Periodic travelling waves in convex Klein-Gordon chains

TL;DR

This paper proves the existence of a two-parameter family of periodic travelling waves in convex Klein-Gordon chains, using a saddle-point approach and discussing numerical methods for computing these wave trains.

Contribution

It introduces a novel existence proof for wave trains in Klein-Gordon chains based on a saddle-point problem with constraints.

Findings

Existence of a two-parameter family of wave trains proven.

Development of a saddle-point based existence proof.

Discussion of numerical computation methods for wave trains.

Abstract

We study Klein-Gordon chains with attractive nearest neighbour forces and convex on-site potential, and show that there exists a two-parameter family of periodic travelling waves (wave trains) with unimodal and even profile functions. Our existence proof is based on a saddle-point problem with constraints and exploits the invariance properties of an improvement operator. Finally, we discuss the numerical computation of wave trains.