Uniqueness of solitary waves in the high-energy limit of FPU-type chains

TL;DR

This paper proves the local uniqueness of high-energy solitary waves in FPU chains by analyzing the linearized operator and showing no additional zero eigenvalues exist beyond those from translation symmetry.

Contribution

It extends previous asymptotic methods to the linearization, establishing a uniqueness result for high-energy solitary waves in FPU chains.

Findings

No additional zero eigenvalues beyond translation symmetry

Linearization analysis confirms local uniqueness

Asymptotic approximation replaces advance-delay equations

Abstract

Recent asymptotic results by the authors provided detailed information on the shape of solitary high-energy travelling waves in FPU atomic chains. In this note we use and extend the methods to understand the linearisation of the travelling wave equation. We show that there are not any other zero eigenvalues than those created by the translation symmetry and this implies a local uniqueness result. The key argument in our asymptotic analysis is to replace the linear advance-delay-differential equation for the eigenfunctions by an approximate ODE.