#### Analytic and geometric theory of the Camassa-Holm equation and Integrable systems

Abstracts of talks

1) S. Abenda (Università di Bologna) and T. Grava (SISSA): Modulation equations for Camassa Holm and reciprocal tranformation

Using the Whitham method, we derive the modulation equations for the Camassa-Holm equation
 ut + 3u ux = uxxt+2uxuxx+uu3x-2nux,
where n is a constant parameter. The Camassa-Holm equation is part of a bi-hamiltonian system with an infinite number of non-local conservation laws. The Camassa-Holm equation can also be written as a local Lagrangian system.

The Whitham averaged equations for a nonlinear evolution system describe slow modulations of parameters of a family of periodic travelling wave solutions (or family of multi-phase quasi periodic solutions which are so far known to exist only for integrable equations). To derive the modulation equation we use the Whitham averaging theory which applies to local Lagrangian systems and we obtain the modulation equations (which are hyperbolic) in the Lagrangian form. Then we construct the corresponding Hamiltonian structure. Despite the CH equation has a nonlocal bi-Hamiltonian structure the corresponding Whitham averaged equations have a local Hamiltonian structure. Furthermore, we show that there exists two local compatible Hamiltonian structure, so that the Whitham averaged equations are bi-Hamiltonian as the original system. Therefore an infinite number of conservation laws can be derived and the equations can be integrated.

Moreover it is known that a reciprocal transformation maps the Camassa-Holm equation to the first negative flow of the Korteweg-de Vries hierarchy. We show that the averaged transformation maps the modulated CH equations to the modulated negative KdV equations.

2) R. Camassa (University of North Carolina, Chapel Hill): Integral and integrable algorithms for a nonlinear shallow-water wave equation
An asymptotic higher-order model of wave dynamics in shallow water is examined in a combined analytical and numerical study, with the aim of establishing robust and efficient numerical solution methods. Based on the Hamiltonian structure of the nonlinear equation, an algorithm corresponding to a completely integrable particle lattice is implemented first. Each ``particle'' in the particle method travels along a characteristic curve. The resulting system of nonlinear ordinary differential equations can have solutions that blow-up in finite time. Conditions for global existence are identified and l1-norm convergence of the method proved in the limit of zero spatial step size and infinite particles. The numerical results show that this method captures the essence of the solution without using an overly large number of particles. A fast summation algorithm is introduced to evaluate the integrals of the particle method so that the computational cost is reduced from O(N2) to O(N), where N is the number of particles. The method possesses some analogies with point vortex methods for 2D Euler equations. In particular, near singular solutions exist and singularities are prevented from occurring in finite time by mechanisms akin to those in the evolution of vortex patches. The second method is based on integro-differential formulations of the equation. Two different algorithms are proposed, based on different ways of extracting the time derivative of the dependent variable by an appropriately defined inverse operator. The integro-differential formulations reduce the order of spatial derivatives, thereby relaxing the stability constraint and allowing large time steps in an explicit numerical scheme. In addition to the Cauchy problem on the infinite line, we include results on the study of the nonlinear equation posed in the quarter (space-time) plane. We discuss the minimum number of boundary conditions required for solution uniqueness and illustrate this with numerical examples.

3)  G. Carlet (University of Cambridge):  Hamiltonian structures for the two-dimensional Toda hierarchy and R-matrices

4)  A. Constantin (Trinity College, Dublin): Geodesic flow on the diffeomorphism group of the circle and on the Virasoro group
We study the geodesic exponential maps corresponding to Sobolev type right-invariant Riemannian metrics mk (k ³ 0) on the diffeomorphism group of the circle and on the Virasoro group. For k ³ 1 , but not for k=0, in the case of the diffeomorphism group and for k ³ 2, but not for k=0,1, in the case of the Virasoro group, each of them defines a smooth chart. In particular, the geodesic exponential map corresponding to the periodic Korteweg-de Vries equation (k=0 on the Virasoro group) is not a local diffeomorphism near the origin whereas the geodesic exponential map corresponding to the periodic Camassa-Holm equation (k=1 on the diffeomorphism group of the circle) is a local diffeomorphism near the origin. These results have been obtained as joint work with Boris Kolev (for the diffeomorphism group of the circle) and with Thomas Kappeler, Boris Kolev, and Peter Topalov (for the Virasoro group).

5) B. Dubrovin
(SISSA): TBA

6) G. Falqui
(SISSA): On a two-component generalization of the CH equation
By using the theory of Lie Poisson affine bihamiltonian pencils, we will derive
the generalization of the CH hierarchy with two dependent variables recently introduced by S-Q. Liu and Y. Zhang (math.DG/0405146).
A few of its properties will be presented

7) Yu. Fedorov (UPC, Barcelona):
Geodesic Billiards on Quadrics and elliptic Calogero--Moser systems

8) D.D. Holm (
Imperial College London and LANL): Weak solution interactions in nonlinear internal waves and in computational anatomy

The depth-averaged 3D Euler equations for shallow water flow are well approximated by 2D equations of geodesic motion for a certain Sobolev norm. These are the 2+1 EPDiff equations, or ``Euler-Poincaré equations on the diffeomorphisms.'' 1+1 EPDiff is the CH equation, whose weak solutions are solitons, called peakons. The initial value problem for 2+1 EPDiff produces soliton-like weak solutions, supported on curves that evolve in the plane. Numerics shows these filamentary solutions supported on delta-functions emerge in the initial value problem (IVP) for any confined smooth initial velocity distribution.

Besides dominating the IVP, these weak EPDiff solutions have three otherinteresting dynamical properties:

• they superpose,
• they form an invariant manifold and
• their nonlinear interactions allow them to reconnect with each other in 2D.
The phenomenon of reconnection seen in the IVP for EPDiff is also observed for oceanic internal waves in synthetic aperture radar (SAR) observations seen from the space shuttle. Thus, in accord with their original derivation, weak solutions of EPDiff provide a simplified 2D description of evolving arrays of interacting internal waves in the Ocean.
Remarkably, the same family of geodesic equations also arises in image processing using the template matching approach to computational anatomy. Here, for example, a measure-valued 2+1 EPDiff solution at two times corresponds to the outlines, or ``cartoons," of an image and its target image under the template-matching map. The nonlinear exchange of momentum seen in the interactions of these ``cartoons" introduces the collison paradigm from soliton dynamics into imaging science. Namely, the optimization problem for image matching corresponds to an evolutionary problem in which image outlines exchange momentum and may reconnect as their positions evolve.

The existence of these measure-valued solutions of EPDiff is guaranteed - for any Sobolev norm, and in any number of spatial dimensions - because the weak solution ansatz is a momentum map for the (left) action of diffeomorphisms on the measure-valued support set of the solutions.
We review the derivation of EPDiff and show numerical and analytical results for its solutions in 1+1, 2+1 and 3+1. (EPDiff - what an equation!)

9)  A.N.W. Hone (University of Kent at Canterbury): Hamiltonian structures for integrable and non-integrable relatives of the Camassa-Holm equation
We consider a family of integro-differential equations depending upon a parameter b as well as a symmetric integral kernel g(x). When b=2 and g is the peakon kernel, g(x)=exp(-|x|), the Camassa-Holm (CH) equation results, while the Degasperis-Procesi (DP) equation is obtained from the peakon kernel with b=3. Although these two cases are integrable, generically the corresponding integro-PDE is non-integrable. However, for b=2 the family restricts to the pulson family of Fringer and Holm, which is Hamiltonian and numerically displays elastic scattering of pulses. On the other hand, for arbitrary b it is still possible to construct a nonlocal Hamiltonian structure provided that g is the peakon kernel or one of its degenerations: we present a proof of this fact using an associated functional equation for the skew-symmetric antiderivative of g. We briefly survey other ways in which CH and DP can be isolated within this family, as follows: work with Wang on bi-Hamiltonian structures; reciprocal transformations and Painleve tests; prolongation algebras; and the perturbative symmetry approach of Mikhailov and Novikov.

10) P. Lorenzoni (Università di Milano Bicocca) and M. Pedroni (Università di Genova): On the bi-Hamiltonian structures of the Camassa-Holm and Harry Dym equations

11)  F. Magri  (Università di Milano Bicocca): In search of separation coordinates
After a brief review of the geometry of separable systems , I present a new algorithm for finding the separation coordinates.The algorithm works for the Kowalevski's top .

11) F. Musso (Università di Roma III): Algebraic contractions of Gaudin models
Starting from the (rational, trigonometric or elliptic) Lax matrix for a N-bodies classical Gaudin model associated to a simple Lie algebra g, we obtain, through a Inonu-Wigner contraction, a new Lax matrix satisfying the same r-matrix structure as the uncontracted one. The whole procedure leads us to define a new integrable system on the dual of the N-th Jet extension of the original Lie algebra g

12) V. Novikov (LITP Moscow and University of Kent at Canterbury): Perturbative Symmetry Approach
One of the main problems in the theory of Integrable systems  is a problem of recognition of Integrable PDE's. There are  several approaches to this issue and one of them is Symmetry  Approach developed by A.B. Shabat et al.  In the symmetry approach the existence of infinite hierarchies of higher symmetries and/or local conservation laws is taken as a definition of integrability. The main aims of the theory is to obtain easily verifiable necessary conditions of integrability, to identify integrable cases and even to give a complete classification of integrable systems of a particular type. The main aim of Perturbative Symmetry Approach is to extend the theory in order  to make it suitable for study of non-local and non-evolution equations.  Our formalism is the development and incorporation of the perturbative  approach of Zakharov and Schulman, symbolic method of Sanders and Wang, and
symmetry approach.
We apply the theory to describe integrable generalisations of Benjamin-Ono type equations and to isolate integrable cases of Camassa-Holm
type equations.

13) O. Ragnisco (Università di Roma III):  Exact  time-discretization of the Lagrange top
An exact time discretization of the Lagrange top will be presented, in the framework of the "Lax-Sklianin" approach. The problem of the interpolating hamiltonian flow will be discussed and hopefully solved.

14) A. Shabat
( and Università di Roma III):  The universal solitonic hierarchy

15) V. Sokolov
(LITP Moscow):  A decomposition problem for the loop algebras and Lax pairs

Decompositions of the loop algebra over semi-simple Lie algebra G into a vector direct sum of subalgebra of all Taylor series and a complementary subalgebra are considered. In the case G=so(3) all possible complementary subalgebras are found. The corresponding Lax pairs for nonlinear PDEs and ODEs are presented.

16) Y. Zhang (
Tsinghua U. Beijing)
:   Infinitesimal deformations of bihamiltonian structures of hydrodynamic type
We consider the problem of classification of deformations of bihamiltonian structures of hydrodynamic type
and its relation to a class of Camassa-Holm type hierarchies of integrable PDEs.