Long time dynamics for the one dimensional non linear Schr\"odinger equation

TL;DR

This paper constructs Gibbs measures for 1D nonlinear Schrödinger equations with harmonic potential, proves their invariance, and establishes global well-posedness and scattering results for critical and super-critical cases.

Contribution

It introduces a novel approach to global dynamics by combining measure construction with well-posedness and invariance proofs for the NLS.

Findings

Gibbs measures are constructed for the 1D nonlinear Schrödinger equation with harmonic potential.

The Cauchy problem is globally well-posed for initial data in the support of these measures.

Gibbs measures are proven to be invariant under the flow of the equation.

Abstract

In this article, we first present the construction of Gibbs measures associated to nonlinear Schr\"odinger equations with harmonic potential. Then we show that the corresponding Cauchy problem is globally well-posed for rough initial conditions in a statistical set (the support of the measures). Finally, we prove that the Gibbs measures are indeed invariant by the flow of the equation. As a byproduct of our analysis, we give a global well-posedness and scattering result for the $L^2$ critical and super-critical NLS (without harmonic potential).