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

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

2007-10-03

arxiv.org/abs/0710.0816v1

Annales de l'Institut Henri Poincar\'e Analyse non lin\'eaire 26, 3 (2009) 959-977

Mathematics

Analysis of PDEs

20 pages

Scientific paper

10.1016/j.anihpc.2008.02.006

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.

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