The stability of nonlinear Schr\"odinger equations with a potential in high Sobolev norms revisited

TL;DR

This paper demonstrates the almost global stability of nonlinear Schrödinger equations with a potential in high Sobolev norms on tori, using Birkhoff normal form techniques, extending previous theoretical results.

Contribution

It revisits and extends the stability analysis of nonlinear Schrödinger equations with potentials in high Sobolev norms using an iterative Birkhoff normal form approach.

Findings

Almost all potentials lead to stability in high Sobolev norms.

Reproves a dynamical consequence of the infinite dimensional Birkhoff normal form theorem.

Extends stability results to a broader class of potentials.

Abstract

We consider the nonlinear Schr\"odinger equations with a potential on $\mathbb T^d$. For almost all potentials, we show the almost global stability in very high Sobolev norms. We apply an iteration of the Birkhoff normal form, as in the formulation introduced by Bourgain \cite{Bo00}. This result reprove a dynamical consequence of the infinite dimensional Birkhoff normal form theorem by Bambusi and Grebert \cite{BG}