Unimodal wave trains and solitons in convex FPU chains

TL;DR

This paper introduces a new existence proof for wave trains and solitons in convex FPU chains, emphasizing unimodal profiles and convergence properties, with implications for numerical approximation and localization.

Contribution

It provides a novel existence proof based on constrained energy maximization and invariance, specifically for convex potentials, refining previous results.

Findings

Existence of unimodal, even wave trains and solitons proven.

Wave trains converge to solitons as periodicity increases.

Numerical methods effectively approximate localized wave solutions.

Abstract

We consider atomic chains with nearest neighbour interactions and study periodic and homoclinic travelling waves which are called wave trains and solitons, respectively. Our main result is a new existence proof which relies on the constrained maximisation of the potential energy and exploits the invariance properties of an improvement operator. The approach is restricted to convex interaction potentials but refines the standard results as it provides the existence of travelling waves with unimodal and even profile functions. Moreover, we discuss the numerical approximation and complete localization of wave trains, and show that wave trains converge to solitons when the periodicity length tends to infinity.