Summary: WKB ANALYSIS FOR THE GROSS­PITAEVSKII EQUATION
WITH NON-TRIVIAL BOUNDARY CONDITIONS AT INFINITY
THOMAS ALAZARD AND R´EMI CARLES
Abstract. We consider the semi-classical limit for the Gross­Pitaevskii equa-
tion. In order to consider non-trivial boundary conditions at infinity, we work
in Zhidkov spaces rather than in Sobolev spaces. For the usual cubic nonlin-
earity, 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.
1. Introduction
We study the semi-classical limit 0 for the Gross­Pitaevskii equation
i tu +
2
2m
u = V u + f |u|2
u,
where x Rn
. In the case of Bose­Einstein condensation (BEC), the external
potential V = V (t, x) models an external trap, and the nonlinearity f describes the

Source: Alazard, Thomas - Département de Mathématiques, Université de Paris-Sud 11

Collections: Mathematics