Asymptotic stability of lattice solitons in the energy space

TL;DR

This paper proves the asymptotic stability of 1-soliton solutions in lattice equations like Toda and FPU, addressing challenges posed by their non-conservation of momentum and bidirectional nature.

Contribution

It introduces new methods to establish stability of lattice solitons, extending existing theories to non-Hamiltonian and bidirectional lattice models.

Findings

Stability of 1-solitons in Toda lattice established.

Improved asymptotic stability results for FPU lattices.

Application of recent decay estimates to lattice soliton stability.

Abstract

Orbital and asymptotic stability for 1-soliton solutions to the Toda lattice equations as well as small solitary waves to the FPU lattice equations are established in the energy space. Unlike analogous Hamiltonian PDEs, the lattice equations do not conserve momentum. Furthermore, the Toda lattice equation is a bidirectional model that does not fit in with existing theory for Hamiltonian system by Grillakis, Shatah and Strauss. To prove stability of 1-soliton solutions, we split a solution around a 1-soliton into a small solution that moves more slowly than the main solitary wave, and an exponentially localized part. We apply a decay estimate for solutions to a linearized Toda equation which has been recently proved by Mizumachi and Pego to estimate the localized part. We improve the asymptotic stability results for FPU lattices in a weighted space obtained by Friesecke and Pego.