A Simple Proof of the Stability of Solitary Waves in the Fermi-Pasta-Ulam model near the KdV Limit

TL;DR

This paper provides a straightforward proof demonstrating that small amplitude, long-wavelength solitary waves in the Fermi-Pasta-Ulam model are linearly and nonlinearly stable by leveraging existing results on related lattice models and the KdV limit.

Contribution

It offers a simple, unified proof of the stability of FPU solitary waves near the KdV limit, combining prior results with a new rescaling approach.

Findings

Small amplitude FPU solitary waves are linearly stable.

FPU solitary waves are nonlinearly, asymptotically stable.

The proof simplifies previous complex stability analyses.

Abstract

By combining results of Mizumachi on the stability of solitons for the Toda lattice with a simple rescaling and a careful control of the KdV limit we give a simple proof that small amplitude, long-wavelength solitary waves in the Fermi-Pasta-Ulam (FPU) model are linearly stable and hence by the results of Friesecke and Pego that they are also nonlinearly, asymptotically stable.