Invariant measures for the defocusing Nonlinear Schrödinger equation

Nikolay Tzvetkov[1]

  • [1] Université Lille I Département de Mathématiques 59 655 Villeneuve d’Ascq Cedex (France)

Annales de l’institut Fourier (2008)

  • Volume: 58, Issue: 7, page 2543-2604
  • ISSN: 0373-0956

Abstract

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We prove the existence and the invariance of a Gibbs measure associated to the defocusing sub-quintic Nonlinear Schrödinger equations on the disc of the plane ℝ 2 . We also prove an estimate giving some intuition to what may happen in 3 dimensions.

How to cite

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Tzvetkov, Nikolay. "Invariant measures for the defocusing Nonlinear Schrödinger equation." Annales de l’institut Fourier 58.7 (2008): 2543-2604. <http://eudml.org/doc/10386>.

@article{Tzvetkov2008,
abstract = {We prove the existence and the invariance of a Gibbs measure associated to the defocusing sub-quintic Nonlinear Schrödinger equations on the disc of the plane $\mathbb\{R\}^2$. We also prove an estimate giving some intuition to what may happen in $3$ dimensions.},
affiliation = {Université Lille I Département de Mathématiques 59 655 Villeneuve d’Ascq Cedex (France)},
author = {Tzvetkov, Nikolay},
journal = {Annales de l’institut Fourier},
keywords = {Nonlinear Schrödinger; eigenfunctions; dispersive equations; invariant measures; nonlinear Schrödinger equation},
language = {eng},
number = {7},
pages = {2543-2604},
publisher = {Association des Annales de l’institut Fourier},
title = {Invariant measures for the defocusing Nonlinear Schrödinger equation},
url = {http://eudml.org/doc/10386},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Tzvetkov, Nikolay
TI - Invariant measures for the defocusing Nonlinear Schrödinger equation
JO - Annales de l’institut Fourier
PY - 2008
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 7
SP - 2543
EP - 2604
AB - We prove the existence and the invariance of a Gibbs measure associated to the defocusing sub-quintic Nonlinear Schrödinger equations on the disc of the plane $\mathbb{R}^2$. We also prove an estimate giving some intuition to what may happen in $3$ dimensions.
LA - eng
KW - Nonlinear Schrödinger; eigenfunctions; dispersive equations; invariant measures; nonlinear Schrödinger equation
UR - http://eudml.org/doc/10386
ER -

References

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  9. S. Kuksin, A. Shirikyan, Randomly forced CGL equation : stationary measures and the inviscid limit, J. Phys A 37 (2004), 1-18 Zbl1047.35061MR2039838
  10. J. Lebowitz, R. Rose, E. Speer, Statistical dynamics of the nonlinear Schrödinger equation, J. Stat. Physics V 50 (1988), 657-687 Zbl1084.82506MR939505
  11. E. Stein, G. Weiss, Introduction to Fourier analysis on euclidean spaces, 32 (1971), Princeton University PressPrinceton N.J.P. N. Zbl0232.42007MR304972
  12. N. Tzvetkov, Invariant measures for the nonlinear Schrödinger equation on the disc, Dynamics of PDE 3 (2006), 111-160 Zbl1142.35090MR2227040
  13. P. Zhidkov, Korteweg de Vries and nonlinear Schrödinger equations : qualitative theory, 1756 (2001), Springer-Verlag, Berlin Zbl0987.35001MR1831831

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