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
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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.
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.
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):
8) D.D. Holm (
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:
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.
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
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 (
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.