The dynamical behaviour near solitary waves in Hamiltonian lattice differential equations

Abstract

In this article we consider Hamiltonian lattice differential equations and investigate the existence of travelling waves near solitary wave solutions. We focus on the case where the solitary wave profile induces a homoclinic solution of the associated traveling wave equation, which is a typical scenario in the Fermi-Pasta-Ulam lattice. We will then show the existence of finitely many scalar bifurcation equations, such that zeros of these equations correspond to multi-pulses or periodic traveling waves of the original lattice equation that are located near the primary solitary wave. Compared to previous works, we will have to overcome technical complications which result from the lack of hyperbolicity of the asymptotic steady state.We use our results to prove the existence of periodic travelling waves accompanying a family of stable solitary waves in the Fermi-Pasta-Ulam lattice, where properties of the latter waves have been recently investigated by Pego and Friesecke [8]. As we will show, these waves persist under small reversible perturbations of the FPU lattice, where they induce a family of solitary waves.