Long time dynamics for the one dimensional non linear Schrödinger equation

Nicolas Burq[1]; Laurent Thomann[2]; Nikolay Tzvetkov[3]

  • [1] Laboratoire de Mathématiques, Bât. 425, Université Paris Sud, 91405 Orsay Cedex, France.
  • [2] Laboratoire de Mathématiques J. Leray, Université de Nantes, UMR CNRS 6629 2, rue de la Houssinière, 44322 Nantes Cedex 03, France.
  • [3] University of Cergy-Pontoise, UMR CNRS 8088, F-95000, Cergy-Pontoise, France.

Annales de l’institut Fourier (2013)

  • Volume: 63, Issue: 6, page 2137-2198
  • ISSN: 0373-0956

Abstract

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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).

How to cite

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Burq, Nicolas, Thomann, Laurent, and Tzvetkov, Nikolay. "Long time dynamics for the one dimensional non linear Schrödinger equation." Annales de l’institut Fourier 63.6 (2013): 2137-2198. <http://eudml.org/doc/275635>.

@article{Burq2013,
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).},
affiliation = {Laboratoire de Mathématiques, Bât. 425, Université Paris Sud, 91405 Orsay Cedex, France.; Laboratoire de Mathématiques J. Leray, Université de Nantes, UMR CNRS 6629 2, rue de la Houssinière, 44322 Nantes Cedex 03, France.; University of Cergy-Pontoise, UMR CNRS 8088, F-95000, Cergy-Pontoise, France.},
author = {Burq, Nicolas, Thomann, Laurent, Tzvetkov, Nikolay},
journal = {Annales de l’institut Fourier},
keywords = {Nonlinear Schrödinger equation; potential; random data; Gibbs measure; invariant measure; global solutions; nonlinear Schrödinger equation},
language = {eng},
number = {6},
pages = {2137-2198},
publisher = {Association des Annales de l’institut Fourier},
title = {Long time dynamics for the one dimensional non linear Schrödinger equation},
url = {http://eudml.org/doc/275635},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Burq, Nicolas
AU - Thomann, Laurent
AU - Tzvetkov, Nikolay
TI - Long time dynamics for the one dimensional non linear Schrödinger equation
JO - Annales de l’institut Fourier
PY - 2013
PB - Association des Annales de l’institut Fourier
VL - 63
IS - 6
SP - 2137
EP - 2198
AB - 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).
LA - eng
KW - Nonlinear Schrödinger equation; potential; random data; Gibbs measure; invariant measure; global solutions; nonlinear Schrödinger equation
UR - http://eudml.org/doc/275635
ER -

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