Dept. of Mathematics and Statistics, University of Surrey
Guildford GU2 5XH, United Kingdom

E-mail: k.blyuss@eim.surrey.ac.uk

Melnikov analysis for multi-symplectic PDE's

Abstract:
In this work we consider a Melnikov method for perturbed Hamiltonian wave equations in order to determine possibile chaotic behaviour in the systems governed by them. The backbone of the analysis is the multi-symplectic formulation of the unperturbed PDE and its further reduction to travelling waves. In the multi-symplectic approach two separate symplectic operators are introduced for the spatial and temporal variables, which allow one to generalise the usual symplectic structure. The systems under consideration include generalised KdV, nonlinear wave equation, Boussinesq equation. These equations are equivariant with respect to abelian subgroups of Euclidean group. It is assumed that the external perturbation preserves this symmetry. Travelling wave reduction for the above-mentioned systems results in a four-dimensional system of ODEs, which is considered for Melnikov type chaos. As a preliminary for the calculation of a Melnikov function, we consider the existence of a fixed point for the perturbed Poincare map and corresponding stable and unstable manifolds. The persistence of a fixed point for the perturbed Poincare map is proved using Lyapunov-Schmidt reduction. Some version of a theorem on persistence of invariant manifolds can also be concluded in this case. The framework sketched will be applied for the analysis of possible chaotic behaviour of travelling wave solutions to the above-mentioned PDEs within the multi-symplectic approach.