RECENT DEVELOPMENTS IN INTEGRABLE CURVEDYNAMICSANNALISA CALINIAbstract. The dynamics of vortex �laments has provided for almost a cen-tury one of the most interesting connections between di�erential geometry andintegrable equations, and an example in which knotted curves arise as solutionsof di�erential equations possessing remarkably rich geometrical structures. Inthis paper we present several aspects of the integrable evolution of a closed vor-tex �lament in an Eulerian uid. Starting with the derivation of the equationsfrom an idealised physical model, we describe their hamiltonian formulation onan appropriate in�nite-dimensional phase space. Then we discuss a transfor-mation discovered by Hasimoto which converts the �lament equation into thecubic nonlinear Schr�odinger equation, unveiling its complete integrability. Wework principally with the evolution equation for the tangent indicatrix of thevortex �lament, more suitably describing the dynamics of closed curves: thisis also a completely integrable soliton equation, originally derived to modelthe evolution of a continuous Heisenberg spin chain.This paper has a double task. On the one hand it aims at providing anintroduction to soliton equations in a concrete geometrical setting. For thispurpose, we discuss an important technique for constructing a large class ofspecial solutions, among which are some interesting torus knots. Also, wemake use of the B�acklund transformation for the Heisenberg spin chain modelto explore the symmetries of a given curve and to construct its homoclinicsolutions.On the other hand, some results and techniques of soliton theory are revis-ited in the light of the di�erential geometry of curves, where their geometricsigni�cance becomes transparent. In this context, we introduce a new or-thonormal framing of a given curve and formulate a geometric construction ofthe Hasimoto transformation. We use this approach to show that the relationbetween the cubic nonlinear Schr�odinger equation and the Heisenberg modelis realised by a Poisson map between two di�erent Poisson structures.1. Background1.1. The physical model. Whirlpools and smoke rings are common examples ofvortex �laments: approximately one-dimensional regions where the velocity distri-bution of a uid has a rotational component. We give below an idealised descriptionof the self-induced dynamics of a closed line vortex in a Eulerian uid, based onBatchelor's approach [4].Let u be the velocity distribution of an incompressible (divu = 0) uid �lling anunbounded region in space. We suppose that the vorticity w = curlu (measuringthe rotational component of the velocity �eld) is zero at points not on the linevortex. Because divu = 0, and assuming the in�nite domain to be simply connected,Date: May 15, 1997.Proceedings of the Miniworkshop on Geometry and Di�erential Equations, Canberra May 5{61995. 1

2 ANNALISA CALINIφ

θfilament

O

R

xx’

B T

N

Figure 1. Idealised portion of a vortex �lament.we can introduce a vector potential A and write u = curlA. The choice of A isnot unique, in fact curlA = curl(A + grad�) for any arbitrary scalar function �.We use this freedom to satisfy divA = 0 and to derive the following vector Poissonequation, w = curl(curlA) = ��A;(1)where � is the Laplacian operator. In an in�nite domain, its solution isA = 14� Z d3x0 w(x0)kx� x0k ;and the corresponding velocity distribution isu = � 14� Z d3x0 (x� x0)�w(x0)kx� x0k3 :(2)We now make an essential simpli�cation: we model the line vortex with a tube ofin�nitesimal cross sectional area dA in which the vorticity w (everywhere tangentto the line vortex) has constant magnitude w. We then haveu = � 14� Z dA�l(x� x0)�w(x)kx� x0k3 = � 14� Z wdA I (x� x0)� �lkx� x0k3 :Introducing the circulation � = H u � �l = R wdA, we obtain the expressionu = � �4� I (x� x0)� �lkx� x0k3 ;(3)which describes a velocity distribution that becomes singular along the line vortex.Evidently, the uid around the line vortex circulates at an increasing speed as itapproaches the core of the vortex, causing local cross sections of the vortex tubeto rotate about its center without translating. Therefore, we need to computethe asymptotic contribution of the non-circulatory part of the velocity at pointsin�nitesimally close to the vortex �lament in order to describe its dynamics.We approximate a portion of the vortex �lament with a circular arc as shownin �gure 1., where the Frenet-Serret frame of the curve (T;N;B) de�nes a localcoordinate system. We choose x = x2N + x3B in the plane normal to the linevortex, x0 = x01T + x02N to be the position vector of a point along the curve, andwe let � =px22 + x23 be the distance of x from the �lament. Near the origin O (for

INTEGRABLE CURVE DYNAMICS 3small values of the angle �) we writex0 ' lT+ 12kl2N; �l = dx0 ' (T+ klN)dl;(4)where l = R� is the arclength and k = R�1 is the curvature of the �lament. In thisapproximation, the integrand in equation (3) becomes(x � x0)� �l(x0)kx� x0k3 ' �klx03T+ x3N� (x2 + 12kl2)B[l2 + �2 � x2kl2 + 14k2l4] 32 dl:This expression is valid at a point x = � cos�N + � sin�B close enough to thevortex �lament, and for l varying in a given interval [�L;L]; introducing the newvariable m = l=� we rewrite equation (3) asu = �4� Z L��L� ��1[cos�B� sin�N] + 12m2kB�1 +m2 � k�m2 cos�+ 14�2k2m4� 32 dl:(5)(We observe that there is no tangential component, the corresponding term in theintegrand being an odd function ofm.) As � ! 0, the denominator of the integrandconverges to (1 +m2) 32 and we obtain the following asymptotic expansionu ' �2�� (sin�B� cos�N) + k �4� log L�B:(6)The O(��1){term represents the rotational component of the motion which doesnot produce a displacement of the vortex �lament, while the logarithmic term causesthe �lament to move in the uid at a large speed in the direction of the binormal.By rescaling the time variable t ! �4� log L� t, and introducing the position vectorto the �lament (x; t), we write the following evolution equation@ @t = kB:(7)We observe that the vector�eld is purely local (a point along the �lament only\feels" the e�ect of nearby points) and non-zero only when the �lament is curved(there is no self-induced motion of a straight line vortex).1.2. The Hamiltonian formulation. Following a treatment contained in [6], wedescribe the vortex �lament equation (7) derived above as a hamiltonian system ona suitable in�nite-dimensional symplectic manifold.We �rst need to specify the appropriate phase space. We will restrict consid-eration to closed curves in space; let then LR3 = f : S1 ! R3g be the space ofsmooth maps from the circle into R3 (the loop space of R3 ). The in�nite groupDi� +(S1) of orientation preserving di�eomorphisms of the circle acts on LR3 sim-ply by changing the parametrisation of any given element. For our purposes wewish to identify knots which \look the same" as global objects, however, we wantto distinguish identical pairs that have opposite orientations. To this end, we in-troduce the quotient space Y = LR3=Di� +(S1) as the space of unparametrisedoriented loops. Y is a very singular space, for example it contains curves with anin�nite number of self-crossings, or with self-tangencies of in�nite order, and atsuch points the tangent space to Y cannot be de�ned. Let bX be the space of loopswith �nitely many self-crossings and contacts of �nite order, then

4 ANNALISA CALINITheorem 1. [6] The space of oriented singular knotsbY = bX =Di� +(S1)is a smooth in�nite-dimensional manifold modelled on the Frech�et space C1(S1;R2 ).The tangent space of bY can be characterised in the following way: a tangentvector at a point in bY is a smooth vector�eld along the curve that is everywherenormal to (we quotient out the tangential component which purely reparametrisesthe curve).We now de�ne a 2-form on T bY which endows bY with a symplectic structure,and allows us to construct hamiltonian ows on it. The following 2-form wasintroduced by J. Marsden and A. Weinstein in their study of vortex dynamics inEulerian uids [42] ! (u;v) = Z 2�0 �d dx � u � v� dx;(8)where u, v are arbitrary tangent vectors at (i.e. two normal vector �elds along )and x is the arclength parameter (we can choose an arbitrary parametrisation of ).The integrand in equation (8) is simply the oriented volume of the parallelepipedconstructed on the triple ( ;u;v).It is immediate to verify that ! is a closed 2-form; in fact, we can carry theexterior di�erentiation inside the integral and observe that the exterior derivativeof a volume form on a 3-dimensional manifold is zero. Nondegeneracy also followseasily.In the case of a �nite-dimensional symplectic manifold M (necessarily of evendimension) a symplectic form ! determines a natural isomorphismTM �! T �MX �! !(X; �)between tangent vectors and one-forms (see for example [2]). In particular, !associates to any smooth function H on M (a hamiltonian) a unique vector�eldXH (and thus a hamiltonian ow on M) de�ned by!(�; XH) = dH(�):(9)In an in�nite-dimensional setting the correspondence TM ! T �M may fail tobe onto, i.e. not for every hamiltonian functional can we construct an associatedhamiltonian vector�eld. This technical point is addressed in [6] for the space bYand we refer the interested reader to that book for details. Here we simply remarkthat, among those functionals which admit hamiltonian vector�elds constructed viathe correspondence (9) (Brylinski calls such functionals supersmooth) is the totallength functional L( ) = Z 2�0 d dx dx;(10)de�ned on arbitrary smooth curves (not necessarily arclength parametrised). TheFrech�et derivative of L( ), restricted to the representative of that is arclengthparametrised is dL( ) = �d2 dx2 :

INTEGRABLE CURVE DYNAMICS 5We check using (9) that the hamiltonian vector�eld for the total length functionalL is given by XL = d dx � d2 dx2 :In fact, using the Marsden-Weinstein symplectic form we compute, for arbitrary �!(�; XL) = Z 2�0 d dx � � ��d dx � d2 dx2 � dx = Z 2�0 ��d2 dx2 � �� dx = dL[ ] (�) :Given the hamiltonian vector�eldXL we write the associated hamiltonian evolutionequation @ @t = @ @x � @2 @x2 ;(11)which can be reduced to the vortex �lament equation (7) using the Frenet equationsfor in the case of nowhere vanishing curvature.Remark. An immediate consequence of the hamiltonian form (11) is the invarianceof the total length during the curve evolution; a stronger result can be provenwith a short calculation: local arclength is also preserved during the motion, i.e.@tk xk = 0, therefore the vortex �lament moves in time without stretching (animplicit assumption in the study of the physical model). It follows that x and t areindependent variables: we will see that the ability to \commute mixed partials" isat the basis of the most interesting properties of the vortex �lament equation.1.3. Completely integrable in�nite-dimensional systems. This special classof nonlinear partial di�erential equations gained its importance in the mid-sixtieswith the coining of the word \soliton" by Kruskal and Zabuski (although \solitonic"behaviour was �rst observed in 1953 in a seldom acknowledged article by Seeger,Donth and Kochend�orfer [51]; we thank an anonymous referee for this historicalremark) and with the development of the inverse scattering method for solving theKorteweg-de Vries equation (KdV) by Gardner, Green, Kruskal and Miura.The several features shared by these nonlinear equations make them the in�nite-dimensional analogues of integrable hamiltonian systems in �nite dimensions.In this section we give a brief exposition of the most important properties ofintegrable PDE's. A good overview of the subject can be found in the book by A.Newell [44] and a very clear exposition is contained in the book by G. L. Lamb [33];we also refer to M. J. Ablowitz and H. Segur's monograph [1].Integrable PDE's possess a class of special solutions widely known as solitons.These are solitary waves in the form of pulses whose behaviour has many particle-like features. During their evolution, solitons propagate without change of shapeand with no energy loss. When two or more solitons with di�erent propagationspeed collide, after a highly nonlinear interaction the pulses emerge with the sameinitial form and no energy is lost in radiation processes in the course of the inter-action.All the known soliton equations can be considered as in�nite-dimensional coun-terparts of completely integrable �nite-dimensional hamiltonian systems in the fol-lowing sense. They possess an in�nite sequence fIkg1k=1 of constants of motion,whose gradients are linearly independent and whose associated hamiltonian owscommute (i.e. the Ik's are pairwise in involution with respect to a given Poisson

6 ANNALISA CALINIbracket on the phase space). For example, the focussing cubic nonlinear Schr�odingerequation i t + xx + 2j j2 = 0;(12)which will play an important role in the course of this article, can be rewritten asa hamiltonian system t = fH; g;with Hamiltonian H [ ] = ZD(j xj2 � j j4) dxwith respect to the following Poisson bracketfF;Gg = i ZD �@F@ @G@ � � @F@ � @G@ � dx:The �rst few invariants areI1 = 12i ZD( x � � � x ) dx; I2 = ZD j j2 dx; I3 = ZD �j xj2 � j j4� dx; : : :They satisfy fIk; Ijg = 0; 8k; j and their gradients are linearly independent.As a consequence of the existence of a hierarchy fIkg1k=1 of invariants, thephase trajectories are restricted to lie on the intersection of the level sets Ik =ck; k = 1 : : :1, which forms a submanifold of in�nite codimension. In the caseof a �nite-dimensional phase space, the preimage of a regular value ~c of the map~I = (I1; : : : ; In) is di�eomorphic to a product of circles and lines (see [2]) and thedynamical system can be described in terms of the linear evolution of a collectionof action-angle variables. For soliton PDE's, the inverse scattering method explic-itly constructs a nonlinear change of variables that linearises the ow. The KdVequation was the �rst soliton equation for which the inverse scattering method wasdeveloped: in this case, the analogues of action-angle variables are the scatteringdata of a related linear Schr�odinger operator. The initial value problem for theKdV equation can thus be solved exactly by mapping the initial condition to itsscattering data, evolving the scattering data according to a linear evolution up totime t, and by reconstructing the solution at time t of the original PDE by meansof the inverse transform.We mentioned that at the heart of the inverse scattering method for the KdVequation is an underlying linear operator L. Peter Lax [38] proved that the spec-trum of the Schr�odinger operator L does not vary in in time, and recast the non-linear PDE into a general framework. If the spectrum of L is independent of time,then its evolution can be written asL(t) = U(t)L(0)U�1(t)(13)for some time-dependent unitary operator U(t). It follows directly from equation(13) that, if some function � solves the eigenvalue problem L� = ��, then � isalso a solution of the linear system �t = B�, where B = UtU�1. Di�erentiatingequation (13) with respect to time gives Lax form of the KdV equationLt = [B;L];where the operators L and B are called a Lax pair for the soliton equation. Laxalso showed the existence of an in�nite sequence of operators B that give rise to

INTEGRABLE CURVE DYNAMICS 7spectrum preserving evolutions of L: the corresponding hierarchy of Lax equationscoincide with the in�nite sequence of commuting hamiltonian KdV ows mentionedabove.In fact, all completely integrable PDE's arise from a Lax pair, i.e. as solvabilityconditions of an auxiliary pair of linear systems. The Lax pair for the focussingcubic NLS equation has the following formFx = UFFt = V F;(14)where F is an auxiliary complex vector-valued function andU = i��3 +� 0 i � i 0 � ; V = (2i�2 � ij j2)�3 +� 0 2i� � + � x2i� � x 0 � ;with �3 = � 1 00 �1 � (the parameter � is called the spectral parameter).We conclude this section with a geometric interpretation of the Lax pair (14)given in [20] that is appropriate to this context. The solvability condition for theoverdetermined system (14) is obtained by di�erentiating the �rst equation withrespect to t, the second equation with respect to x and equating the mixed partialderivatives, giving the equation@U@t � @V@x + [U; V ] = 0:(15)We can check that equation (15) is identically true for every � provided the NLSequation (12) is satis�ed. We can give to this equation and to the associatedlinear system (14) a natural geometrical interpretation. The system (14) can beregarded as equations of parallel transport in the trivial vector bundle R2 � C 2 ,where the vector function F takes values in the �bre C 2 and the matrices U andV are interpreted as local connection coe�cients. Then, equations (14) express thefact that the covariant derivative of F is zero and equation (15) is equivalent tothe (U; V ){connection being a at connection on R2 � C 2 . For this reason, Laxequations of the form (15) are also called the zero curvature formulation of thesoliton equation.1.4. The Hasimoto transformation. In 1972, R. Hasimoto [26] constructed thecomplex function = kexp(i R x �ds) of the curvature k and the torsion � of aspace curve, and showed that if the curve evolves according to the vortex �lamentequation (7), then solves the focussing cubic nonlinear Schr�odinger equationi t + xx + 12 j j2 = 0;(16)which can be reduced to equation (12) of section 1.2 by rescaling to 2 . Themost direct way to prove this result is to rewrite the �lament equation in terms ofa new orthonormal frame (T;U;V), whereU = cos(Z x �ds)N� sin(Z x �ds)B;V = sin(Z x �ds)N+ cos(Z x �ds)B:

8 ANNALISA CALINIThe role of (T;U;V) will be the main theme of section 4, for our purposes here wejust need the Darboux equations for the new frameTx = k1U+ k2V; Ux = �k1T; Vx = �k2T;(17)with k1 = k cos(R x �ds), k2 = k sin(R x �ds). The �lament equation (7) can berewritten in the following form t = �k2U+ k1V:(18)Di�erentiating (18) twice with respect to the arclength parameter x, we obtain txx = �k2xxU+ k1xxV + (k1k2x � k2k1x)T:(19)On the other hand, xx = k1U+ k2V from which we derive xxt = k1tU+ k2tV + k1Ut + k2Vt:(20)In order to compare equations (19) and (20), we observe that Tt = �k2xU+ k1xV(this follows directly from di�erentiating (18) with respect to x), and that Ut �V =�U �Vt. Using the Darboux equations for the new frame we compute(Ut �V)x =Uxt �V +Ut �Vx =� k1k1x � k2Ut �T = �k1k1x + k2U �Tt = �k1k1x � k2k2x:It follows that Ut � V = �(k21 + k22)=2 + A(t), where A(t) is some arbitrary x-independent function. By equating the right-hand sides of (19) and (20), we derivethe following pair of equationsk1t + �12(k21 + k22)�A(t)� k2 = �k2xxk2t � �12(k21 + k22)�A(t)� k1 = k1xxwhich can be written asi t + xx + �12 j j2 �A(t)� = 0:(21)in terms of the complex function = k1+ ik2, and can be reduced to the focussingnonlinear Schr�odinger equation (16) by changing to exp(�i R tA(s)ds) . There-fore, the vortex �lament evolution is described by a completely integrable solitonequation. A consequence of this fact is the existence of an in�nite hierarchy ofglobal geometric functionals that remain constant throughout the evolution andthat are obtained by reexpressing the constants of motion of the NLS equation interms of the curvature and torsion of the corresponding curve. The �rst few globalinvariants of the �lament equation are listed below (the �rst two are not obtainedfrom NLS invariants, as we will see in the next section)I�1 = Z 2�0 k skds; I0 = Z 2�0 �ds; I1 = Z 2�0 12k2ds;I2 = Z 2�0 k2�ds; I3 = Z 2�0 �(kx)2 + k2�2 � k4� ds : : : :J. Langer and R. Perline [34] unveiled the geometrical signi�cance of the Hasimototransformation by showing that the NLS equation and the vortex �lament equationcan be regarded as the same hamiltonian system written with respect to di�erent

INTEGRABLE CURVE DYNAMICS 9Poisson structures. We will explain this interesting result using an intermediatedynamical system introduced in the next section.1.5. The Continuous HeisenbergModel. If we di�erentiate the vortex �lamentequation (11) once with respect to the variable x and permute t and x derivatives(the fact that arclength is locally preserved allows us to do that), we obtain thefollowing equation for the unit tangent vector T,Tt = T�Txx:(22)This equation was derived by M. Lakshmanan [32] and it is known as the Continu-ous Heisenberg Model (HM). It describes the integrable evolution of the continuumapproximation of a discrete spin chain with nearest neighbour interaction (the ana-logue of the localised potential for the vortex �lament dynamics).The Lax pair for the Heisenberg Model is more conveniently written using ma-trices rather than spin vectors. If we represent the unit vector TT = (t1; t2; t3) withthe hermitian trace zero matrixS = � t3 t1 � it2t1 + it2 �t3 � S2 = I;the equation for the continuous spin chain can then be expressed in the followingcommutator formSt = 12i [S; Sxx] ; S(x+ 2�; t) = S(x; t):(23)It can be easily checked that (23) is the compatibility condition of the followingpair of linear systems Fx =i�SFFt =(2i�2S � �SxS)F;(24)where � is the complex spectral parameter. The main aspects of the completeintegrability of HM, the Hamiltonian formulation, its inverse scattering transformand the hierarchy of conserved functionals are discussed in [20].Of particular interest to us is its relation with the vortex �lament equationand the nonlinear Schr�odinger equation. In a short communication J. Langer andR. Perline [35] point out that the HM is to be regarded as the intermediate modelbetween these two evolution equations. In [20] the HM and the NLS equation areshown to be related by a gauge transformation between the respective Lax pairs;since we will provide several geometrical interpretations of this gauge equivalence,we brie y describe it here as a purely algebraic fact and leave its veri�cation to thereader.Proposition 1.1 (The gauge transformation). Let V be the unitary matrix whichsolves the NLS Lax pair (14) at � = 0 and which satis�es the additional relationV SV �1 = �3. Then, if F is an eigenvector for the Lax pair (24) of the HM, thevector V F solves the linear problem for the NLS equation.In the course of this article, we will also discuss various properties of the in�nitegeometry of the HM as they manifest themselves in the context of curve dynamics.We conclude by mentioning two of the advantages of using the HM for modellingclosed vortex �laments (besides being itself a model of curve evolution on the 2-sphere). The space of periodic curves is characterised by the following invariant

10 ANNALISA CALINIsubspace of the HM phase spaceP0 = �T : Z 2�0 T(x; t)dx = 0; T(2�; t) = T(0; t)� :On the other hand, establishing which subspace of the NLS phase space correspondsto closed curve is a di�cult task: deriving a simple set of conditions on the curvaturek and the torsion � of a space curve for the curve to be closed is still an openproblem.Moreover, the curve is easily reconstructed as (x; t) = R xT(s; t)ds+C (whereC can be chosen to be time-independent), while the reconstruction of the curve fromits curvature and torsion (and thus from a solution of the NLS equation) involvessolving an inverse problem (the Frenet equations of the curve).2. Special solutions: methods of algebraic geometry.Soliton equations, when considered on periodic domains, admit large classes ofspecial solutions which are the analogues of solitary waves for rapidly decreasinginitial data on an in�nite domain. These solutions are named N-phase solutionssince they depend on x and t through a �nite number of linear phases. Like thesolitary waves, they interact nonlinearly in a particle-like manner and superimposenonlinearly to produce more general solutions of the boundary value problem.N-phase solutions of soliton equations such as KdV, NLS and general AKNSsystems were �rst constructed by S. P. Novikov and I. M. Krichever [31]. Theirmethod is a rediscovery of classical Riemann surface techniques developed by J. L.Burchnall and T. W. Chaundy [7] and by H. E. Baker [3] for classifying commutativealgebras of scalar di�erential operators.A di�erent characterisation of N-phase solutions was given by P. D. Lax (inparticularly for the KdV equation) [39]: N-phase solutions are critical points fora linear combination of integrals of motion. This characterisation, although non-constructive except for a very low number of phases, provides an interesting pointof view: N-phase solutions lie on low-dimensional sets which topologically are N-tori in function space and which foliate the phase space in a way similar to a�nite-dimensional integrable system.In this section we construct the explicit formulas for N-phase solutions of theHM and the associated curves using methods of algebraic geometry. These curvesare interesting from a geometrical point of view: they are critical points of theglobal geometric invariants R 2�0 �ds; R 2�0 k2ds; R 2�0 k2�ds : : : (among them we �ndthe classical elastic curves). We �nd the construction of the N-phase solutions of theHM itself interesting, since it gives a clear interpretation of the gauge transformationbetween the HM and the NLS equation.2.1. The reconstruction of the curve. Instead of taking the anti-derivative ofthe tangent vector, we derive a formula for the position vector of the curve in termsof the eigenfunction of the HM linear problem. In the context of N-phase solutions,this provides one with a straightforward way to obtain the curve. A analogousformula (involving the NLS eigenfunctions) was obtained by A. Sym [54] via adi�erent argument. We start with the following result, whose proof follows fromstandard ODE theory.

INTEGRABLE CURVE DYNAMICS 11Proposition 2.1. The solution of the linear problem@F@x = i�SF(25)is analytic in � 2 C for an analytic initial condition F(0;�).Let � be the fundamental solution matrix of equation (25). Since it is analyticin the eigenvalue parameter, we can di�erentiate both sides of the linear systemwith respect to � and evaluate them at � = 0, obtaining the following formula forS: S = �i @@x @�@� �����=0 ��1j�=0:(26)This expression is a perfect derivative since the columns of the fundamental solutionmatrix at � = 0 are independent of x. Integrating (26) with respect to x we obtainthe hermitian matrix � =: Z x S(s)ds = �i @�@� �����=0 ��1j�=0;(27)which represents the position vector of the curve in terms of the eigenfunctionmatrix � of the HM linear problem.2.2. The Baker-Akhiezer function. Multi-phase solutions are associated to aset of data on a Riemann surface. We start with a hyperelliptic Riemann surface� of genus g described by the equationy2 = 2g+2Yi=1 (�� �i):(28)We mark the two points 1+, 1�, which are permuted by the holomorphic invo-lution �(�; y) = (�;�y) exchanging the two sheets. We also choose a set of g + 1distinct points D0 = P1 + � � � + Pg+1 placed in a generic position (a non-specialdivisor) and not containing 1�.Let �(P ) : �! C [ f1g be the hyperelliptic projection to the Riemann sphere;this is a choice of meromorphic function on � whose pole divisor is 1+ +1� (i.e.whose poles are the preimage of 1 via the map �). In neighbourhoods of 1� wechoose the local parameters k� such that(k�)�1 = (�(P ))�1near 1� respectively.The main idea of Krichever is to construct a function (�) on � which is uniquelyde�ned by a prescribed behaviour at its singularities and which turns out a poste-riori to be the simultaneous solutions of a pair of linear systems. The compatibilitycondition, i.e. the zero curvature representation of these two linear operators, is acompletely integrable non-linear equation for the coe�cients. Thus, the construc-tion of such a function provides one both with the non-linear equation, an initialcondition and its solution.De�nition 2.1. A Baker-Akhiezer function associated to (�;D0;1�) is a function on � which:� is meromorphic everywhere except at1� and whose set of poles on �nf1�gis D0,

12 ANNALISA CALINI� has essential singularities at1� that locally are of the form (k�) � Cep(k�),where C is a constant and p(k) an arbitrary polynomial with complex coe�-cients.For our purposes, we recall the following result [31],Proposition 2.2. Suppose that the following technical condition holds:Condition 1. The divisor P1+ � � � +Pg+1�1+�1� is not linearly equivalentto a positive divisor.Then, if p(k) = ikx + iQ(k)t, with x and t complex parameters with jxj; jtjsu�ciently small and Q(k) a given polynomial, the linear vector space of Baker-Akhiezer functions associated to (�;D0; k�) is 2-dimensional and it has a uniquebasis 1; 2 with the following normalised expansions at 1�: j(x; t;�) = eik�x+iQ(k�)t 1Xn=0 �j�n (x; t)k�n� ! ; j = 1; 2(29)where �j�n are functions of the parameters x and t and �1+0 = 1, �1�0 = 0, �2+0 = 0,�2�0 = 1.Remarks. We make two remarks before proving the proposition.1. For a non-special divisor of degree d (a formal integer linear combination ofpoints on � counted with multiplicity) the Riemann-Roch formula (see for example[24]), states that the dimension h0(D) of the linear space of meromorphic functionson � whose pole divisor is D ish0(D) = � 1; d � g;d� g + 1; d > g:(30)2. Condition 1 can be rephrased as: there exists no non-constant meromorphicfunction with pole divisor P1 + ::: + Pg+1 which vanishes simultaneously at 1+and at 1�.Proof. Uniqueness: Suppose there are two functions 1 and 2 which satisfy theprescription; then their ratio 1= 2 is a meromorphic function (the essential sin-gularities mutually cancel) whose poles are contained in the zero divisor of 2.The condition of \non-speciality" assures that the dimension of the space of suchfunctions is 2(= g + 1 � g + 1, according to the Riemann-Roch formula). As re-gards the normalisation, since h0(D0 �1+) = h0(D0 �1�) = 1, we can choosetwo such functions vanishing respectively at 1+ and at 1� respectively; becauseh0(D0 �1+ �1�) = 0 (Condition 1), they must be independent.Existence: We follow an argument given in [46] and exhibit a pair of independentfunctions on � with the correct singularities by constructing them explicitly asratios of Riemann Theta functions (a nice survey of the features which are relevantin this context can be found in [15]). Leta1; ::: ; ag ; b1; ::: ; bg;be a canonical homology basis for the Riemann surface � such that ai�aj = 0; bi�bj =0; ai � bj = �ij , and!1; ::: !g; Iai !j = �ij ; i; j = 1; :::; g

INTEGRABLE CURVE DYNAMICS 13be g normalised holomorphic di�erentials. We introduce the period matrix B,Bij = Ibi !j ; i; j = 1; ::: ; gand construct the associated Riemann Theta function�(z) = Xn2Zg exp i� (< n;Bn > +2 < n; z >) ; z 2 C g :The essential behaviour at 1 is introduced by means of the unique normaliseddi�erentials of the second kind � and �, which satisfy the following conditions:1. � and � have a single pole at 1 with local expansions� � idk; � � idQ(k):(these are dictated by the polynomial dependence of the exponent on the localparameter).2. (normalisation) Iai � = 0; Iai � = 0:We can now build the following function of P 2 � ( where P0 2 � is a base point)~ = exp x Z PP0 � + t Z PP0 �! �(A(P ) + Ux+Wt�K �A(D))�(A(P ) �K �A(D)) :(31)In this expression D is a divisor speci�ed below and K is the Riemann constant(see [15]) chosen so that �(A(P )�K �A(D)) has zeros precisely at D, andA gXk=1Qk! := gXk=1 Z QkP0 !is the Abel map (! is the vector of holomorphic di�erentials). The map A associatesto a divisorPgk=1Qk on � a point of the complex torus C g =� (the Jacobian of theRiemann surface Jac(�)), where � is the 2g-dimensional lattice spanned by thecolumns of the matrix (I jB).The vectors U and W have been introduced to make ~ a well-de�ned functionon �. The only indeterminacy is the path of integration which can be modi�edby adding any integer combination of homology cycles. Taking in account thevanishing condition for � and �, this produces the overall factor (the mk' s areintegers) exp gXk=1 �mk �x Ibk � + t Ibk ���mk (xUk + tWk)�! :This is equal to 1 if we de�ne the components of the \frequency vectors" U and Wto be Uk = 12�i Ibk �; 12�iWk = Ibk �:At last we are left to choose the pole divisor D of degree g and to �x the nor-malisation. As discussed above, two independent functions are uniquely pickedby requiring that one vanishes at 1+ and the other at 1�; for this purpose weintroduce the followingDe�nition 2.2. Let D� be the unique positive divisors linearly equivalent to D0�1�.

14 ANNALISA CALINIThe choices D = D+ and D = D� in formula (31) produce two independentfunctions, whose poles are in D+ and D� respectively. In order to make the poledivisor be D we multiply ~ � by a meromorphic function g�(P ) whose zeros liein D� +1� and whose poles lie in the original divisor D. We �nally obtain thecorrect Baker-Akhiezer functions 1;2 =exp"x Z PP0 � � �1� !+ t Z PP0 � � �1� !#��(A(P ) + Ux+Wt�K �A(D�))�(A(P ) �K �A(D�)) ��(A(1�)�K �A(D�))�(A(1�) + Ux+Wt�K �A(D�)) � g�(P )g�(1�) :(32)The constant terms �1� and �1� in the expansion of the argument of the exponentialat 1� have been subtracted to make the leading coe�cient of the meromorphicpart of the eigenfunction matrix be the identity.2.3. The gauge transformation revisited. Given the unique basis of the vectorspace of Baker-Akhiezer functions guaranteed in the previous theorem, we can buildunambiguously a function and a corresponding pair of linear operators which willbe identi�ed with the Lax pair for the Continuous Heisenberg Model. We have thefollowing result [8]:Proposition 2.3. If Q(k) = 2k2 and (x; t;P ) is the matrix of Baker-Akhiezerfunctions constructed in proposition 2.2(x; t;P ) = � 1(x; t;P ) 1(x; t; �(P )) 2(x; t;P ) 2(x; t; �(P )) � ;then the columns of the matrix�(x; t;�) = (x; t; 0)�1(x; t;�)(33)are linearly independent simultaneous solutions of the following pair of linear sys-tems @F@x = i�SF@F@t = (2i�2S � �SxS)F;where the matrix S is independent of � and is given byS(x; t) = (x; t; 0)�1�3(x; t; 0);(34)with �3 = � 1 00 �1 � :The value of at � = 0 de�nes the normalisation of the eigenfunction � at theessential singularity. We have in factlim�!1�(x; t;�) = (x; t; 0)�1:(35)Moreover, the choice of � = 0 as the point at which the HM Baker-Akhiezer functionis normalised is arbitrary, provided it is not chosen to be at the essential singularity.

INTEGRABLE CURVE DYNAMICS 15It is shown in [46] that the matrix (x; t;�) constructed in proposition 2.2 whenQ(k) = 2k2 solves the Lax Pair (14) for the focusing nonlinear Schr�odinger equa-tion. These considerations lead to the following simple interpretation of the gaugetransformation between the HM and the NLS equation:Corollary 1. Given the matrix (x; t;�) of Baker-Akhiezer functions for the NLSLax Pair (14), the matrix obtained by renormalising to be the identity matrix atan arbitrary �0 6= 1 solves the HM Lax Pair at �� �0. Moreover the potential Scan be reconstructed by means of equation (35) with 0 replaced by �0.The proof of Proposition 2.3 is contained in the following lemma [8],Lemma 2.1. Given the Baker-Akhiezer matrix function �(x; t;�) normalised asin Proposition 2.2, there exists a unique pair of matrix di�erential operators L1and L2 of the following formL1 = 1X�=0U�(x; t) @�@x� ; L2 = 2X�=0V�(x; t) @�@x� ;(36)such that �(x; t;�) solves simultaneously the linear systemsL1� = ��; L2� = @�@t :(37)Proof. In order to determine the coe�cients of the linear operators L1 and L2we examine the asymptotic behaviour of both columns of � at 1�. We de�neA(x; t) =: �1(x; t; 0) and look at an expansion for � in the global coordinate � ofthe form�(x; t;�) � A(x; t) + 1Xn=1 1�nXn(x; t)! ei(�x+2�2t) 00 e�i(�x+2�2t) !(38)The operator L1 is uniquely determined by the requirement(L1 � � I)�(x; t;�) = O� 1�� ei(�x+2�2t) 00 e�i(�x+2�2t) ! :(39)In fact, by substituting expression (38) in equation (39) and requiring that theO(1) and O(�) terms are equal to zero, one �nds the following recursive system ofequations: U1 = �iA�3A�1U0 = X1A�1 � U1AxA�1 � iU1X1�3A�1(40)which determines U0 and U1 as functions of A and X1. Since �(x; t; 0) = I, themultiplicative term U0 in the expression of L1 must vanish. We check this in thefollowing way.We consider the matrix of normalised Baker-Akhiezer functions (x; t;�) con-structed in proposition 2.2 and its asymptotic expansion at 1,(x; t;�) � I + 1Xn=1 1�nZn(x; t)! ei(�x+2�2t) 00 e�i(�x+2�2t) ! ;Because � is obtained by normalising as in (34), then Zn = A�1Xn; n = 1; ::: .If we substitute the asymptotic expansion of into an equation of the same form

16 ANNALISA CALINIas (39), we can show that satis�es the following linear eigenvalue problem,�i�3@@x � (�3Z1�3 � Z1) = �;(41)In particular, the coe�cient A�1(x; t) = (x; t; 0) satis�es (41) at � = 0 which werewrite as dA�1dx � i[Z1; �3]A�1 = 0:(42)We now solve for U0 in the system (40), use equation (42) and obtainU0 = �A�dA�1dx � i[Z1; �3]A�1� = 0:(43)The coe�cients of L2 are determined in the same way. The requirement(L2 � @@t )�(x; t;�) = O( 1� ) ei(�x+2�2t) 00 e�i(�x+2�2t) ! ;(44)together with the asymptotic expansion (38), determines the following recursivesystem of equations for the coe�cients Vi:V2 = �2iA�3A�1V1 = 2X1A�1 � 2V2AxA�1 � iV2X1�3A�1(45) V0 = At � V1(Ax + iX1�3)� V2(Axx + 2iX1x�3 �X2):In order to simplify the expressions for V1 and V2 we use the the equation for thetime evolution of (x; t;�), which we can derive by substituting the asymptotic ex-pansion of in the time linear system (36). We obtain the following �-independentequation�2i�3@2@x2 + 2[Z1; �3]�3 @@x + 2i[Z2; �3]� 2i[Z1; �3]�3Z1�3 � 2�3Z1x�3 = @@t :Using the fact that A�1(x; t) = (x; t; 0) satis�es this last equation we substitutethe expression for At into (45) and obtainV2 = �2iA�3A�1V1 = �i(A�3A�1)xV0 = 0:For this choice of Ui's and Vi's the right hand side of the relation(Lj � �I)�(x; t;�) = O( 1� ) ei(�x+2�2t) 00 e�i(�x+2�2t) ! ;(46)gives a pair of Baker-Akhiezer functions whose asymptotic expansions have leadingcoe�cients vanishing at 1�. Because they are unique, they must be identicallyzero. Therefore �(x; t; �) solves simultaneously the equations(Lj � �Id)�(x; t; �) = 0; j = 1; 2which, by introducing the quantity S = A�3A�1, become the Lax Pair for theContinuous Heisenberg Model.

INTEGRABLE CURVE DYNAMICS 172.4. N-phase curves. In the previous section, we built a basis of Baker-Akhiezereigenfunctions for the linear problem of the Heisenberg Model and we expressedthe potential S purely in terms of the leading coe�cient of the expansion of itsmeromorphic part at � = 1. We now derive a formula for the correspondingN-phase curves. It is clear from what we have discussed so far that the Baker-Akhiezer function for the Heisenberg Model is speci�ed by g+1 poles, two essentialsingularities over 1 and its normalisation at 0; it is also clear that normalising ata di�erent point (so far as it does not coincide with one of the poles or the zeros)will a�ect neither the essential singularity nor the divisor of the meromorphic part.Moreover if the pole divisor D is in general enough position, Condition 1 can bereplaced by the following equivalentCondition 2. The divisor P1 + ::: + Pg+1 � 0+� 0� is not linearly equivalent toa positive divisor.Here 0+ and 0� = �(0+) are the two points on � corresponding to � = 0 whichare exchanged by the hyperelliptic involution. Condition 2 and D being non-specialassure that, among the meromorphic functions with poles in D, there is only onewhich vanishes at 0+ and just one which vanishes at 0� (this assures the existence oftwo distinct functions with poles in D), but there exists no non-constant functionwhich vanishes at both points (this guarantees the independence of the two andtherefore the ability to realise any normalisation at � = 0). We introduce thefollowing quantities:a) D0+ (resp. D0�), the unique e�ective divisor which is linearly equivalent toD � 0� (resp. D � 0+).b) h�(P ), a meromorphic function whose divisor is (h�) = D0� + 0� �D.By an argument identical to the one described in Proposition 2.2, we obtain thefollowing result,Proposition 2.4. The linear problem associated to the Heisenberg Model is solvedby a vector function with the following components�� =exp"x Z PP0 � � �0�!+ t Z PP0 � � �0�!#��(A(P ) + Ux+Wt�K �A(D0�))�(A(P ) �K �A(D0�)) ��(A(0�)�K �A(D0�))�(A(0�) + Ux+Wt�K �A(D0�)) � h�(P )h�(0�) :(47)Moreover, the corresponding eigenfunction matrix�(P ) = � �+(P ) �+(�(P ))��(P ) ��(�(P )) �(48)is normalised to be the identity matrix at � = 0.In formula (47), (�; �; P0;!) are the same as in proposition 2.2. The terms �0� =R 0�P0 � and �0� = R 0�P0 � have been introduced to produce the correct normalisationat � = 0.Finally, we construct the N-phase curves. We use the reconstruction formula de-rived in section 3:1 (neglecting the additive constant associated to the normalization

18 ANNALISA CALINIof � at x = 0) �(x; t) = �i @�(x; t; �)@� �����=0:The derivatives of the entries of � can be expressed in terms of the followingfunctions (and by the sheet information),@��@� �����=0 = a�x+ b�t+ @@P log �(A(P ) + Ux+Wt�K �A(D0�))����P=0� + c�;where the c�'s are constant in x and t depending on the Riemann surface data,a� = ��(0), and b� = ��(0). We show that both coe�cients of the linear termsin x and t are zero. Let � be real for simplicity (we are just interested in whathappens at � = 0). In this case the fundamental solution � of the spatial linearproblem is a unitary matrix. We introduce the Transfer MatrixT (x; �) = �(x+ 2�; �)��1(x; �)which takes a solution evaluated at a given x to its value after one spatial period.It is an easy exercise to show that its trace�(�) = Tr [T (x; �)]is invariant both with respect to x and t. If we de�ne the Floquet eigenfunctionsby means of the following formula��(x; �) = ei�(�)xf�(x; �);where f�(x; �) are bounded, periodic functions, one can easily check that the fol-lowing relation holds between the Floquet exponent �(�) and �(�),d�d� = 12� d�d�p�2 � 4 :By comparing the expression of the Baker-Akhiezer eigenfunction with the one forthe Floquet eigenfunction, we can identify the di�erential � with the derivative ofthe Floquet exponent, �(�) = d�d� (�):As a consequence of the form of the linear system we have the following symmetry�(x;��) = �(�x; �);from which there followsT (2�;��) = T�1(2�; �) = �T T (2�; �):Since the trace of a unitary matrix is real, we have�(��) = �(�); � 2 R:Therefore �(�) is an even function of � and the Floquet exponent � = � must vanishat 0. In order to show that the coe�cient b vanishes as well, we use the symmetryd�d� (x;��) = �d�d� (�x; �), where � is any of the components of the eigenmatrix.This formula must hold for all times and in particular in the limit as t � 1. Atleading order we have bt = �bt, it follows that b = 0.

INTEGRABLE CURVE DYNAMICS 19Finally we obtain the following compact formula for the components of the her-mitian matrix � (its left column is the vector (�+;��)T representing the positionvector of the curve,�� = �i @@P log �(A(P ) + Ux+Wt+ �0)����P=0� ;(49)where we absorbed the information about the divisor and the Riemann constant inthe initial condition �0.Among the N-phase curves we �nd planar circles (1-phase solutions that are theanalogues of plane wave solutions of the NLS equation) which will play an importantrole in section 5. Below, we show a pair typical 2-phase curves, constrained extremaof the total squared curvature functional. Solutions to the associated variationalproblem in the case of unconstrained total torsion (the classical elastic curves) werecompletely classi�ed by J. Langer and D. Singer [37], and shown to belong to certainclasses of torus knots. The curves in �gure 2., a 5-2 and a 9-2 torus knot respectively,are critical points of the total squared curvature with constrained total torsion andtotal length (the most general 2-phase solutions), a vertical ribbon around each ofthem has been added to make the over and under crossings evident.

(a) (b)Figure 2. A 5-2 torus knot (a) and a 9-2 torus knot (b) are amongthe 2-phase solutions of the �lament equation3. Natural framesIf the vortex �lament equation describes the evolution of an arclength parametrisedcurve in R3 , the continuous Heisenberg Model describes the corresponding evolu-tion of its unit tangent vector. In the course of this section we will explain how, inorder to understand the geometry of the solution space of HM, we need to considernot just the unit tangent vector, but orthonormal frames of the original curve.The space of unit tangent vectors of space curves is the 2-dimensional sphereof radius 1. The space of associated orthonormal frames can be identi�ed with itsunit tangent bundle T1S2. Then, for a given curve in R3 , choosing a orthonor-mal frame means choosing a way to lift the spherical curve described by the unittangent vector into the bundle. In this section we study the horizontal lifting as-sociated to the canonical invariant connection on T1S2 and introduce the notion ofnatural frame. We show that the horizontal lifting is the geometrical realisation of

20 ANNALISA CALINIthe gauge transformation between the HM and the NLS equation. Moreover, weuse this concrete construction to characterise the di�erential of the gauge transfor-mation in terms of the second symplectic operator for the NLS equation; we willshow as a consequence that the HM and the NLS equation can be regarded as thesame hamiltonian system written with respect to di�erent symplectic structuresbelonging to the same hierarchy of hamiltonian ows.3.1. The Circle Bundle of S2. We start with recalling a few facts about the2-dimensional sphere and its associated circle bundle. The unit sphereS2 = �x 2 R3 j x21 + x22 + x23 = 1is an example of a symmetric homogeneous space. Given a connected Lie group Gand a nontrivial group automorphism � such that �2 = Id, the associated symmetrichomogeneous space is the orbit space G=H , where H is the identity component ofthe set of elements h 2 G which are invariant under the action of �, i.e. such that�(h) = h. In the case of the 2-sphere, G is the Lie group of rotations SO(3;R)and � is \conjugation by T": �(g) = T�1gT , g 2 SO(3;R), where T is the matrixT = � I �1 �. For this choice of the matrix T , H becomes the subgroupof elements of the form h = � � 1 � which can be identi�ed with the groupSO(2;R).The symmetric homogeneous space SO(3;R)=SO(2;R) is naturally di�eomor-phic to the 2-dimensional sphere, we show it by constructing the following transi-tive action of SO(3;R) on S2. Let fe1; e2; e3g be the standard orthonormal basisin R3 , the map SO(3;R) ��! S2g �! g � e3induces a di�eomorphism between SO(3;R)=SO(2;R) and S2 since the subgroupH �xes the vector e3. Moreover, given the canonical projection p : SO(3;R) !SO(3;R)=SO(2;R) �= S2, the automorphism � induces the involution �o(px) =p�(x); x 2 SO(3;R), for which the distinguished point o = pe is an isolated �xedpoint (e denotes the identity element of SO(3;R)).We now de�ne the circle bundle of S2 as the space of all unit tangent vectors tothe sphere: T1S2 = �(x; v) : x 2 S2; v 2 TxS2; kvk = 1 ;(k � k is the Euclidean norm in R3 ) together with the projection � : T1S2 ! S2,�(x; v) = x with �bre ��1(x) = S1.Remarks. 1. There is a natural action of the unit circle which \�xes" the base pointx and rotates v in the tangent plane to S2 at x described by the smooth mapS1 � T1S2 �! T1S2h � (x; v) �! (x; hv):This action is free, i.e. with no �xed points.2. We can always construct a local cross section, i.e. a smooth map ��1 :��1(U) ! U � S1 for some neighbourhood U of each point x 2 S2. To do so,we choose a smooth unit vector�eld e(U) (for example, we can take the vector�eld@u=k@uk in a local coordinate system (u; v)) and de�ne �(x; h) = (x; he(x)). Since

INTEGRABLE CURVE DYNAMICS 21the S1-action is free and e(x) is smooth and never zero in U , � is smooth andinvertible and so is ��1.3. The properties described above de�ne a principal �bre bundle with �bre S1and base S2. The classical result that there does not exist a non-vanishing vector�eld on S2 (and therefore a global cross-section) implies that T1S2 is a non-trivialbundle.4. The transitive, free action of SO(3;R) on T1S2 described by the map g�(x; v) =(gx; gv); g 2 SO(3;R), identi�es the space of orthonormal frames with the circlebundle of S2: SO(3;R) �= T1S2.3.2. The Canonical Connection. We summarise the notion of a connection ina principal �bre bundle and the construction of the canonical invariant connectionfor T1S2. The content of this section can be found in various references, ([30]is comprehensive, [52] contains a simple treatment of the circle bundle of a 2-dimensional Riemannian manifold).As in Euclidean space there is a natural way to parallel-translate and comparevectors at di�erent points, likewise in a general manifold a choice of a connectionprescribes a way of translating tangent vectors \parallel to themselves" and tointrinsically de�ne a directional derivative. In the case of a principal bundle P withstructure group G over a manifold MG �! P??y�Mwe best explain the role of a connection when thinking of lifting a vector�eld v 2TM to a vector�eld ~v 2 TP in a unique way. For each p 2 P , let Gp be the subspaceof TpP consisting of all the vectors tangent to the vertical �bre. The lifting of vwill be unique if we require ~v(p) to lie in a subspace of TpP complementary toGp. A smooth and G-invariant choice of such a complementary subspace is calleda connection on P . More precisely,De�nition 3.1. A connection on a principal bundle P is a smooth assignment ofa subspace Hp � TpP; 8p 2 P such that:1. TPp = Gp �Hp2. Hgp = (Lg)�Hp, 8g 2 G (Lg is the left-translation in G).Given a connection, the horizontal subspace Hp is mapped isomorphically by d�onto T�pM . Therefore the lifting of v is the unique horizontal ~v which projectsonto v. An equivalent way of assigning a connection is by means of a Lie algebravalued 1-form � (the connection form). If A 2 g (the Lie algebra of G), let A�be the vector�eld on P induced by the action of the 1-parameter subgroup etA.Since the action of G maps each �bre into itself, then A� is tangent to the vertical�bre at each point. For each X 2 TpP , �(X) is the unique A 2 g such that A� isequal to the vertical component of X . It follows that �(X) = 0 if and only if X ishorizontal.Proposition 3.1. A connection 1-form � has the following properties:1. �(A�) = A2. (Lg)�� = Adg�; 8g 2 G (Ad is the adjoint representation of G).The �rst property follows immediately from the de�nition of connection 1-form;for a proof of the second property we refer to [30].

22 ANNALISA CALINIWe are now ready to construct an invariant connection on T1S2. The groupinvolution � de�ned in the previous section induces a Lie algebra automorphism ong = so(3; R) (denoted with the same letter �). The automorphism � determinesthe following direct sum decomposition of gg = h� k;where h = fX 2 gj �X = Xg is the subalgebra of the invariant subgroup H =SO(2; R) and k = fX 2 gj �X = �Xg.Let � be the canonical 1-form of SO(3;R), i.e. the left-invariant g-valued 1-formde�ned by �(A) = A; A 2 g;thenTheorem 2. The h-component � of the canonical 1-form � of SO(3;R) de�nes aleft-invariant connection on T1S2.Proof. Since � is left-invariant, we restrict to left-invariant vector�elds X on T1S2.Then �(X)=0 if and only if X 2 k and �(Y ) = Y if Y 2 h. � is left-invariantalso with respect to the action of H = SO(2; R) �= S1; therefore � satis�es bothproperties 1 and 2 of a connection form and it is invariant under the full action ofSO(3;R).Structure equations. Let (V;E1; E2) be the canonical basis for the Lie algebraso(3; R) such that V spans the vertical space h and (E1; E2) span the horizontalsubspace k. We have[V;E1] = E2; [V;E2] = �E1; [E1; E2] = V:Setting � = �V + !1E1 + !2E2 and using the Maurer-Cartan equationd�(X;Y ) = �12�([X;Y ]); X; Y 2 g;we obtain the following structure equations for the dual basis (�; !1; !2)d� = �!1 ^ !2; d!1 = � ^ !2; d!2 = �� ^ !1:Remarks. 1. If the Riemannian metric on S2 is inherited from restricting the Eu-clidean metric on R3 to S2, then the invariant connection constructed above co-incides with the Riemannian connection on the frame bundle of S2 (the uniqueconnection which leaves the metric invariant and has zero torsion).2. We can give an invariant de�nition of the 1-forms !1, !2 using the Riemannianstructure, the left-invariance of the connection and the isomorphism d� : H(x;v) !TxS2 between the horizontal subspace at (x; v) and the tangent space to S2 at x.We identify ToS2 with the horizontal subspace h. Then (E1; E2) may be taken to bethe orthonormal basis (e1; ie1) of ToS2, where ie1 is obtained by a 90� rotation of e1within the tangent plane. Then for t 2 T(o;e1)T1S2, !1j(o;e1) (t) and !2j(o;e1) (t) arethe components of the projection d�(t) relative to the orthonormal basis (e1; ie1),we have for example !1j(o;e1) (t) =< d�(t); e1 >. Since !1; !2 are invariant withrespect to the S1-action, we can de�ne them on all tangent vectors at o of the formv = he1; h 2 S1. The left-invariance of the metric under the full group actionde�nes them everywhere:!1j(x;v)(t) =< d�(t); v >; !2j(x;v)(t) =< d�(t); iv >; t 2 T(x;v)T1S2:

INTEGRABLE CURVE DYNAMICS 233.3. Horizontal Lifting, Natural Curvatures, Gauge Transformation. Let (s) be a smooth curve in R3 with nonvanishing curvature, s 2 [0; 2�] be its ar-clength parameter and (T;N;B) be its Frenet frame. The unit tangent vector Tdescribes a smooth curve c(s) on the unit sphere. In this section we construct thehorizontal lifting of c(s) in the bundle T1S2, we introduce the notion of natural cur-vatures and discuss a geometric interpretation of the gauge transformation betweenthe HM and NLS equations de�ned in section 1.5.Given the curve c(s) = T(s) in S2, we construct the unique horizontal lifting~co = (T; v), with v(0) = N(0) (i.e. we require that the lifting to the Frenet frameand ~co agree at s = 0). In order to compute the vector�eld tangent to ~co, we worklocally in a coordinate patch U � S2 and use the map � : U � S1 �! ��1(U)to identify ��1(U) with the product space U � S1. In a local patch, there existsmooth real-valued functions �(s), �(s) for which (T;N) �= (T; ei�(s)) and~co(s) �= (T(s); h(s)); with h(s) = ei(�(s)+�(s)) 2 S1:If @� is the unit tangent vector �eld to S1 and r : R ! S1 is the exponential mapr(�) = ei�, then dhds = d(� + �)ds dr� dds� = d(� + �)ds @�. Therefore the velocity�eld of ~co is d~cods = �kN;�d�ds + d�ds� @�� :The lifting ~co is horizontal if and only if the component of its tangent vector�eldalong the vertical �bre vanishes, i.e. when d�ds = �d�ds . We will show in the nextsection (see equation (59) )that d�ds = � ; using this result and taking count of theinitial condition, we require �(s) = � Z s0 �(u)du:and we obtain the following expression for the corresponding horizontal lifting~co(s) = �T(s); e�i R s0 �(u)duN� = �T; cos(Z s0 �du)N� sin(Z s0 �du)B�(50)(where we have introduced the binormal vector B = iN). Equation (50) de�nesthe following orthonormal framing of the curveU = cos(Z s0 �du)N� sin(Z s0 �du)BV = sin(Z s0 �du)N+ cos(Z s0 �du)B:(51)This is known as the natural frame of the curve (see [5] for some history anddiscussion); it is indeed \natural" in the sense that it is the lifting of a parallelvector�eld along T(s) to the orthonormal frame bundle endowed with its canonicalconnection. The natural frame varies along the curve according to the following

24 ANNALISA CALINIsystem of linear equationsdTds = k cos(Z s0 �du)U + k sin(Z s0 �du)VdUds = �k cos(Z s0 �du)T(52) dVds = �k sin(Z s0 �du)T:Correspondingly, the components of the projection of the velocity �eld of the hor-izontal lifting with respect to U and V are called the natural curvatures, (we usethe invariant expressions of !1,!2 derived at the end of the previous section)kU = !1j~co (d~cods ) =< kN; cos(Z s0 �du)N� sin(Z s0 �du)iN >= k cos(Z s0 �du)kV = !2j~co (d~cods ) =< kN; cos(Z s0 �du)iN+ sin(Z s0 �du)N >= k sin(Z s0 �du):We can now interpret the gauge transformation that relates HM and NLS in ageometric way: the procedure described above de�nes a map from curves in S2 tocurves in the complex planeH : T(s) �! (s) = k(s) exp (i Z s0 �du):The real and imaginary component of the complex function are the components ofthe projection of the horizontal vector �eld onto TTS2. If T satis�es the HeisenbergModel equation, then the complex function (s) = k(s)ei R s0 �du is a solution of thecubic non-linear Schr�odinger equation. We summarise this discussion in theProposition 3.2. For frozen t, the gauge transformation between the HM and NLSequations is the composition of the parallel transport of a vector v(0) along the curvec(s) described by T with its projection onto TcS2.Remarks. 1. If we parallel transport a di�erent vector (there is a whole circleof choices) the change is re ected in a constant phase factor. The correspondingcomplex function has the form = k exp [i(R s0 �du+ �0)] for some real constant�0, which is still a solution of the NLS equation. Thus, choosing a particular initialtangent vector is equivalent to de�ning a map from the space of smooth curves inS2 to the space of smooth complex-valued functions quotiented by the action of S1.2. A natural frame (a choice of a parallel vector�eld) is always de�ned, alsowhen the curvature k(s) vanishes, unlike the Frenet frame. In fact, once the vectorv(0) is chosen, its horizontal lifting is unique.3. We also remark that while the Frenet lifting of a closed curve is always closed,the horizontal lifting need not be. Its holonomy is the element of the �bre S1 whichtakes the initial value of the lifting to its value at s = 2�, thereforehol(~co) = exp (i I �du):Thus, the condition for a closed lifting (i.e. for trivial holonomy) is the followingquantisation condition for the total torsionI �du = 2�j; for j 2 Z:

INTEGRABLE CURVE DYNAMICS 25In particular all closed liftings of parallel vectors along a solution of the HM aremapped to periodic solutions of the NLS equation, since (2�) = (0) exp (i H �du)while a general lifting is mapped to a quasi-periodic solution. A special case iswhen the curve itself lives on a sphere of radius r, then its total torsion vanishes[12] and its associated natural frame is closed.3.4. Comparison between the HM and NLS Phase Spaces. In this sectionwe compare the phase spaces of the HM and the NLS equation by computing ofthe di�erential of the gauge transformation H. Let L(S2) = �c : S1 ! S2 be thespace of smooth closed spherical curves, and let = f : R ! C g be the space ofsmooth complex-valued functions. We introduce the mapsL(S2) H�! ??y�=S1where H(T) = k exp (i R s0 �du), and where � is the projection onto the quotientspace =S1. In this way the composition h = � � H maps closed curves in S2 tocomplex periodic functions (we factor out the holonomy of the horizontal lifting).Next we compute the di�erential of h = � � H following a procedure used byJ. Langer and R. Perline [34] in deriving a compact expression for the di�erentialof the Hasimoto map. We express dh in terms the second symplectic operatorfor the NLS equation and relate it to the recursion operator for NLS. As a con-sequence, we show that the gauge transformation is a Poisson map which carriesthe Marsden-Weinstein Poisson structure for HM to the second Poisson structurefor NLS inducing a corresponding shift in the hierarchy of hamiltonian vector�elds.Firstly, we recall a few facts about Poisson brackets.De�nition 3.2. A Poisson bracket on a manifold M is a bilinear skew-symmetricoperation which endows the space of smooth functions on M with a Lie algebrastructure; i.e. it is a bilinear map de�ned on smooth functions on Mf ; g : C1(M)� C1(M);�!Msatisfying the following properties:ff; gg = �fg; fg (Skew symmetry)ff; ghg = gff; hg+ ff; ggh (Leibnitz rule)ff; fg; hgg+ fh; ff; ggg+ fg; fh; fgg = 0 (Jacobi Identity)Remarks. 1. If M is a Riemannian manifold (i.e. if TM is endowed with an innerproduct < ; >m) then any skew-symmetric linear operator J(m) on tangent vec-tor�elds induces a skew-symmetric bilinear map (not necessarily a Poisson bracket)of the form ff; gg(m) =< Jrf;rg > (m); m 2M:(53)(in this case the Leibnitz rule is automatically satis�ed).2. If equation (53) de�nes a non-degenerate Poisson bracket, then the 2-form< J ; > is a symplectic form on M (the Jacobi identity is equivalent to such formbeing closed) and the operator J is called a symplectic operator. For example, thesymplectic structure introduced on L(S2) in section 1.2 is obtained in this way by

26 ANNALISA CALINIusing the inner product< X; Y >L(S2) (T) = 12� Z 2�0 (X � Y )(s)dsand the skew-symmetric operatorJ jTX = T�X T 2 S2; X 2 TTL(S2):The resulting Poisson structure is known as the Marsden-Weistein Poisson bracketff; ggMW(T) = 12� Z 2�0 rg � (T�rf)(s)ds:If f : M ! N is a di�erentiable map, we can obtain an explicit formula forits di�erential that is suitable for computations, by using the notion of directionalderivative. Let g : N ! R be an arbitrary di�erentiable function on N , thenthe action of the vector �eld V on the real-valued function g � f is described byVm(g � f) =: V [g � f ](m), where the right hand side is the directional derivativeof g � f along V at the point m. Thus, we write the following expression fordf : TM ! TN : df(V )[g] = V (g � f):Before computing the di�erential of H we need the following variational formulas:Lemma 3.1. Consider a family of arclength parametrised curves (w; s) : (��; �)�S1 ! R3 and let k and � be the curvature and torsion of (0; s). Let T(w; s) be thefamily of spherical curves described by the the unit tangent vector of and denotewith W = Tw j(0;s) the variation vector�eld along T. Then, the variation of k and� along W are given byW (k) = < Ws;N >(54) W (�) = < rNW;B >s +k < W;B > :(55)In formulas (54) and (55) r is the symbol of covariant di�erentiation and thesubscript s denotes the derivative with respect to s (i.e. the covariant derivativerkN along the velocity vector of T).Proof. Since the velocity �eld Tsjw=0 = kN of T commutes with the variation W ,we have the sequence following identities0 = [W;V ] = rW (kN)�rkNW =W (k)N+ krWN� krNW:(56)Given k2 =< kN; kN >, we take the covariant derivative of both sides of thisequation in the direction of W and use (56) to obtain2kW (k) = 2 < rW (kN); kN >= 2k < rkNW;N >= 2k2 < rNW;N > :Formula (54) follows directly together with the following useful relations,[W;N] = � < rNW;N > N(57) rWN = rNW� < rNW;N > N =< rNW;B > B:(58)As for W (�), we write �2 =< �B; �B >, where �B = rkNN. Using the chain ruleand formula (56) we compute2�W (�) =2� < rW (rkNN);B >=2� < �rkNrW +r[W;kN] +R(W;kN)�N;B > :

INTEGRABLE CURVE DYNAMICS 27We have used equationrXrY Z �rYrXZ �r[X;Y ]Z = R(X;Y )Z;where R(X;Y ) is the Riemann curvature tensor. For the unit sphere we haveR(X;Y )Z =< Z; Y > X� < Z;X > Y , there followsW (�) = < rkNrWN+ kW� <N;W > kN;B >=< rkN (< rNW;B > B) ;B > +k < W;B >=kN(< rNW;B >) + k < W;B >=< rNW;B >s +k < W;B > :Remark. In the set-up of the previous section N belongs to the vertical �bre overT in T1S2. Then, in a local representation N is identi�ed with some element ei�of S1, where � is a smooth real-valued function. Thus, the action of the vector�eld W on N can be written in terms of the unit tangent vector along the vertical�bre @� as rWN = W (�)@�. Comparing this expression with formula (58) andidentifying B with @� (B is a unit vector tangent to the �bre at N), we computeW (�) =< rNW;B >. In particular, if W = kN, the velocity �eld of the curve, weobtain rkN(�) = �s = �:(59)Next we compute the di�erential of the map h = � � H and reexpress it in termsof the second Poisson structure for NLS. We �rst recall the following result whichcan be found in [40] and [20]:Theorem 3. There exist two compatible symplectic operators for the periodic NLSequation: ~J� = i�(60) ~K � = �s + 14 �Z s0 + Z s2�� [� � � �� ]du;(61)with respect to the inner product< �1; �2 >j = 12� Z 2�0 (�1 ��2 + ��1�2) du �1; �2 2 T :(62)The associated recursion operator ~R = ~K ~J�1 generates the following in�nite hi-erarchy of Poisson structuresff; ggn =< ~Rn ~Jrf;rg >; f; g 2 (63)and the sequence of Hamiltonian vector�eldsXn = ~RnX0; X0 = xRemark. It is easy to check that the NLS equation can be written as a hamilton-ian system with respect to two di�erent hamiltonians using the operators ~J and~K. Whenever this happens and the two operators are compatible ( ~J and ~K arecompatible if ~J + ~K de�nes a Poisson bracket) we say that ~J and ~K de�ne a bi-hamiltonian structure on . A bihamiltonian formulation appears to be one of theunifying properties of soliton equations. Given a bihamiltonian structure, one canformally construct an in�nite sequence of constant of motions; it is still an open

28 ANNALISA CALINIquestion though, whether every completely integrable equation admits a bihamil-tonian structure (this very question does not yet have a rigorous answer for theperiodic HM itself!). The reader who is interested in this aspect of integrability isreferred to [45, 40, 10].Finally we compute the di�erential of H(T) = k exp (i R s0 �du). Using the chainrule and lemma 3.1 we writedH(W ) =W (k)ei R s0 �du + ik�Z s0 W (�)du� ei R s0 �du =(< Ws;N > +i < Ws;B >) ei R s0 �du + i Z s0 < W; kB > du+ ic ;c is a constant of integration, and since the kernel of d� contains vector�elds of theform c , we can write equivalentlydH(W ) = (< Ws;N > +i < Ws;B >) ei R s0 �du + i2 �Z s0 + Z s2�� < W; kB > du:We rewrite W = gU + hV on the basis of the natural frames and introduce thecomplex vector �eld �(W ) = g + ih. Using the Darboux equations (52) for thecomponents of the natural frame, we can rewrite the above expression asdH(W ) = dds�(W ) + 14 �Z s0 + Z s2�� [�(W ) � � ��(W ) ]du:(64)The right-hand side of (64) is nothing but the second Poisson operator K forthe NLS equation acting on the tangent vector �(W ); we summarise this in thefollowingProposition 3.3. The di�erential of the composed map h : L(S2) ! =S1, h =� � H is given by: dh = d� � ~Kh(T) � �:(65)An important consequence of formula (65) is that h preserves Poisson structures,in particular it carries the Marsden-Weinstein Poisson bracket into the second Pois-son bracket for the NLS equation. In order to show this we need the followingtechnical result:Lemma 3.2. r(f � h)jT = dh�(rf)jh(T) ; f 2 =S1� � J � �� = ~J:Here �� is the adjoint of � with respect to the inner product (62), J is thesymplectic operator T� of HM and ~J is simply multiplication by i. The proof ofthis lemma is a one line computation (in a similar context see [34]) and it is left tothe reader.Corollary 2. The map h is a Poisson map and the following relation holdsff � h; g � hgMW(T) = ff; gg2(h(T));(66)for f; g 2 =S1.

INTEGRABLE CURVE DYNAMICS 29Proof.ff � h; g � hgMW(T) = < Jr(f � h);r(g � h) > (T) =< J � dh�rf; dh�rg > (h(T)) =< dh � J � dh�rf;rg > (h(T)) =< d� � ~K � � � J � �� � ~K� � d��rf;rg > (h(T)) =< ~K � ~J � ~K�d��rf; d��rg > (h(T)) =< ~R2 ~Jr(f � �);r(g � �) > (H(T)) =ff � �; g � �g2(H(T)) = ff; gg2(h(T)):(We used the skew-symmetry of ~K and the identity ~J�1 = ~J .) Therefore, we canview the NLS equation and the HM as the same hamiltonian system written withrespect to two di�erent Poisson structures which belong to the same integrablehierarchy. 4. B�acklund transformation and immersed knotsA fundamental algebraic property of soliton equations is the fact that they ariseas compatibility conditions for a pair of linear operators. In section 3.2 we dis-covered a deep relation among the in�nite family of multi-phase solutions, thenonlinear equation and its associated Lax pair in the case of the NLS and HMequations. The zero-curvature formulation of integrable partial di�erential equa-tions is at the basis of the inverse scattering method for �nding exact solutions(see for example [20]). Moreover, the hierarchy of conserved quantities and therelated conservation laws can be constructed in a purely algebraic way through theknowledge of the Lax pair, independently of global properties such as the bound-ary conditions and the di�erentiability class of the solutions. Another seeminglyuniversal feature of soliton equations, also intimately connected to the presenceof a Lax pair, is the existence of B�acklund transformations. A B�acklund formulaproduces more complicated solutions of the nonlinear equation from a simpler so-lution for which the pair of linear systems can be solved exactly. The close relationamong inverse scattering, the in�nite number of conservation laws and B�acklundtransformations was �rst discussed in 1974 in an elementary and intriguing paperby M. Wadati, H. Sanuki and K. Konno [55]. In their 1983 papers [21, 22, 23],H. Flaschka, A. Newell and T. Ratiu developed a general algebraic framework inwhich to interpret the unifying properties of soliton equations. The central ideais to view the hierarchy of integrable equations as a sequence of commuting owson an in�nite-dimensional loop algebra (a Kac-Moody Lie algebra, see the originalpapers or Chapter 5 in [44]), and to think of the solution as a function of an in�nitenumber of independent variables, none of which playing a distinguished role (x isnow just one of the time variables).In this section, we will simply focus on the use of B�acklund transformationsto construct new interesting solutions of soliton equations from simpler solutions.It is instructive to start with describing the B�acklund formula for the nonlinearSchr�odinger equation iqt = qxx + 2jqj2q;(67)

30 ANNALISA CALINIto be considered together with its associated Lax pair@@xFNS = �i��3 + i� 0 �qq 0 ��FNS@@tFNS = �(2i�2 � ijqj2)�3 +� 0 2i��q + �qx2i�q � qx 0 ��FNS:(68)The B�acklund transformation for equation (67) is obtained in the following way(the proof is a direct veri�cation, see [50] for its derivation in the context of gaugetransformations). Let ( +; �) be two independent solutions of the linear system(68) at (q; �):We construct the following quantities (where c� are complex arbitraryconstants) = c+ + + c� �NNS = � 1 � � 2 2 � 1 �GNS = NNS� �� � 00 �� �� �NNS�1:Then F(1)NS(x; t;�; �) = GNS(�; �; )FNS(x; t;�)(69)solves equation (68) at (q(1); �), whereq(1) = q + 2(� � ��) 1 � 2j 1j2 + j 2j2(70)is the corresponding new solution of the NLS equation.B�acklund transformations can be used to generate homoclinic orbits of a givensolution. The importance of understanding the homoclinic manifolds of N-phasesolutions that possess nonlinear instabilities is discussed in the work of N. Ercolani,G. Forest and D. McLaughlin [16, 17, 18]. Unstable N-phase solutions, just likesaddle points and periodic orbits of saddle-type in �nite-dimensional integrablesystems, reside on singular level sets in phase space. A complete topological de-scription of the singularities of the foliation of the space of N-phase solutions canbe achieved by means of B�acklund transformations. B�acklund formulas produceexplicitly the homoclinic manifolds and thus label the underlying level sets. More-over, using B�acklund transformations one is able to construct both the tangent andnormal vector�elds to the level set of a given solution and to generate families ofsolutions residing on the same level set. In fact [17], a �nite number of iteratedB�acklund transformations generates the entire level set of a given N-phase solution.These ideas are hereby applied to curves in space, a context in which the relevantstructures can be more easily visualised. We will use B�acklund formulas in a two-fold way. Firstly, we will construct curves which are homoclinic to a given curve andconjecture that such homoclinic curves play a role in separating di�erent classes ofknots. Secondly, we will use the theory of B�acklund transformations to understandthe symmetries of a given curve. All the computations will be carried out in thesimplest case of planar circles, where interesting results already arise.

INTEGRABLE CURVE DYNAMICS 314.1. The B�acklund formula for the Heisenberg Model. We �rst derive theB�acklund transformation for the HM equation and the expression for the corre-sponding transformed curve [8]. To this end, we use the B�acklund formula (69) andthe expression of the gauge transformation between the HM and NLS.Proposition 4.1 (HM B�acklund Transformation). Let � = c+�+ + c��� be acomplex linear combination of linearly independent solutions of the Lax pair (24)at (S; �). We construct the matrix of gauge transformationG(�; �;�) = N � ���� 00 ������ �N�1(71)with N = � �1 ���2�2 ��1 � :(72)Then, if F solves the linear system (24) at (S; �), the vectorF(1)(x; t;�; �) = G(�; �;�)F(x; t;�)(73)solves (24) at (S(1); �) and the new solution of the HM equation is given by theB�acklund formulaS(1)(x; t) = N � e�i� 00 ei� �N�1S(x; t)N � ei� 00 e�i� �N�1(74)with ei� = �j�j :Remarks. 1. The matrix U = N � ei� 00 e�i� �N�1 is unitary, hence the solutionS(1) is hermitian and has zero trace (and its determinant is 1). The correspondingvectorT(1) is therefore obtained by a rotation of the tangent vectorT of the originalcurve by a constant angle � = �j�j depending only on the eigenvalue parameter �,around the instantaneous axis of rotation� 2Re(�1 ��2)j�1j2 + j�2j2 ;� 2Im(�1 ��2)j�1j2 + j�2j2 ; j�1j2 � j�2j2j�1j2 + j�2j2�T :2. If � 2 R, then S(1) = S and the B�acklund transformation is the identity. If � ischosen such that the corresponding eigenfunction � is periodic or antiperiodic inthe x-variable (such � is called a periodic/antiperiodic eigenvalue), then the corre-sponding solution is periodic (i.e. the transformed curve is closed). For any otherchoice of � 2 C , formula (74) produces quasi-periodic solutions of the nonlinearequation.Proof. [8] The gauge formula allows us to write the eigenfunction of the NLS Laxpair at (q; �) in terms of the solution of the HM linear system at (S; �) as FNS = V F,where V is the gauge matrix introduced in section 1.5. The eigenvector F(1)NS of theNLS Lax pair at (q(1); �) can similarly be expressed as F(1)NS = V (1)F(1), where V (1)is the new gauge matrix. Since V (1) solves the linear system for NLS at (q(1); 0),it can be written as V (1) = GNSj�=0V and we can write the new eigenfunction forthe HM F(1) = V �1 (GNSj�=0)�1GNSV F:(75)

32 ANNALISA CALINIWemake the important observation that the expression of F(1) is independent of thegauge transformation, and therefore it does not depend on the NLS eigenfunctions.In order to show this we use the fact that V is a unitary matrix with determinant1, and computeV �1NNS = V �1� 1 � � 2 2 � 1 � = � �1 ���2�2 ��1 � = N;(76)with = V � and � = c+�++ c��� a complex linear combination of independenteigenfunctions of the linear system (24). Using (76) to simplify equation (75) weobtain expression (73) for the new eigenfunction. Substituting the expression forF(1) in the equation @@xF(1) = i�S(1)F(1);(77)we derive the following formula for the transformed potentialS(1) = 1i� @G@x G�1 +GSG�1:(78)The dependence on � is only apparent, by computing the x-derivative of G andrearranging terms we can reduce (78) to the form of equation (74).In order to derive the formula for the corresponding curve in space, we use the rep-resentation of the curve in terms of the fundamental solution of the linear problem.�new = �i @@� �����=0 �new��1newj�=0 + C;(79)where C is a hermitian x-independent. Using expression (73) for �new, we �nd (upto an x-independent translation)Corollary 3. The B�acklund transformation for the �lament equation is describedby the following formula (in matrix representation):�new = �+ Im(�)j�j2 V(80)with V = N�3N�1(81)Formula (80) reduces to the identity when the eigenvalue � is real, and it producesbounded solutions of the �lament equation (not necessarily periodic in the spacevariable x) for any complex �. In fact, directly from the formula for the new curvewe deduce the following con�nement result,Proposition 4.2. k�k � jIm(�)jj�j2 � k�newk � k�k+ jIm(�)jj�j2(82)( k � k is the standard Euclidean norm in R3). The transformed curve is con�nedto the interior of a sphere if jIm(�)jj�j2 � maxx k�(x)k or to the interior of a torus ifjIm(�)jj�j2 < maxx k�(x)k.

INTEGRABLE CURVE DYNAMICS 33Sym et al. [11] obtained this same con�nement result (which is of interest in thecontext of vortex dynamics) by reconstructing the curve from a basis of eigenfunc-tions of the NLS Lax pair and by using directly the NLS B�acklund transformation.4.2. An application to planar circles. The simplest solutions which possesshomoclinic instabilities are planar circles. Their tangent vector�elds are �xed pointsof the Continuous Heisenberg Model, therefore the expression of the associatedcurve can be chosen to be time-independent. We consider k copies of a circle lyingin the (x; y)-plane with non zero curvature k 2 Z (an integral curvature is requiredfor the circle to be closed). If x denotes the arclength parameter, the hermitianmatrix which represents the unit tangent vector isS0(x; t) = � 0 e�ikxeikx 0 �(83)Following [16, 17, 18], we make use of the Floquet spectrum of the linear operatorL(S; �) = �� 1 00 �1 � ddx + i�S(84)(the spatial part of the HM Lax pair) in order to characterise the level set on whichS0 resides. The fundamental solution matrix �(x; y;S; �) is de�ned by the followingproperties L(S; �)� = 0;�(x; x;S; �) = � 1 00 1 � :(85)We introduce the transfer matrix �(x + 2�; x;S; �) as the transformation whichtakes a solution of (84) evaluated at x to its value after a period 2�. It is easy tocheck that the transfer matrix has determinant 1, as a consequence its eigenvaluesare completely characterised in terms of its trace�(S; �) = Trace [�(x+ 2�; x;S; �)] :(86)The quantity �(S; �) is called the Floquet discriminant of the linear operator L.Recalling that the spectrum of a linear operator is the set of eigenvalues for whichthe corresponding eigenfunction is bounded in x, then the spectrum �(L) of L isthe set of �'s for which the eigenvalues of the transfer matrix lie on the unit circle.In terms of the trace of the Floquet discriminant,�(L) = f� 2 C j�(S; �) 2 R and � 2 � �(S; �) � 2g :The discriminant has the following properties (property 1 follows from the analyt-icity of the fundamental solution of the linear system, the veri�cation of property2 is a simple computation):Theorem 4. If S solves the Continuous Heisenberg Model then,1. �(S; �) is an analytic function of �.2. �(S; �) is invariant under the evolution, i.e.ddt�(S(t); �) = 0:(87)

34 ANNALISA CALINIRemark. Since the discriminant is a constant of motion which depends analyticallyon the complex parameter �, � provides us with a 1-complex parameter family ofinvariants. In fact, one can extract the usual hierarchy of constant of motions fromthe coe�cients of the asymptotic expansion of the discriminant at a distinguishedvalue of � [20]. Moreover, we can characterise the level set on which a givensolution S0 resides as its isospectral set (the set of all potentials which have thesame spectrum as S0).We introduce the following distinguished points of �(L):critical points �c: dd��(S; �)�����=�c = 0;periodic (antiperiodic) points ��:�(S; �)j�=�� = �2;multiple points �m: �(S; �)j�=�m = �2dd��(S; �)�����=�m = 0The periodic (antiperiodic) points are associated with periodic (antiperiodic) eigen-functions. The B�acklund transformation at one such point produces a solutionwhich is periodic in x. The existence of complex multiple points (we will consideronly double points, i.e. critical points of multiplicity 2) is related to the presenceof linear instabilities. The level set of a solution whose spectrum contains com-plex double points is saddle-like and the corresponding homoclinic orbits can beconstructed by means of B�acklund transformations. We refer the reader to refer-ences [16, 17, 18] for the discussion of these general results. We will instead inferthese conclusions for the simplest example of planar circles and their associatedB�acklund transformations.The HM Lax pair can be easily solved when S is the matrix S0 representing amultiple covered circle as in (83). The associated Floquet discriminant is�(S0; �) = �(k; �) = (�1)k2 cos(��) with � =pk2 + 4�2;and the corresponding spectrum is (see �gure 4.2)�(S0; �) = f� 2 C j cos(��) 2 Rg = R [ fi� j � 2 R ; �k=2 � � � k=2g:The critical points are the set of solutions ofdd�� = � sin(��) 4�pk2 + 4�2 = 0:All of them are double points except for � = 0 which is a critical point of multiplicity4. There is a �nite number of complex double points (in this case they are purelyimaginary) given by�n = � i2pk2 � n2 n = 1; 2; : : : ; k � 1:We now implement the B�acklund transformation derived in the previous section

INTEGRABLE CURVE DYNAMICS 35complex double point

band of spectrum

-ik/2

ik/2 planeλ−

Figure 3. The spectral con�guration of S0and construct the corresponding new solutions of the HM. We choose two linearlyindependent solutions of the linear problem at (S0; �) as follows,�� = e� i2 �(x+2�t) �2 �k+� e� i2 kxe i2kx ! ; �+ = e i2 �(x+2�t)� e� i2 kx��k2 e i2kx � :�+ and �� are called Baker eigenfunctions and are uniquely characterised by theirexpression in terms of an exponential phase factor multiplied by a bounded (peri-odic) vector function.We distinguish two classes of B�acklund transformations which generate qualita-tively di�erent solutions in the level set of S0:Type I B�acklund Transformation. If we take � in formula (80) to coincide with oneof the Baker eigenfunctions at the eigenvalue �, we obtain a periodic solution forevery choice of � 2 C . Moreover such solution belongs to the same class S0; inparticular, for � = �+ we obtain the following family of planar circles�1(x; �) =� 0 ik e�ikx� ikeikx 0 �+Im(�)j�j2(4j�j2 + j� � kj2) � 4j�j2 � j� � kj2 4�(�� � k)e�ikx4��(� � k)eikx 4j�j2 � j� � kj2 �(88)parametrised by the complex parameter �.Remark. In order to interpret the expression of the new curve as the e�ect of theaction of a symmetry, we compute an in�nitesimal B�acklund transformation asIm(�) ! 0. In this limit, �1 � �0 approximates a family of vector�elds along theoriginal curve �0 of the formV(x; �) = c1(�)(0; 0; 1)T + c2(�)S0:(89)where ci's are constants depending on �. Formula (89) describes a family of Killingvector�elds (in�nitesimal isometries) for the the original circle: the sum of a con-stant translation �eld and a rotation. Already in this simple example we see that

36 ANNALISA CALINIa type I B�acklund transformation realises the symmetries of the level set of the so-lution S0. Such symmetries manifest themselves at the curve level as in�nitesimalisometries.Type II B�acklund transformation. We now take � = c��� + c+�+ to be a generalcomplex linear combination of Baker eigenfunctions. In this case, the transformedcurve is closed if � is a periodic/antiperiodic eigenvalue. When � is chosen to beone of the complex double points � = i�n = � i2pk2 � n2, n = 1; : : : ; k� 1, settingc+=c� = �ei�, we obtain the following formula:�2(x; t;�n) = � 0 ik e�ikx� ikeikx 0 �+1�n � f(x; t)� f(0; t) g(x; t)� g(0; t)g(x; t)� g(0; t) �(f(x; t)� f(0; t)) � ;wheref(x; t) = � 12�n h0(t)kh(t) + 8��2n sin(nx+ �) ;g(x; t) = �2i�nk e�ikx � 4 �n�n cos(nx+ �) + ink sin(nx+ �)kh(t) + 8��2n sin(nx+ �) e�ikx ;h(t) = [k + n+ (k � n)�2] sinh(n�t) + [k + n� (k � n)�2] cosh(n�t):Since limt!�1 �2(x; t) = �0(x), the new curve �2 is a homoclinic orbit for theoriginal solution, (this also implies that multiple covers of circles are unstable solu-tions of the �lament equation). Moreover we conclude that �2 resides on the samelevel set as �0, indeed spectral con�gurations such as the one depicted in Figure 3describe both the target solution and its homoclinic orbits.Figure 4 shows a sequence of time frames of the evolution of the curve �2 homo-clinic to a 6-fold circle. The new curve possesses 5 points of self-intersection whichpersist throughout the dynamics. It appears to be a general feature of these singu-lar curves that the number of such \stable" self-intersections equals the ordering ofthe complex double point.An interesting question is whether the presence of self-intersections has a topo-logical meaning. A more general related question is whether the curves producedby a type II B�acklund transformation play a role in distinguishing di�erent knotclasses of multiphase solutions, and whether the Floquet spectrum can be used tocharacterise topological invariants. Figures 5b, 5c show typical spectral con�gu-rations of initial data close to a planar circle with a single complex double point(Figure 5a). The spectra can be computed using standard perturbation analysis(we consider only spatially even solutions, the general case being far more com-plicated [19]). The degenerate double point splits in a \cross" (b) or a \gap" (c)con�guration characteristic of di�erent 3-phase solutions which in turn correspondto complicated, likely knotted curves in space. Figure 5 illustrates the role of the ho-moclinic orbit to the planar circle as a separatrix between di�erent types of 3-phasesolutions, how this should be interpreted at the curve level and whether this factcarries a topological signi�cance is an open problem being currently investigatedby the author. A positive answer would support the use of the Floquet spectrumin the topological classi�cation of N-phase curves.

INTEGRABLE CURVE DYNAMICS 37t = �23 t = �13 t = �7t = �3 t = 0 t = +5Figure 4. Evolution of the B�acklund transformation of a 6-foldplanar circle. k = 6; n = 5; � = 1; � = 0

(a) (c)(b)Figure 55. Future directionsWe conclude with a description of current research activities and a list of openquestions.To my knowledge, the following groups have been working on related topics: A.Sym et al. (see e.g. [11, 54, 53] have worked on applications of soliton theory to vor-tex �lament dynamics and on the characterisation of soliton surfaces generated bythe evolution of vortex �laments in the shape of solitons. H. K. Mo�at, R. L. Riccaand collaborators (at Cambridge and University College, London) have focused on

38 ANNALISA CALINIapplications of knot theory to uid mechanics (see e. g. [49, 47, 48]. The workof J. P. Keener and J. J. Tyson [29, 28] on scroll waves in excitable media and onvortex �laments in ideal uids has important points of contacts with the author'swork.The group at Case Western Reserve University including J. Langer, D. Singer,T. Ivey and myself, together with collaborator R. Perline at Drexel University, haveworked on higher-dimensional generalisations of the �lament equation hierarchy andon the topological implications of related B�acklund transformations. A. Doliwa andP. M. Santini have addressed similar questions about integrable curve evolutions[13] on n-dimensional spheres and have also discussed the discrete problem in thecontext of polygonal spherical curves [14]. On the same theme of polygonal curves,J. Millson at the University of Maryland and M. Kapovich at Utah State Univer-sity [27] have studied the topology of the space of polygonal knots, its symplecticstructure and an associated integrable evolution. The continuous counterpart hasbeen studied by J. Millson and B. Zombro [43], with a focus on the symplecticgeometry of smooth curves in Euclidean space and on the bihamiltonian structureof the periodic Heisenberg Model.There are several open directions of research.1. The formulation and study of more realistic models of vortex �lament dynam-ics which take account of the �nite structure of the vortex core, the longitudinalstretching of the �lament, nonlocal interactions and the e�ect of �nite viscosity.2. Discrete models of integrable curve dynamics (i.e. evolution equations of polyg-onal curves): discrete soliton equations such as the Ablowitz-Ladik model or inte-grable discretisations of the Heisenberg Chain are among them.3. Other integrable hierarchies are associated to curve dynamics, among them theKdV, mKdV and the Sine-Gordon equations: a natural question is whether anysoliton equation can be associated with a curve evolution.4. Closely related to the previous question is the construction of a general frame-work in which to describe integrable hierarchies of curve dynamics in a space ofarbitrary dimension; a suitable set-up appears to be the Kac-Moody Lie algebraframework where a soliton equation is regarded as a dynamical system on an in�nite-dimensional loop algebra. Since this work was written Langer and Perline haveshown that dynamical systems associated with Hermitian symmetric Lie algebrascan be used to model the evolution of arclength parametrized curves in Euclideanand other spaces, see [36] for details.5. In the context of higher phase solutions of the �lament equation, what do thehigher N-phase curves look like? is it possible to use the Floquet spectrum of thespatial part of the associated Lax pair to classify their knot type? The authoris currently working on implementing the general expression for N-phase curves interms of Riemann theta functions discussed in this work. Recently, in an interestingarticle by P. G. Grinevich and M. U. Schmidt [25], conditions for the closure of thecorresponding curves in terms of the NLS spectral problem are derived (generaliz-ing the relation between complex double points and close B�acklund transformationsthat we explicitly computed in the elementary case of planar circles).6. Also, does the B�acklund transformation carry any topological information? Apartial answer has been provided (since the date of this article) by the author andher collaborator T. Ivey [9] with respect to closed constant torsion elastic rods, thatcan be associated with the Sine-Gordon equation. B�acklund transformations areshown to provide a variety of topological phenomena such as knotting, unknottingand self-intersections. Questions about the mechanisms of topological changes are

INTEGRABLE CURVE DYNAMICS 39gaining attention in a variety of �elds (e.g. [41]).7. Mo�at and Ricca posed an analogous question for the sequence of global geomet-rical invariants: whether they restrict the possible knot types of the correspondingcurves and whether they themselves have a topological interpretation.8. The Marsden-Weistein Poisson structure is realised naturally at the level ofcurves as a 90-degree rotation of any normal vector�eld to the curve and it is asso-ciated to the almost-complex structure on the loop space of R3 : an open questionis whether the higher Poisson structures have a natural geometrical realisation, andwhether the zero-curvature formulation of the sequence of NLS ows (and of generalsoliton equations) can be given a direct geometric interpretation.AcknowledgementsThe original results in this article are based on my dissertation work at theUniversity of Arizona. Many thanks go to my advisor Nick Ercolani for hours andhours of stimulating discussions and exciting mathematics. I am also grateful to mycollaborators in the \geometry group" at Case Western Reserve University: JoelLanger, David Singer and Tom Ivey.References[1] M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM,Philadelphia, 1981.[2] V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York,1980.[3] H. E. Baker, Proc. Royal Soc. London Ser. A, 118 (1928), pp. 573{580.[4] G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, 1967.[5] A. Bishop, There is more than one way to frame a curve, Amer. Math. Monthly, 82 (1975),pp. 246{251.[6] J.-L. Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantization,Birkh�auser, Basel, 1992.[7] J. L. Burchnall and T. W. Chaundy, Proc. London Math. Soc., 21 (1922), pp. 420{440.[8] A. Calini, Integrable Curve Dynamics, PhD thesis, University of Arizona, 1994.[9] A. Calini and T. Ivey, B�acklund transformations and knots of constant torsion, (1996).submitted.[10] P. Casati, F. Magri, and M. Pedroni, Bihamiltonian manifolds and Sato's equations, inIntegrable Systems: the Verdier Memorial Conference, vol. 115 of Progress in Mathematics,Birkh�auser, 1993.[11] J. Cielinski, P. Gragert, and A. Sym, Exact solutions to localized approximation equationmodelling smoke ring evolution, Phys. Rev. Lett. A, 57 (1986), p. 1507.[12] M. do Carmo, Di�erential Geometry of Curves and Surfaces, Prentice Hall, 1976.[13] A. Doliwa and P. M. Santini, An elementary characterization of the integrable motions ofa curve, Phys. Lett. A, 185 (1994), pp. 373{384.[14] , Integrable dynamics of a discrete curve and the Ablowitz-Ladik hierarchy, J. Math.Physics, (1995). in press.[15] B. A. Dubrovin, Theta functions and nonlinear equations, Russian Math. Surveys, 36 (1981),pp. 11{92.[16] N. M. Ercolani, M. G. Forest, and D. W. McLaughlin, Geometry of the modulationalinstability, I: Local analysis. preprint, 1987.[17] , Geometry of the modulational instability, II: Global analysis. preprint, 1987.[18] , Geometry of the modulational instability, III: Homoclinic orbits for the periodic Sine-Gordon equation, Physica D, 43 (1990), pp. 349{384.[19] N. M. Ercolani and D. W. McLaughlin, Towards a topological classi�cation of integrablePDE's, in The Geometry of Hamiltonian Systems, Springer-Verlag, 1991.[20] L. D. Faddeev and L. A. Takhtajain, Hamiltonian Methods in the Theory of Solitons,Springer and Verlag, 1980.

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INTEGRABLE CURVE DYNAMICS 41[51] Seeger, H. Donth, and A. Kochend�orfer, Theorie der Versetzungen in eindimension denAtomreiher iii Versetzungen, Eigenbewegungen und ihre Wechselworkung, Zeit. Physik, 134(1953), pp. 173{193.[52] I. M. Singer and J. A. Thorpe, Lecture Notes on Elementary Topology and Geometry,Springer-Verlag, 1967.[53] A. Sym, Soliton surfaces I-V, Lettere al Nuovo Cimento, 33, 36, 39, 40, 42 (1982-84), pp. 394{400.[54] , Vortex �lament motion in terms of Jacobi Theta functions, Fluid Dynamics Research,3 (1988), pp. 151{156.[55] M. Wadati, H. Sanuki, and K. Konno, Relationships among inverse method, B�acklundtransformation and an in�nite number of conservation laws, Prog. Theor. Phys., 53 (1975),pp. 419{436.University of Charleston Dept. of Mathematics 66 George St., Charleston SC 29424USAE-mail address: calini@stelvio.cofc.edu