  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 Poincare 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.
