Long time dynamics for the one dimensional non linear Schrödinger equation

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).

56 pages

References in corpus (2)

• The cubic nonlinear Schrödinger equation in two dimensions with radial data
• Invariant measures for the defocusing NLS

Cited by in corpus (9)

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• On Gibbs measure and weak flow for the cubic NLS with non-localised initial data
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• Gibbs measure for the periodic derivative nonlinear Schrödinger equation