Near-Integrability of Periodic Klein-Gordon Lattices

Ognyan Christov

doi:10.3390/sym11040475

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 ϕ 4 form. First, we prove that the system in consideration is non-integrable in Liouville sense. The proof is based on the Morales-Ramis-Simó 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 use 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 that almost all low-energetic motions of the periodic KG lattice are quasi-periodic. We also prove that the KG lattice with Dirichlet boundary conditions (that is, with fixed endpoints) admits an integrable, nondegenerate normal forth order form. Then, the KAM theorem applies as above.

Highlights

IntroductionThe fourth order normal form H = H2 + H 4 of KG lattice with fixed endpoints is completely integrable and KAM nondegenerate
Let us introduce the Klein-Gordon (KG) lattice described by the Hamiltonian H= ∑ j ∈Z h p2 j + iC ( q j +1 − q j )2 + V ( q j ), p j = qj . (1)The constant C > 0 measures the interaction to nearest neighbor particles and V ( x ) is a nonlinear potential
We prove that the KG lattice with Dirichlet boundary conditions admits an integrable, nondegenerate normal forth order form

Summary

Introduction

The fourth order normal form H = H2 + H 4 of KG lattice with fixed endpoints is completely integrable and KAM nondegenerate This result and KAM theorem show the existence of large-measure set of low-energy quasi-periodic solutions of KG lattice with fixed endpoints. We already mentioned the works of Rink who proved rigorously that the periodic FPU Hamiltonian is a perturbation of a nondegenerate Liouville integrable Hamiltonian, namely the normal form of order 4 [8] He described the geometry of even FPU lattice in [12], and rigorously proved Nishida’s conjecture stating that almost all low-energetic motions in FPU with fixed endpoints are quasi-periodic [9]. We finish with some remarks and possible directions to extend our results

Resonances and Symmetries

Proof of Theorem 1

The Birkhoff Normal Form

KG Lattice with Fixed Endpoints

Discussion