Near-Integrability of Low-Dimensional Periodic Klein-Gordon Lattices

Christov, Ognyan

doi:10.1155/2018/7023696

Cited by 1 publication

(3 citation statements)

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“…. , 6 show that resonant third order terms do not appear in the corresponding normal forms (see [3]). Hence, these normal forms remain integrable for the latter potential.…”

Section: Discussionmentioning

confidence: 99%

“…It is clear that there are plenty of resonances when a ∈ Q. Moreover, there are certain irrational values of a > 0 for which fourth order resonances exist in the low-dimensional periodic KG lattices (see [3]). Such values of a are difficult to control in the higher dimensions, that is why from now on we put a = 1 ω k = 1 + 4 sin 2 kπ N .…”

Section: Resonances and Symmetriesmentioning

confidence: 99%

“…The actions I k can be extended to global action-angle variables (I k , ϕ k ) (symplectic polar coordinates) and the KAM condition is straightforward (compare with [3]). Proof of Theorem 2.…”

Section: Resonances and Symmetriesmentioning

confidence: 99%

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Near-Integrability of Periodic Klein-Gordon Lattices

Christov

2019

Symmetry

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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 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.

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“…. , 6 show that resonant third order terms do not appear in the corresponding normal forms (see [3]). Hence, these normal forms remain integrable for the latter potential.…”

Section: Discussionmentioning

confidence: 99%

Section: Resonances and Symmetriesmentioning

confidence: 99%