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

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.

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• Existence and asymptotic stability of quasi-periodic solution of discrete NLS with potential in $\mathbb{Z}$
• Revisiting Multi-breathers in the discrete Klein-Gordon equation: A Spatial Dynamics Approach
• Nonlinear instabilities of multi-site breathers in Klein-Gordon lattices
• Existence, linear stability and long-time nonlinear stability of Klein-Gordon breathers in the small-amplitude limit
• Asymptotic stability of soliton for discrete nonlinear Schrödinger equation on one-dimensional lattice
• Dispersive estimate for quasi-periodic Schrödinger operators on 1-$d$ lattices