Exponential Stability Estimates for the 1D NLS

TL;DR

This paper establishes exponential and sub-exponential stability time estimates for parameter-dependent nonlinear Schrödinger equations on the circle using a Birkhoff Normal Form approach under Diophantine conditions.

Contribution

It introduces a flexible Birkhoff Normal Form theorem for 1D NLS equations near the origin under Diophantine conditions, leading to new stability time estimates.

Findings

Proves exponential stability times in Sobolev spaces.

Establishes sub-exponential stability in Gevrey classes.

Utilizes a novel Birkhoff Normal Form theorem for parameter-dependent NLS.

Abstract

We study stability times for a family of parameter dependent nonlinear Schr\"odinger equations on the circle, close to the origin. Imposing a suitable Diophantine condition (first introduced by Bourgain), we prove a rather flexible Birkhoff Normal Form theorem, which implies, e.g., exponential and sub-exponential time estimates in the Sobolev and Gevrey class respectively.