Solitary waves on FPU lattices: II. Linear implies nonlinear stability

Abstract

In part I of this series (1999 Nonlinearity 12 1601-27), we showed that a two-parameter family of supersonic solitary waves exists in one-dimensional nonintegrable lattices with Hamiltonian H = ∑j∊(½pj2 + V(qj + 1-qj)), with a generic nearest-neighbour potential V. Part II is devoted to the evolution of nearby initial data. A natural notion of stability of solitary waves is `orbital plus asymptotic stability'. Perturbations should - modulo adjustments of wave speed and phase - stay small for all time, and fall behind the wave in the sense of decaying in an appropriate weighted-norm centred on the wave.