# Near-integrability of periodic Klein-Gordon lattices

## TL;DR

This paper proves the non-integrability of the periodic Klein-Gordon lattice but shows that its resonant normal form is integrable, leading to the existence of many quasi-periodic solutions at low energy levels.

## Contribution

It demonstrates the non-integrability of the periodic KG lattice and establishes the integrability of its resonant normal form, revealing invariant tori and quasi-periodic solutions.

## Key findings

- The KG lattice is non-integrable in Liouville sense.
- The resonant Birkhoff normal form of the KG Hamiltonian is integrable.
- Existence of many invariant tori and quasi-periodic solutions at low energy levels.

## Abstract

In this paper we study the Klein-Gordon (KG) lattice with periodic boundary conditions. It is an $N$ degrees of freedom Hamiltonian system with linear inter-site forces and nonlinear on-site potential, which here is taken to be of the $\phi^4$ form. First, we prove that the system in consideration is non-integrable in Liuville sense. The proof is based on the Morales-Ramis theory. Next, we deal with the resonant Birkhoff normal form of the KG Hamiltonian, truncated to order four. Due to the choice of potential, the periodic KG lattice shares the same set of discrete symmetries as the periodic Fermi-Pasta-Ulam (FPU) chain. Then we show that the above normal form is integrable. To do this we utilize the results of B. Rink on FPU chains. If $N$ is odd this integrable normal form turns out to be KAM nondegenerate Hamiltonian. This implies the existence of many invariant tori at low-energy level in the dynamics of the periodic KG lattice, on which the motion is quasi-periodic. We also prove that the KG lattice with Dirichlet boundary conditions (that is, with fixed endpoints) admits an integrable, KAM nondegenerated normal forth order form, which in turn shows that almost all low-energetic solutions of KG lattice with fixed endpoints are quasi-periodic.