Periodic travelling wave solutions of discrete nonlinear Klein-Gordon lattices

TL;DR

This paper establishes the existence of periodic travelling wave solutions in discrete nonlinear Klein-Gordon lattices using fixed point and variational methods, identifying energy thresholds and bounds on wave velocity.

Contribution

It introduces new existence proofs for travelling waves in Klein-Gordon lattices for both hard and soft potentials, employing fixed point and variational techniques respectively.

Findings

Existence of solutions for hard potentials with energy and velocity bounds.

Existence of solutions for soft potentials with kinetic energy thresholds.

Identification of energy thresholds for wave existence.

Abstract

We prove the existence of periodic travelling wave solutions for general discrete nonlinear Klein-Gordon systems, considering both cases of hard and soft on-site potentials. In the case of hard on-site potentials we implement a fixed point theory approach, combining Schauder's fixed point theorem and the contraction mapping principle. This approach enables us to identify a ring in the energy space for non-trivial solutions to exist, energy (norm) thresholds for their existence and upper bounds on their velocity. In the case of soft on-site potentials, the proof of existence of periodic travelling wave solutions is facilitated by a variational approach based on the Mountain Pass Theorem. Thresholds on the averaged kinetic energy for these solutions to exist are also derived.