Near-integrability of low dimensional periodic Klein-Gordon lattices

TL;DR

This paper investigates the integrability of low-dimensional periodic Klein-Gordon lattices, demonstrating non-integrability for two particles but integrability for lattices with up to six particles using normal form analysis and KAM theory.

Contribution

It provides the first rigorous proof of non-integrability for two-particle cases and establishes integrability for lattices with up to six particles through normal form methods.

Findings

Two-particle lattice is non-integrable.

Lattices with up to six particles are integrable via normal forms.

KAM theory applies to these integrable cases.

Abstract

The low dimensional periodic Klein-Gordon lattices are studied for integrability. We prove that the periodic lattice with two particles and certain nonlinear potential is non integrable. However, in the cases of up to six particles, we prove that their Birkhoff-Gustavson normal forms are integrable, which allows us to apply KAM theory.