Asymptotic stability of breathers in some Hamiltonian networks of weakly coupled oscillators

TL;DR

This paper proves the asymptotic stability of localized periodic solutions (breathers) in weakly coupled anharmonic oscillator chains using normal form transformations and dispersion analysis, under specific potential conditions.

Contribution

It establishes the first rigorous proof of asymptotic stability for breathers in Hamiltonian oscillator networks with a zero of order 8 in the potential.

Findings

Breathers are asymptotically stable in energy space.

Normal form theory effectively simplifies the system near breathers.

Dispersion mechanisms ensure decay of perturbations.

Abstract

We consider a Hamiltonian chain of weakly coupled anharmonic oscillators. It is well known that if the coupling is weak enough then the system admits families of periodic solutions exponentially localized in space (breathers). In this paper we prove asymptotic stability in energy space of such solutions. The proof is based on two steps: first we use canonical perturbation theory to put the system in a suitable normal form in a neighborhood of the breather, second we use dispersion in order to prove asymptotic stability. The main limitation of the result rests in the fact that the nonlinear part of the on site potential is required to have a zero of order 8 at the origin. From a technical point of view the theory differs from that developed for Hamiltonian PDEs due to the fact that the breather is not a relative equilibrium of the system.