WKB analysis for the Gross–Pitaevskii equation with non-trivial boundary conditions at infinity

  • Thomas Alazard

    CNRS & Université Paris-Sud, Mathématiques UMR CNRS 8628, Bât. 425, 91405 Orsay cedex, France
  • Rémi Carles

    CNRS & Université Montpellier 2, Mathématiques UMR CNRS 5149, CC 051, Place Eugène Bataillon, 34095 Montpellier cedex 5, France

Abstract

We consider the semi-classical limit for the Gross–Pitaevskii equation. In order to consider non-trivial boundary conditions at infinity, we work in Zhidkov spaces rather than in Sobolev spaces. For the usual cubic nonlinearity, we obtain a point-wise description of the wave function as the Planck constant goes to zero, so long as no singularity appears in the limit system. For a cubic-quintic nonlinearity, we show that working with analytic data may be necessary and sufficient to obtain a similar result.

Cite this article

Thomas Alazard, Rémi Carles, WKB analysis for the Gross–Pitaevskii equation with non-trivial boundary conditions at infinity. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 3, pp. 959–977

DOI 10.1016/J.ANIHPC.2008.02.006