Long time stability of small amplitude Breathers in a mixed FPU-KG model

TL;DR

This paper proves the long-term stability of small amplitude Breathers in a mixed FPU-KG model by applying a normal form transformation and stability analysis in the anti-continuous limit.

Contribution

It introduces a novel stability analysis for Breathers in a mixed FPU-KG model using resonant normal forms and extends stability results to the true model.

Findings

Existence and orbital stability of Breathers in the normal form.

Long time stability of Breathers in the original FPU-KG models.

Application of normal form techniques to mixed nonlinear lattice models.

Abstract

In the limit of small couplings in the nearest neighbor interaction, and small total energy, we apply the resonant normal form result of a previous paper of ours to a finite but arbitrarily large mixed Fermi-Pasta-Ulam Klein-Gordon chain, i.e. with both linear and nonlinear terms in both the on-site and interaction potential, with periodic boundary conditions. An existence and orbital stability result for Breathers of such a normal form, which turns out to be a generalized discrete Nonlinear Schr\"odinger model with exponentially decaying all neighbor interactions, is first proved. Exploiting such a result as an intermediate step, a long time stability theorem for the true Breathers of the KG and FPU-KG models, in the anti-continuous limit, is proven.