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∊ℤ(½pj2 + V(qj + 1-qj)), with a generic nearest-neighbour potential V. Part IIis devoted to the evolution of nearby initial data. A natural notion ofstability 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 decayingin an appropriate weighted-norm centred on the wave.Here we show that stability of this type follows as a consequence of a corresponding property for the linearized evolution equation. A maindifficulty is that the solitary-wave manifold is two dimensionalbut only one conserved functional (the Hamiltonian) is available tocontrol the evolution parallel to the manifold. Unlike analogous Hamiltonian PDEs,the lattice admits no Noether invariant corresponding to translation,due to lack of continuous translational invariance. One ingredient in overcoming this difficulty is toexploit conservation under the linearized flowof the symplectic form introduced in part I.Parts III and IV of this series will be devoted to verifying the requiredlinear stability property for near-sonic solitary waves.