Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Long time dynamics for the one dimensional non linear Schr\"odinger
  equation   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).

6
The categories of this paper are: astro-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: nlin
22
The categories of this paper are: chao-dyn
8
The categories of this paper are: hep-ex
35
The categories of this paper are: mtrl-th
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The categories of this paper are: gr-qc
11
The categories of this paper are: hep-lat
28
The categories of this paper are: alg-geom
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The categories of this paper are: funct-an
32
The categories of this paper are: atom-ph
24
The categories of this paper are: dg-ga
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The categories of this paper are: cond-mat
3
The categories of this paper are: physics
34
The categories of this paper are: chem-ph
7
The categories of this paper are: hep-th