Long time dynamics for the one dimensional non linear Schrödinger equation

Details Long time dynamics for the one dimensional non linear Schrödinger equation Long time dynamics for the one dimensional non linear Schrödinger equation

2010-02-22

arxiv.org/abs/1002.4054v1

Mathematics

Analysis of PDEs

56 pages

Scientific paper

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).

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