The Hamiltonian structure of the nonlinear Schrödinger equation and the asymptotic stability of its ground states

Details The Hamiltonian structure of the nonlinear Schrödinger equation and the asymptotic stability of its ground states The Hamiltonian structure of the nonlinear Schrödinger equation and the asymptotic stability of its ground states

2009-10-20

arxiv.org/abs/0910.3797v6

Mathematics

Analysis of PDEs

This is the rvised version

Scientific paper

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.

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