# Traveling wave solutions for the FPU chain: a constructive approach

## TL;DR

This paper develops a computer-assisted method to construct traveling wave solutions for the FPU chain, accommodating large derivatives and sign changes, without requiring an attractive quadratic potential, focusing on supersonic solutions.

## Contribution

It introduces a novel constructive approach that does not depend on the potential having an attractive quadratic term, broadening the class of solutions for the FPU chain.

## Key findings

- Constructed traveling wave solutions with large derivatives and sign changes.
- Method applies to non-attractive quadratic potentials.
- Solutions are predominantly supersonic.

## Abstract

Traveling waves for the FPU chain are constructed by solving the associated equation for the spatial profile $u$ of the wave. We consider solutions whose derivatives $u'$ need not be small, may change sign several times, but decrease at least exponentially. Our method of proof is computer-assisted. Unlike other methods, it does not require that the FPU potential has an attractive (positive) quadratic term. But we currently need to restrict the size of that term. In particular, our solutions in the attractive case are all supersonic.