The Hamiltonian structure of the nonlinear Schr\"odinger equation and the asymptotic stability of its ground states

TL;DR

This paper proves that ground states of the nonlinear Schrödinger equation, which are orbitally stable, are also asymptotically stable under certain conditions, using Hamiltonian and normal form techniques.

Contribution

It extends the stability analysis of NLS ground states by establishing their asymptotic stability through Hamiltonian normal form and Darboux transformations.

Findings

Ground states are asymptotically stable under specified conditions.

Application of Birkhoff normal form to NLS Hamiltonian.

Use of Darboux Theorem to handle non-canonical coordinates.

Abstract

In this paper we prove that ground states of the NLS which satisfy the sufficient conditions for orbital stability of M.Weinstein, are also asymptotically stable, for seemingly generic equations. Here we assume that the NLS has a smooth short range nonlinearity. We assume also the presence of a very short range and smooth linear potential, to avoid translation invariance. The basic idea is to perform a Birkhoff normal form argument on the hamiltonian, as in a paper by Bambusi and Cuccagna on the stability of the 0 solution for NLKG. But in our case, the natural coordinates arising from the linearization are not canonical. So we need also to apply the Darboux Theorem. With some care though, in order not to destroy some nice features of the initial hamiltonian.