Melnikov Analysis for Multi-Symplectic PDEs

In this work the Melnikov method for perturbed Hamiltonian wave equations is considered in order to determine possible chaotic behaviour in the systems. 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 perturbations of generalised KdV equation, 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 prove the persistence of a fixed point for the perturbed Poincaré map by using Lyapunov–Schmidt reduction. The framework sketched will be applied to the analysis of possible chaotic behaviour of travelling wave solutions for the above-mentioned PDEs within the multi-symplectic approach.