Near-Integrability of Periodic Klein-Gordon Lattices

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 ${\varphi }^4$ 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.