Long-time stability of small FPU solitary waves

TL;DR

This paper extends the justification of small-amplitude wave approximations in the FPU lattice to longer times by analyzing the global well-posedness and boundedness of the generalized KdV equation, leading to insights on metastability.

Contribution

It provides a method to extend the validity of the small-amplitude approximation over longer times by linking KdV well-posedness to FPU wave stability.

Findings

Extended approximation validity to logarithmic time scales.

Established nonlinear metastability of FPU solitary waves.

Connected KdV orbital stability with FPU wave behavior.

Abstract

Small-amplitude waves in the Fermi-Pasta-Ulam (FPU) lattice with weakly anharmonic interaction potentials are described by the generalized Korteweg-de Vries (KdV) equation. Justification of the small-amplitude approximation is usually performed on the time scale, for which dynamics of the KdV equation is defined. We show how to extend justification analysis on longer time intervals provided dynamics of the generalized KdV equation is globally well-posed in Sobolev spaces and either the Sobolev norms are globally bounded or they grow at most polynomially. The time intervals are extended respectively by the logarithmic or double logarithmic factors in terms of the small amplitude parameter. Controlling the approximation error on longer time intervals allows us to deduce nonlinear metastability of small FPU solitary waves from orbital stability of the KdV solitary waves.