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 nonintegrable. 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 in most cases.
Highlights
In this article we deal with the periodic Klein-Gordon (KG)lattice described by the Hamiltonian H = ∑ [ pj2 j∈Z/nZ 2 + C 22 V]
We prove that the periodic lattice with two particles and certain nonlinear potential is nonintegrable
This paper presents partial results on integrability of normal forms of the periodic KG lattices
Summary
Motivated by the works of Rink and Verhulst [3, 4], who presented the periodic FPU chain as a perturbation of an Advances in Mathematical Physics integrable and KAM nondegenerated system, namely, the truncated Birkhoff-Gustavson normal form of order 4 in the neighborhood of an equilibrium, our aim is to verify whether this can be done for the low-dimensional KG lattices. We summarize our second result concerning the integrability of truncated resonant normal forms of the periodic KG lattices up to six particles in the following. As a consequence from this result, we may conclude for the low-dimensional KG lattices when KAM theory applies that there exist many quasi-periodic solutions of small energy on a long time scale (see Section 2 and for more detailed explanation [3]) and chaotic orbits are of small measure. The proof of Theorem 2 is based on the Ziglin-MoralesRamis theory and since it is more algebraic in nature, it is carried out in the Appendix
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