularity results would require additional work (in this case Hi ,Gi are only O(1/|k|) ask → ±∞).

4.3. Affine equation in exponentially weighted spaces. The problem now is to extendLemma 4.3 to the case (f1, f2) ∈ Eα0 (R2), with α > 0 sufficiently close to 0. This hasbeen done in [22] by constructing a suitable distribution space, but the following lemmagives an alternative proof (see [34]).

Lemma 4.4. Consider Banach spaces D,Y and X such that: D ↪→ Y ↪→ X.Let L be a closed linear operator in X, of domain D, such that the equation

dU

dt= LU + f (65)

admits for any fixed f ∈ C0b (Y) a unique solution U = Kf in C0

b (D)⋂C1b(X), with in

addition K ∈ L(C0b (Y), C

0b (D)). Then there exists α0 > 0 such that if 0 ≤ α < α0, for

all f ∈ Eα0 (Y) the system (65) admits a unique solution in Eα0 (D)⋂Eα1 (X) with

‖U‖Eα0 (D) ≤ C(α)‖f ‖Eα0 (Y). (66)

Proof. Let f ∈ Eα0 (Y). We set: f (t) = f (t)cosh(αt) ∈ C0

b (Y) and U (t) = U(t)cosh(αt) . The

property U ∈ Eα0 (D)⋂Eα1 (X) is equivalent to U ∈ C0

b (D)⋂C1b(X). Furthermore, we

havedU

dt= LU + f − α tanh(αt)U .

This equation is equivalent to

U + αK(tanh(αt)U) = Kf . (67)

Equation (67) can be written

(I + αT )U = Kf ,

where T U = K(tanh(αt)U). We have then T ∈ L(C0b (D)) and ‖T ‖ ≤ ‖K‖. If 0 ≤

α < 1‖K‖ , I + αT is invertible in C0

b (D) and we have ‖(I + αT )−1‖ ≤ 11−α‖K‖ .

Therefore, (67) is equivalent to

U = (I + αT )−1Kf ∈ C0b (D)

68 G. James, Y. Sire

and ‖U‖C0b (D)

≤ ‖K‖1−α‖K‖‖f ‖C0

b (Y). Then, we have U = cosh(αt)U ∈ Eα0 (D) and

‖U‖Eα0 (D) ≤ ‖U‖C0b (D)

≤ ‖K‖1 − α‖K‖‖f ‖C0

b (Y)= 2

‖K‖1 − α‖K‖‖f ‖Eα0 (Y). (68)

This ends the proof. �Applying this result to our problem yields the following.

Proposition 4.5. There exists α0 > 0 such that for all F = (0, 0, f1, f2, 0, 0)T withf1, f2 ∈ Eα0 (R) and α ∈ [0, α0], the affine linear system (40) has a unique solu-tion Uh ∈ Eα0 (Dh)

⋂Eα1 (Hh). Moreover, the operator Kh : Eα0 (R

2) → Eα0 (Dh),(f1, f2) → Uh is bounded (uniformly in α ∈ [0, α0]).

4.4. Center manifold reduction. The above analysis shows that the assumptions of The-orem 3 of reference [46] (p. 133) are satisfied. Hence the reduction on a center manifoldis possible and we have the following result.

Theorem 4.1. Fix (T0, γ0) ∈ � and k ≥ 1. There exists a neighborhood U × V of(0, γ0, T0) in D × R

2 and a map ψ ∈ Ckb(Dc × R2,Dh) such that the following proper-

ties hold for all (γ, T ) ∈ V (with ψ(0, γ, T ) = 0,Dψ(0, γ0, T0) = 0).

• If U : R → D solves (11) and U(t) ∈ U ∀t ∈ R then Uh(t) = ψ(Uc(t), γ, T ) forall t ∈ R and Uc is a solution of

dUc

dt= LUc + PF(Uc + ψ(Uc, γ, T )). (69)

• If Uc : R → Dc is a solution of (69) with Uc ∈ Uc = PU ∀t ∈ R, then U =Uc + ψ(Uc, γ, T ) is a solution of (11).• The map ψ(., γ, T ) commutes with R and S. Moreover, the reduced system (69) isreversible under R and equivariant under S.

5. Study of the Reduced Equation

According to normal form theory (see e.g. [21]), one can perform a polynomial changeof variables Uc = Uc + Pγ,T (Uc) close to the identity which simplifies the reduced Eq.(69) and preserves its symmetries. In this section, we compute this normal form at order3 and give an explicit expression of a particular coefficient, which sign is essential forthe bifurcation of small amplitude homoclinic orbits.

5.1. Normal form computation. The linear operator L restricted to the eight-dimen-sional subspace Dc (denoted as Lc) has the following structure in the basis (V0, V0, V1,

V2, V0,¯V 0, V1, V2):

Lc =

iq0 1 0 0 0 0 0 00 iq0 0 0 0 0 0 00 0 iq1 0 0 0 0 00 0 0 iq2 0 0 0 00 0 0 0 −iq0 1 0 00 0 0 0 0 −iq0 0 00 0 0 0 0 0 −iq1 00 0 0 0 0 0 0 −iq2

.

Travelling Breathers in Klein-Gordon Chains 69

Moreover, the reversibility symmetryR and the symmetryS have the following structure.One has

R =

0 0 0 0 1 0 0 00 0 0 0 0 −1 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 11 0 0 0 0 0 0 00 −1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 0

.

Moreover, if (T0, γ0) ∈ �2k we have

S = diag(1, 1,−1, 1, 1, 1,−1, 1)

and (T0, γ0) ∈ �2k+1 yields

S = diag(−1,−1,−1, 1,−1,−1,−1, 1).

Consequently, our reduced equation has many similarities with the one considered in[22], Sect. 6 (case of a (iq0)

2(iq2) resonance). The only differences are an extra pair ofsimple purely imaginary eigenvalues ±iq1 for the linearized operator, and the additionalsymmetry S. More precisely, the truncated normal form considered in [22] has a sym-metry similar to S (which follows in fact from a phase invariance), but this symmetry isbroken for the full system. In our case, Theorem 4.1 ensures that the full reduced systemis equivariant under S.

It follows that the normal form has a structure similar to the one obtained in [22].To compute the normal form, we exclude points of�which are close to points where

sq0 +rq1 +r ′q2 = 0 for s, r, r ′ ∈ Z and 0 < |s|+|r|+|r ′| ≤ 4 (such values correspondto strong resonances), and denote this new set as �0. The normal form computation isvery similar to [22] (Sect. 6 and Appendix 2), to which we refer for details.

In what follows we set Uc = AV0+BV0+CV1+DV2+AV 0+B V0+C V 1+DV 2.The normal form of (69) at order 3 is given in the following lemma.

Lemma 5.1. The normal form of (69) at order 3 reads

dA

dt= iq0A+ B + iAP(u1, u2, u3, u4)

+O((|A| + |B| + |C| + |D|)4),dB

dt= iq0B + iBP(u1, u2, u3, u4)+ AS(u1, u2, u3, u4)

+O((|A| + |B| + |C| + |D|)4), (70)dC

dt= iq1C + iCQ(u1, u2, u3, u4)+O((|A| + |B| + |C| + |D|)4),

dD

dt= iq2D + iDT (u1, u2, u3, u4)+O((|A| + |B| + |C| + |D|)4),

where

u1 = AA, u2 = CC, u3 = DD, u4 = i(AB − AB)

70 G. James, Y. Sire

and P,S,Q, T are polynomials with smooth parameter dependent real coefficients, for(T , γ ) in the neighborhood of �0.We have

P(u1, u2, u3, u4) = p1(γ, T )+ p2u1 + p3u2 + p4u3 + p5u4,

S(u1, u2, u3, u4) = s1(γ, T )+ s2u1 + s3u2 + s4u3 + s5u4,

Q(u1, u2, u3, u4) = q1(γ, T )+ q2u1 + q3u2 + q4u3 + q5u4, (71)

T (u1, u2, u3, u4) = t1(γ, T )+ t2u1 + t3u2 + t4u3 + t5u4,

where p1, s1, q1, t1 vanish on �0.

The truncated normal form (obtained by neglecting terms of orders 4 and higher) isintegrable with the following first integrals:

AB − AB, |B|2 −∫ |A|2

0S(x, |C|2, |D|2, i(AB − AB))dx, |C|2, |D|2. (72)

Note that if one fixes |C| = 0 and |D| = 0, the truncated normal form yields theclassical 1:1 resonance [24].

In what follows we describe some solutions of the truncated normal form. We shallconcentrate on the description of homoclinic solutions to the equilibrium 0, to a periodicor a quasi-periodic orbit, which may exist when Lc has 4 eigenvalues with nonzero realparts (perturbation of ±iq0). The existence of these homoclinic orbits is linked to thesign of the coefficient s2 in the polynomial S. The following section is devoted to itscomputation.

5.2. Computation of the coefficient s2. We choose (T0, γ0) ∈ �0. Equation (11) can beexpanded as

dU

dt= L0U + (γ − γ0)L

(1)U + (T − T0)L(2)U +M2(U,U)+M3(U,U,U)+ ....,

(73)

whereL0 is the linear operator for (T0, γ0) ∈ �0 andL(i) are linear operators. Moreover,Mj is a j−linear symmetric map satisfying

M2(U,U) = aT 20 (0, 0, u2

1, u22, 0, 0)T , (74)

M3(U,U,U) = bT 20 (0, 0, u3

1, u32, 0, 0)T . (75)

Using the Taylor expansion of the center manifold at (0, γ0, T0) we find

U = AV0 + BV0 + CV1 +DV2 + AV 0 + B V0 + C V 1 +DV 2 +∑(γ − γ0)

m(T − T0)nAr0Br0Cr1Dr2As0 Bs0 Cs1Ds2φ

(m,n)

r0 r0r1r2s0 s0s1s2. (76)

Using this expression and the normal form in Eq. (73), we find by identification at ordersA2, |A|2, A|A|2 (we omit the index (m, n) = (0, 0) in the notations)

(2iq0I − L)φ20000000 = M2(V0, V0),

−Lφ10001000 = 2M2(V0, V0),

ip2V0 + s2V0 + (iq0I − L)φ20001000 = 2M2(V0, φ20000000)+ 2M2(V0, φ10001000)+3M3(V0, V0, V0).

Travelling Breathers in Klein-Gordon Chains 71

The first two equations have a unique solution given by expressions (45), (46). Thelast equation yields the following compatibility condition (expression (45) reduces theproblem to a two-dimensional system)

(2 − q0

tan(q0/2))s2 = T 2

0 (6b + 8a2 − 4a2T 20

2γ0T20 cos(q0)− T 2

0 (1 + 2γ0)+ 4q20

). (77)

The other coefficients in (71) could be computed by identification in a similar way.

5.3. Description of small amplitude solutions for the normal form system. This sectiondescribes some reversible homoclinic solutions of the truncated normal form given inLemma 5.1. The problem of their persistence for the full system is discussed in differentcases.

We choose (γ, T ) ≈ (γ0, T0) ((T0, γ0) ∈ �0), in such a way that the linearizedoperator L has four symmetric eigenvalues close to ±iq0 and having non-zero real parts(s1(γ, T ) > 0 in (71)). We shall distinguish S-invariant and non-S-invariant solutions,where S is the permutational symmetry (17). We recall that S-invariant solutions corre-spond to travelling waves.

5.3.1. Solutions bifurcating at (T0, γ0) ∈ �2k .

• S-invariant homoclinic solutions and persistence problemsWe consider the normal form system (70) restricted to the invariant subspace Fix(S).In this case we have C = 0 and recover the (iq0)

2(iq2) resonance case as in [22].The subspace Fix(S) contains in particular the stable and unstable manifolds of 0.Provided s2(γ0, T0) < 0 and (γ, T ) ≈ (γ0, T0), the truncated normal form systemadmits homoclinic orbits to 0 withD = 0. In addition there exist homoclinic solutionsto small periodic orbits with D �= 0. These solutions are given by (α ≈ 0),

A(t) = r0(t)ei(q0t+ψ(t)+θ), B(t) = r1(t)e

i(q0t+ψ(t)+θ), D(t) = αei(q2t+ϕ2(t)+θ2),

where

r0(t) = (2(s1 + s4 α

2)

−s2 )1/2(cosh(t (s1 + s4α2)1/2))−1,

r1(t) = dr0

dt(t),

ψ(t) = (p1 + p4 α2)t + 2

p2

s2(s1 + s4α

2)1/2 tanh(t (s1 + s4α2)1/2),

ϕ2(t) = (t1 + t4α2) t + t2

∫ t

0r2

0 (τ ) dτ ,

and θ, θ2 ∈ R.

These orbits are reversible under R if one chooses θ and θ2 equal to 0 or π . In thiscase, the problem of their persistence for the full vector field (with additional nonreso-nance conditions on the eigenvalues) has been treated by Lombardi in [28]. Reversiblehomoclinic solutions to periodic orbits persist above a critical tail size α = αc, whichis exponentially small with respect to |A(0)| (size of “central” oscillations). This yieldsexact travelling wave solutions of the Klein-Gordon system [22], which converge towards

72 G. James, Y. Sire

periodic waves at infinity and have a larger amplitude at the center of the chain. On thecontrary, reversible homoclinic orbits to 0 should not persist generically for the fullnormal form (70) when higher order terms are taken into account [28]. In what followswe explain this statement in more detail and give a brief account of persistence andnonpersistence results obtained in [28].

Consider the normal form (70) restricted to the invariant subspace C = 0. We fix(T0, γ0) ∈ �0 ∩ �2k , with additional nonresonance conditions on the eigenvalues (see[28], p. 359) which are generically realized.We assume s2(γ0, T0) < 0 and s1(γ, T ) > 0.For simplicity we fix γ = γ0 and let T ≈ T0 vary. In the linearized system, 4 hyperboliceigenvalues have small real parts ±ν = O(|T−T0|1/2)with ν > 0 (we shall use ν insteadof T −T0 as a small parameter),O(1) imaginary parts ±iω0(ν) (ω0(0) = q0), and thereis in addition one pair of O(1) purely imaginary eigenvalues ±iω2(ν) (ω2(0) = q2).Using the following scaling (see [28], p. 364)

A(t) = σ ν A(νt), B(t) = σ ν2 B(νt), C(t) = ν3/2 C(νt), D(t) = ν3/2 D(νt)

with σ = (−2/s2)1/2, the normal form (70) can be written

dY

dt= N(Y, ν)+ R(Y, ν), (78)

where Y = (A, B,¯A,

¯B, C,

¯C, D,

¯D)T . The linearized system has the eigenvalues ±1±

iω0/ν and ±iω2/ν (a slow hyperbolic part coincides with fast oscillatory parts in thissystem). MoreoverN is a cubic polynomial in Y ,R contains higher order terms in Y andis O(ν) as ν → 0. The truncated system (with R = 0) has explicit reversible solutions±h homoclinic to 0, being O(1) as ν → 0 thanks to the scaling (the unscaled solutionshave been given above). The rescaled solution h has simple poles z = ±iπ/2 in thecomplex plane (one has r0(t) = 1/ cosh t in the above notations).

We start with some comments on the generic nonpersistence of reversible homoclinicorbits to 0 [28]. Setting Y = h+y, where the perturbation y is assumed reversible underR and homoclinic to 0, (78) can be rewritten in the form

dy

dt−DYN(h(t), ν) y = f (y, h(t), ν). (79)

Applying the Fredholm alternative, one obtains a compatibility condition (linked to theeigenvalues ±iω2/ν and reversibility) having the form

∫ +∞

0〈y∗(t), f (y(t), h(t), ν)〉 dt = 0, (80)

where the dual vector y∗ reads

y∗(t) = (0, 0,−ieiψr (t), 0, 0, ie−iψr (t)),

and ψr has the form ψr(t) = ω2t/ν + ν n(ν) tanh t . This yields a condition of the type

I (ν) = Im∫ +∞

0e−iω2t/ν g(y(t), h(t), t, ν) dt = 0, (81)

consisting of a bi-oscillatory integral in which the approximate homoclinic solution halso rotates at the high frequency ω0/ν.

Travelling Breathers in Klein-Gordon Chains 73

The usual way to check if I (ν) vanishes is to split the integral in two parts I (ν) =Me (ν)+ J (ν), where the Melnikov function

Me (ν) = Im∫ +∞

0e−iω2t/ν g(0, h(t), t, ν) dt (82)

depends on the explicitly known function h and is usually expected (at least in classicalperturbation theory) to be the leading part of I (ν). One finds as ν → 0 (see [28], p. 397)

Me (ν) = ν−3/2 e−c/ν (�1 +O(ν))

(c > 0), hence Me (ν) is exponentially small. However, Me (ν) is not the leading partof I (ν) in our case. Indeed, fine estimation techniques [28] yield

I (ν) = ν−3/2 e−c/ν (�+O(ν1/4)) (83)

with � �= �1 in general. The reason is that h is the leading part of Y on R, but not nearthe poles of Y (close to z = ±iπ/2), and the leading part of Y near the poles is preciselythe relevant part for computing I (ν).

More precisely, the coefficient � in (83) is given by a complex integral, involv-ing a (not explicitly known) solution on the stable manifold of Y = 0, extended inthe complex plane and approximated near the poles ±iπ/2 at leading order (see [29,28]). As a consequence, analytical computations of � seem very difficult but numer-ical ones might be achieved. Moreover, an additional difficulty for obtaining estimate(83) has to be pointed out. Since center manifolds are not analytic in general (not evenC∞), one cannot work with the (a priori) non-analytic reduced Eq. (70). In order topreserve analyticity, one works directly with the evolution problem (11), splitted into aninfinite-dimensional hyperbolic part coupled with the normal form (70), whose prin-cipal part remains unchanged (see [28], p. 331). The same techniques as in the finite-dimensional case apply, because Lemma 4.3 and Eq. (60) give the necessary optimalregularity properties for the hyperbolic part of the linearized system (see [28], Sect. 8).

According to expression (83), if � �= 0 (which should be satisfied except for excep-tional choices of (T0, γ0) and V ) and T − T0 is sufficiently small, reversible homoclinicorbit to 0 close to ±h do not exist. Consequently, reversible homoclinic orbits to 0 shouldnot persist generically for the full normal form. This result needs several comments.

Firstly, it might happen that� = 0 for isolated values of (T0, γ0) ∈ �2k . In that case,one might expect the existence of a curve I (T , γ ) = 0 in the parameter plane (with(T , γ ) ≈ (T0, γ0)) on which the compatibility condition (81) is satisfied and reversiblehomoclinic orbits to 0 exist. However this situation is non-generic in the parameter plane.

Moreover, the above analysis only concerns reversible homoclinic orbits, and non-reversible homoclinic orbits to 0 might exist. In addition, homoclinic solutions aresearched in a small neighborhood of h in L∞(R), and reversible homoclinic orbitswith several loops (which do not satisfy this criteria) might exist as it is mentioned in[28]. Consequently, � �= 0 only implies the nonexistence of homoclinic orbits to 0 of acertain type when ν is small enough.

We end with some precisions about persistence of reversible solutions homoclinic toperiodic orbits. One can show [28] that for ν small enough and α in an interval of thetype

α ∈ (K1e−a/ν,K2), (84)

74 G. James, Y. Sire

(a > 0), Eq. (78) admits reversible solutions of the form

Yα,ν(t) = y(t)+ h(t)+Xα,ν(t + ϕ tanh (λ t)), (85)

where y is homoclinic to 0 and Xα,ν is a reversible time-periodic solution of (78) withamplitudeα. The frequency ofXα,ν is close toω2/ν and its principal part (in the unscaledform) has been given above (case A = B = C = 0 in the truncated normal form (70)).Very roughly speaking, looking for a solution of the form (85) yields a compatibilitycondition of the type

α sin ϕ = Im∫ +∞

0e−iω2t/ν G(y(t), h(t), t, α, ν, ϕ) dt, (86)

which holds for a suitable choice of the phase ϕ = ϕ(α, ν) provided (84) is satisfied,due to the exponential smallness of the right side of (86) (see [28], Sect. 9.3 for moredetails).

• Non S-invariant solutionsProvided s2(γ0, T0) < 0 and (γ, T ) ≈ (γ0, T0), the truncated normal form admitshomoclinic solutions to small quasi-periodic orbits, which are not invariant underS due to the additional component C(t). These solutions are given by (α, β ≈ 0,β �= 0)

A(t) = r0(t)ei(q0t+ψ(t)+θ), B(t) = r1(t)e

i(q0t+ψ(t)+θ),C(t) = βei(q1t+ϕ1(t)+θ1), D(t) = αei(q2t+ϕ2(t)+θ2), (87)

where (s = s1 + s4 α2 + s3 β

2)

r0(t) = (2s

−s2 )1/2(cosh(t s1/2))−1,

r1(t) = dr0

dt(t),

ψ(t) = (p1 + p4 α2 + p3 β

2)t + 2p2

s2s1/2 tanh(t s1/2),

ϕ1(t) = (q1 + q4α2 + q3 β

2) t + q2

∫ t

0r2

0 (τ ) dτ ,

ϕ2(t) = (t1 + t4α2 + t3 β

2) t + t2

∫ t

0r2

0 (τ ) dτ ,

and θ, θ1, θ2 ∈ R. This family of solutions does not include homoclinic orbits to 0,since the latter are S-invariant. These orbits are reversible under R if one chooses θ ,θ1 and θ2 equal to 0 or π , and reversible under R1 = R S if one chooses θ1 = ±π/2and θ , θ2 equal to 0 or π .The persistence of these orbits for the full vector field is still an open problem. In thereversible cases this may be analyzed using techniques developed by Lombardi [28]for the (iq0)

2iq2 resonance (see the above paragraph on S-invariant solutions), butthe extra pair of eigenvalues ±iq1 makes the problem more difficult.For β � |A(0)|, solutions (87) of the truncated normal form correspond to approx-imate solutions of the Klein-Gordon system, consisting of a travelling wave super-posed on a small oscillatory mode (mainly visible at the tail).

Travelling Breathers in Klein-Gordon Chains 75

5.3.2. Solutions bifurcating at (T0, γ0) ∈ �2k+1.

• S-invariant solutionsWe consider solutions of the truncated normal form (70) on the invariant subspaceFix(S). These solutions satisfy A = B = C = 0. They are periodic, given byD(t) = αeiω

∗t+θ2 withω∗ = q2 + t1 + t4α2. Their persistence for the full vector field(restricted to Fix(S)) follows from the Devaney-Lyapunov theorem. These solutionscorrespond to spatially periodic travelling waves of the Klein-Gordon system, whichhave been obtained in [22].

• Non S-invariant solutionsFor s2(γ0, T0) < 0 and (γ, T ) ≈ (γ0, T0), the truncated normal form admits homo-clinic solutions to small quasi-periodic orbits, given by Eq. (87). Their persistencefor the full vector field is still an open problem. For reversible solutions this problemmay be treated using the techniques developed by Lombardi [28], but in the presentcase an extra pair of purely imaginary eigenvalues makes the problem more difficult.

• The existence of homoclinic orbits to 0 reversible underRwould be only possible withtwo compatibility conditions satisfied. The situation is similar to Sect. 5.3.1 ((iq0)

2iq2resonance for S-invariant solutions), except one obtains in the present case one com-patibility condition for each pair of simple purely imaginary eigenvalues. Here thelinearized system has 4 hyperbolic eigenvalues with small real parts ±ν (we shall useν as a small parameter) and O(1) imaginary parts ±iω0(ν) (ω0(0) = q0), and thereare in addition two pairs of O(1) purely imaginary eigenvalues ±iω1(ν),±iω2(ν)

(ωj (0) = qj ). Using the same notations as in Sect. 5.3.1, compatibility conditionstake the form of oscillatory integrals

I1(ν) = Im∫ +∞

0e−iω1t/ν g1(y(t), h(t), t, ν) dt = 0, (88)

I2(ν) = Im∫ +∞

0e−iω2t/ν g2(y(t), h(t), t, ν) dt = 0. (89)

As in Sect. 5.3.1, h(t)+y(t) denotes a reversible homoclinic orbit to 0 of the rescaledreduced equation. Its principal part h(t) is explicit and given (in the unscaled form)by (87) (with C = D = 0 and θ equal to 0 or π ). The existence of homoclinic orbitsto 0 reversible under R1 = R S would imply two compatibility conditions similar to(88)–(89) (one has θ = ±π/2 in (87) and one takes the real part of the integral in(88)).Instead of homoclinic orbits to 0, we conjecture the persistence of reversible ho-moclinic orbits to exponentially small 2−dimensional tori, originating from the twoadditional pairs of simple imaginary eigenvalues.As we shall see, solutions (87) of the truncated normal form correspond to approxi-mate solutions of the Klein-Gordon system, consisting of a pulsating travelling wavewith oscillations of size |A(0)| at the center.

5.3.3. Persistence result in a particular case. We consider the case when the potentialV in (1) is even (case a = 0 in (10)). Due to the additional invariance xn → −xn of (1),Eq. (11) is also invariant under −S. Fixed points of −S correspond to solutions of (1)satisfying

xn+1(τ ) = −xn(τ − T

2).

76 G. James, Y. Sire

In this case we have xn(τ ) = (−1)n+1x1(τ − (n−1)T2 ) and x1 satisfies a simpler scalar

advance-delay differential equation

d2x1(τ )

dτ 2 + V ′(x1(τ )) = −γ (x1(τ + T/2)+ 2x1(τ )+ x1(τ − T/2)). (90)

For (T0, γ0) ∈ �2k+1, the symmetry −S has the following structure on the centralsubspace

−S = diag(1, 1, 1,−1, 1, 1, 1,−1).

We consider the normal form (70) on the invariant subspace Fix(−S), which corre-sponds to fixingD = 0. In particular, the stable and unstable manifolds of 0 are includedin Fix(−S).

By considering the flow on Fix(−S), we recover the (iq0)2(iq1) resonance case

treated in [28] and summarized in Sect. 5.3.1. Under non-resonance assumptions ( q1q0

�= pq

for p + q ≤ 5, q1q0

�∈ 2N

and q1q0

�∈ N), reversible homoclinic solutions to periodic orbitsgiven by (87) (with D = 0) persist for the full vector field above a critical tail sizeβ = βc, which is exponentially small with respect to |A(0)| (size of “central” oscilla-tions). These solutions are either reversible under R (for θ , θ1 equal to 0 or π in (87))or R1 = R S = −R (for θ , θ1 equal to ±π/2). As we shall see in Sect. 6, these orbitsyield exact travelling breather solutions of the Klein-Gordon system, superposed on anexponentially small oscillatory tail.

Homoclinic orbits to 0 reversible underR do not persist for the full normal form (70)if the compatibility condition (88) (corresponding to the pair of eigenvalues ±iq1) isnot satisfied [28]. A similar condition holds for reversible solutions under −R. Note thatthe compatibility condition (89) is automatically satisfied by fixing a = 0 in V , thanksto the symmetry −S of (70). Indeed, the stable manifold of 0 has no D-component andthe D-component of the full normal form (70) vanishes for D = 0, which implies thevanishing of g2 in (89).

As in Sect. 5.3.1 for S-invariant solutions, there might be a discrete collection ofcurves I1(T , γ ) = 0 in the parameter plane (with (T , γ ) ≈ �2k+1) on which the rel-evant compatibility condition would be satisfied and reversible homoclinic orbits to 0would exist.

In the next section, we study the sign of the crucial normal form coefficient s2 for(γ0, T0) ∈ �0 (homoclinic orbits are found for s2 < 0).

5.4. Sign of the bifurcation coefficient s2. In the following, we determine the sign ofthe coefficient s2(γ0, T0) as a function of the parameters (T0, γ0) ∈ �k and parametersa,b in the potential (see (10)). We recall that the homoclinic solutions (87) exist fors2(γ0, T0) < 0 and (T0, γ0) ∈ �0.

5.4.1. Case of an even potential (a = 0) For a = 0 we have

s2 = 3T 20 b

1 − q02 tan(q0/2)

.

Let us define

Z(q0) = 1 − q0

2 tan(q0/2). (91)

Travelling Breathers in Klein-Gordon Chains 77

We have then

sign(s2) = sign(b)sign(Z(q0)). (92)

By Lemma 3.2, one has Z = 0 at cusp points of the bifurcation curve � (these pointshave been removed from the parameter set �0). Consequently, the sign of s2 dependson the parameter position with respect to the cusps. More precisely, Z < 0 on the rightbranch of �k and Z > 0 on the left one. It follows that s2 has the sign of b on the leftbranch of �k , and the sign of −b on the right branch.

As a conclusion, if the potential V is hard (b < 0) the homoclinic solutions of thetruncated normal form described above exist for parameter values near the left branchof each “tongue” �k restricted to�0. If V is soft (b > 0), homoclinic solutions exist forparameter values near the right branch. We sum up the situation in Fig. 2.

5.4.2. General case (a �= 0). We now consider the general case a �= 0. We introducethe parameter η = b

a2 and recall the expression of s2,

(1 − q0

2 tan(q0/2))s2 = T 2

0 a2(3η + 4 − 2T 2

0

2γ0T20 cos(q0)− T 2

0 (1 + 2γ0)+ 4q20

). (93)

One can obtain a simpler expression for s2. Indeed, one can prove the identity

q20 = T 2

0 (1 + 2γ0)− 2 cos (q0/2) (−1)mT 20 γ0 (94)

using successively

γ

0 π2 T4 8π6π π 10π

−−−−−− − − − −

−−

−−

−−−−− −

−−

−

−−−−−−−−−

−−

−−

−−

−

−−

−−−−−−−−−−−−

−−

−−

.

............ ................

...................

... Existence of homoclinic orbits

− − − −

Existence of homoclinic orbits

for b > 0

for b < 0

Γ1

Γ2

Γ3

Γ4

Γ5

−

−−−

− −

−−

−−

−−

−−

−−

−

−−−−−

−−−

−−

−−

−−−

−−

−−−−

...........

..........

Fig. 2. Regions in the parameter space where small amplitude homoclinic orbits exist in the case a = 0(even potentials)

78 G. James, Y. Sire

q20 = −4

q0

tan(q0/2)+ T 2

0 (1 + 2γ0) (95)

(see Eq. (31)) and

2q0

sin(q0/2)= (−1)m T 2

0 γ0 (96)

(see Eqs. (28)–(30)). Identity (94) allows us to simplify the right side of (93). Indeed,we obtain by substitution

2γ0T20 cos(q0)−T 2

0 (1+2γ0)+4q20 = T 2

0 (3+6γ0+2γ0 cos(q0)−8γ0(−1)m cos(q0/2)),

which simplifies in

2γ0T20 cos(q0)− T 2

0 (1 + 2γ0)+ 4q20 = T 2

0 (3 + 16γ0 sin4(q

4− mπ

2)).

Consequently, one can write s2 in the form

(1 − q0

2 tan(q0/2))s2 = T 2

0 a2(3η + 4 − 2

3 + 16γ0 sin4(q04 − mπ

2 )) for (γ0, T0) ∈ �m.

We study the sign of s2 when (T0, γ0) covers the left or the right branch of the “tongue”�m. To this end, we fixm ≥ 1 and introduce the subset�lm of�m such thatZ(q0) > 0 (leftbranch) and the subset�rm of�m such thatZ(q0) < 0 (right branch). Note that (T0, γ0) ∈�lm is equivalent to q0 ∈ (q, qmax), where q ∈ (2mπ, 2(m + 1)π) denotes the pointsatisfying Z(q) = 0 (corresponding to the cusp of �m) and qmax ∈ (2mπ, 2(m+ 1)π)is obtained by fixing T = 0 in Eq. (27) or (29) (γ goes to infinity and T = 0 at this valueof q). Similarly, having (T0, γ0) ∈ �rm is equivalent to fixing q0 ∈ (2mπ, q) (see Fig. 3).

T(q),−( γ( )q)−

T

γ

Γml

Γm

r

q π2 m

q q max

Fig. 3. Definition of �lm and �rm

Travelling Breathers in Klein-Gordon Chains 79

We denote by Fm the quantity

Fm(q0) = 4 − 2

3 + 16γ0(q0) sin4(q04 − mπ

2 ), (97)

and s2 writes(

1 − q0

2 tan(q0/2)

)s2 = T 2

0 a2(3η + Fm(q0)). (98)

Note that Fm is a strictly increasing function of q0 for q0 ∈ (2mπ, qmax). Moreover, wehave Fm(2mπ) = 10

3 , Fm(qmax) = 4 and 103 < Fm(q) < 4. We deduce the following

results.

• Case (T0, γ0) ∈ �lm (m ≥ 1)In this case, we have

sign(s2) = sign(3η + Fm(q0)). (99)

If η > −Fm(q)3 , we have s2 > 0 on �lm (this is the case in particular for b ≥ 0). For

− 43 < η < −Fm(q)

3 , s2 is negative only on a piece of �lm. Finally, if η < − 43 then s2

is negative on �lm.• Case (T0, γ0) ∈ �rm (m ≥ 1)

In this case, we have

sign(s2) = −sign(3η + Fm(q0)). (100)

If η > − 109 , we have s2 < 0 on �rm (this is the case in particular for b ≥ 0). For

−Fm(q)3 < η < − 10

9 , s2 is negative only on a piece of �rm. Finally, if η < −Fm(q)3

then s2 is positive on �rm.

We illustrate our analysis for the particular curve �1 (the other curves yield qualitativelysimilar results). Figure 4 describes the sign of s2 depending on q0 and η. Figure 5 indi-cates the regions on �1 where s2 < 0. We recall that the homoclinic solutions (87) existfor s2(γ0, T0) < 0, (T0, γ0) ∈ �0 and (T , γ ) ≈ (T0, γ0) outside of the “tongue” �k .

6. Homoclinic Solutions for the Klein-Gordon System

In this section we construct approximate (leading order) travelling breather solutions ofthe Klein-Gordon system with reversible homoclinic solutions of the truncated normalform. In addition we obtain exact solutions in the case of even potentials.

We choose (γ, T ) ≈ (γ0, T0) ((T0, γ0) ∈ �0), in such a way that the linearized oper-ator L has four symmetric eigenvalues close to ±iq0 and having non-zero real parts. Inaddition we require s2(γ0, T0) < 0. In this case, the truncated normal form admits differ-ent types of homoclinic solutions (A,B,C,D) described in Sect. 5.3. In the sequel werestrict our attention to reversible solutions underR orR1 = R S, for which a persistencetheory has been developed [28].

According to (76), reversible approximate solutions of (11) are given by

U ≈ AV0 + BV0 + CV1 +DV2 + c.c., (101)

80 G. James, Y. Sire

s

s

s

s

2

2

2

2

<

<

>

>

0

0

0

0

qmax

q

q−2π

η

−10/9

−4/3

−1.15

0

Fig. 4. Sign of s2 in the case of general potentials for (T0, γ0) ∈ �1. Note that in this case qmax ≈ 11.2and q ≈ 9

−−−−−

−

−−

−

−− − −

− − −

−−−−−

−−−

−−−

−

−−−

−

−−−

−

− −

−

−

−−−−

−

−

T

TT

Case

Case

γ γ

γ γ

η > −10/9 <Case

<Case −4/3<η<−4/3

η < −10/9

η/3

−−

−

−

T

−−

−−

−F ( q ) /3−1

−F ( q ) 1

−

Fig. 5. Parts of �1 where s2 < 0 (bold line). The dashed regions correspond to the existence of smallamplitude homoclinic solutions given by (87). Note that −F1(q)

3 ≈ −1.15

where A,B,C,D have the form (87). One fixes θ, θ1, θ2 equal to 0 or π if U is revers-ible under R. If (T0, γ0) ∈ �2k+1 and U is reversible under R1, one has θ, θ1 = ±π/2and θ2 equal to 0 or π . For (T0, γ0) in �m ∩�0, (101) yields the approximate solutionsof (7),

Travelling Breathers in Klein-Gordon Chains 81

(u1(t)

u2(t)

)≈ A(t)

((−1)m

1

)+ C(t)

(−11

)+D(t)

(11

)+ c.c.

Coming back to the original variables (using Eq. (6)), we obtain

xn(τ ) ≈ [ (−1)nm A+ (−1)n C +D ] (τ

T− n− 1

2)+ c.c. (102)

As ξ = τT

− n−12 → ±∞ one has

A(ξ) ∼ A0 e−a |ξ | ei(q0ξ±φ+θ), C(ξ) ∼ β ei(q1ξ±φ1+θ1), D(ξ) ∼ α ei(q2ξ±φ2+θ2)

with a > 0. Approximate solutions given by (102) converge for C,D �= 0 towardsquasiperiodic solutions as ξ → ±∞, and have larger oscillations at the center forα, β � |A(0)|.

Homoclinic solutions bifurcating in the neighborhood of �2m ∩ �0 can be seen assuperpositions of a travelling wave of permanent form xTW (τ) = (A+D)( τ

T− n−1

2 ) anda pulsating travelling wave xT P (τ ) = (−1)n C( τ

T− n−1

2 ). If β � |A(0)|, the pulsatingpart xT P is mainly visible at the wave tail. Note that pure travelling waves (with C = 0,D �= 0) exist in the full system (1) [22].

In addition, homoclinic solutions bifurcating in the neighborhood of �2m+1 ∩ �0can be seen as superpositions of a pulsating travelling wave xT P (τ ) = (−1)n (A +C)( τ

T− n−1

2 ) and a travelling wave of permanent form xTW (τ) = D( τT

− n−12 ). For

α, β � |A(0)|, the wave mainly consists of a O(|A(0)|) pulsating part localized at thecenter and a small quasiperiodic tail.

If we fix γ = γ0 and expand (102) for δ = |T − T0| ≈ 0, α ≈ 0, β ≈ 0 we obtainfor bounded values of τ, n,

xn+1(τ ) ≈ δ1/2 A(δ1/2 (n− vg τ)) ei(k0n−ω0τ) + β ei(k1n−ω1τ) + α ei(k2n−ω2τ) + c.c.,

(103)

where k0 = q02 − mπ , ω0 = q0

T0, k1 = q1

2 − π , k2 = q22 , ωi = qi

T0, vg = 2

T0, A has

the form A(ξ) = c1(cosh(c2ξ))−1 and phase shifts have been included in c1, α, β for

notational simplicity. We note that (ωi, ki) satisfies the equation

ω2 = 1 + 4γ0 sin2 k

2(104)

due to the fact that qi satisfies the dispersion relation (19). One recognizes in Eq. (104)the usual form of the dispersion relation of Eq. (1) linearized at xn = 0. MoreoverEq. (103) shows that our approximate solutions can be seen as superpositions of modu-lated plane waves, and one can check that vg is the group velocity ω′(k0) (use Eqs. (28),(30)). Note that, due to condition (2) (p = 2), only specific wave vectors k1, k2 in theoscillatory tail are selected among the whole set of possible ones.

Without further symmetry assumptions (evenness of V , or restriction to travellingwave solutions with C = 0 as in [22]), the persistence of solutions (102) for Eq. (1)is still an open problem, which should be tackled using the finite dimensional reducedsystem (70).

From the analysis of Sect. 5.3, we conjecture that the particular reversible solutionsdecaying to 0 at infinity (C = D = 0) should not persist generically in the Klein-Gordonsystem (1). To make a more precise statement, fix V (x) = 1

2x2 − a

3x3 − b

4x4 and assume

82 G. James, Y. Sire

(T , γ ) close to �0. We conjecture that a solution of (11) reversible under R or R1,homoclinic to 0 and close to an approximate solution (101) withC = D = 0 might onlyexist if (T , γ, a, b) is chosen on a discrete collection of codimension-l submanifoldsof R

4 (l > 0). The codimension depends on the number of pairs of purely imaginaryeigenvalues (i.e. the number of resonant phonons) in our parameter regime and symme-try assumptions. In the present case (with two pairs of purely imaginary eigenvalues, inaddition to weakly hyperbolic ones), we expect l = 2 if (T0, γ0) ∈ �2m+1 ∩ �0 (caseof travelling breather solutions) and l = 1 when (T0, γ0) ∈ �2m ∩�0 (case of solitarywave solutions, which have the additional invariance under S). The codimension is equalto the number of compatibility conditions obtained with the normal form (70) for eachtype of homoclinic bifurcation (see Sect. 5.3).

Instead of solutions decaying to 0 at infinity, we conjecture for (T0, γ0) ∈ �2m+1∩�0the persistence of reversible solutions homoclinic to quasi-periodic waves (since weconjecture the persistence of reversible homoclinic orbits to 2−dimensional tori in thenormal form (70)). Reversible approximate solutions (102) should constitute the prin-cipal part of travelling breather solutions of (1) superposed on a small quasi-periodicoscillatory tail.

The following theorem summarizes the above results in the case of travelling breathersolutions.

Theorem 6.1. Assume s2(γ0, T0) < 0 defined by Eq. (77) for a fixed (T0, γ0) ∈ �0⋂

�2k+1 and consider (γ, T ) ≈ (γ0, T0) such that the linear operator L in (11) has foursymmetric eigenvalues close to ±iq0 and having non-zero real parts. Then the reducedEq. (69) written in the normal form (70) and truncated at order 4 admits small ampli-tude reversible solutions (under R or R S) homoclinic to 2-tori. Such solutions shouldcorrespond to the principal part of travelling breather solutions of system (1), super-posed at infinity on an oscillatory (quasiperiodic) tail, and given at leading order by theexpression

xn(τ ) ≈ [ (−1)n A+ (−1)n C +D ] (τ

T− n− 1

2)+ c.c, (105)

where A,C,D are defined in Eq. (87) (with θ2 equal to 0 or π , θ, θ1 = ±π/2 forR S-reversible solutions, and θ, θ1 equal to 0 or π for R-reversible solutions).

In addition to leading order approximate solutions, we obtain exact travelling breathersolutions superposed on a small oscillatory tail in the case of even potentials. This resultfollows directly from the center manifold reduction theorem (Theorem 4.1) and theanalysis of the reduced equation (see Sect. 5.3.3).

Theorem 6.2. Assume s2(γ0, T0) < 0 defined by Eq. (77) for a fixed (T0, γ0) ∈ �0⋂

�2k+1 and consider (γ, T ) ≈ (γ0, T0) such that the linear operator L in (11) has foursymmetric eigenvalues close to ±iq0 and having non-zero real parts. Moreover assumethat the potential V is even.

Equation (11) is invariant under the symmetry −S defined in (17). If (T0, γ0) liesoutside some subset of �0

⋂�2k+1 having zero Lebesgue measure (corresponding to

resonant cases), the full reduced Eq. (69) restricted to Fix(−S) admits small amplitudereversible solutions (under ±R) homoclinic to periodic orbits. These solutions corre-spond to exact travelling breather solutions of system (1) superposed at infinity on anoscillatory (periodic) tail. Their principal part is given by

xn(τ ) = (−1)n[A+ C ] (τ

T− n− 1

2)+ c.c. + h.o.t, (106)

Travelling Breathers in Klein-Gordon Chains 83

whereA,C are given by Eq. (87) (with θ, θ1 = ±π/2 for reversible solutions under −R,and θ, θ1 equal to 0 or π for reversible solutions under R). For a fixed value of (γ, T )(and up to a time shift), these solutions occur in a one-parameter family parametrizedby the amplitude β of oscillations at infinity. The lower bound of these amplitudes isO(e−c/µ1/2

), where µ = |T − T0| + |γ − γ0|, c > 0.

Remark. The lower bound of the amplitudes should be generically nonzero, but mayvanish on a discrete collection of curves in the parameter plane (T , γ ).As a consequence,in a given system (1) (with fixed coupling constant γ and symmetric on-site potentialV ), exact travelling breather solutions decaying to 0 at infinity (and satisfying (2) forp = 2) may exist in the small amplitude regime, for isolated values of the breathervelocity 2/T .

We conclude by comparing our findings to a previous work. The existence of modu-lated plane waves in Klein-Gordon chains has been studied by Remoissenet [36] usingformal multiscale expansions. Under this approximation, the wave envelope satisfiesthe nonlinear Schrodinger (NLS) equation. In this problem a rigorous analysis of thevalidity of NLS equation (on large but finite time intervals) has been performed in [19].

The condition obtained by Remoissenet for the existence of NLS solitons (for thespecific wave number k = k0 = q0

2 − mπ ) is exactly the condition s2 < 0 derived inSect. 5.3. Indeed, the condition obtained by Remoissenet is PQ > 0, where

Q = T0

2q0(4a2 − 2a2

3 + 16γ0 sin4( k02 )

+ 3b), (107)

P = γ0T0

2q0(cos(k0)− γ0T

20

q20

sin2(k0)). (108)

Using the same equations as in Sect. 5.4.2 one can express P and Q as a function ofγ0, T0, q0,

Q = T0

2q0(4a2 − 2a2T 2

0

−T 20 (1 + 2γ0)+ 2γ0T

20 cos(q0)+ 4q2

0

+ 3b), (109)

P = 1

2q0T0(−4 + γ0T

20 (−1)m cos(q0/2)). (110)

The coefficient P is Z(q0) (defined in (91)) multiplied by a negative constant (use Eq.(96)). Similarly, the expression into brackets in Q is exactly the same as the one in thenormal form coefficient s2. Consequently, the product PQ differs from s2 by a negativemultiplicative factor, and thus PQ > 0 is equivalent to s2 < 0.

Acknowledgements. We wish to thank Gerard Iooss for helpful comments. We are grateful to SergeAubryfor his hospitality at the Laboratoire Leon Brillouin (CEA Saclay, France) and stimulating discussions.This work has been supported by the European Union under the RTN project LOCNET (HPRN-CT-1999-00163).

84 G. James, Y. Sire

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Communicated by A. Kupiainen