Long time dynamics for the one dimensional non linear Schrödinger equation

Nicolas Burq; Laurent Thomann; Nikolay Tzvetkov

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arXiv · 1002.4054

Long time dynamics for the one dimensional non linear Schrödinger equation

Nicolas Burq·Laurent Thomann·Nikolay Tzvetkov

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Abstract

In this article, we first present the construction of Gibbs measures associated to nonlinear Schrödinger 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).

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Nicolas Burq, Laurent Thomann, Nikolay Tzvetkov. 2010-02-22. Long time dynamics for the one dimensional non linear Schrödinger equation. https://arxiv.org/abs/1002.4054

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