 
Summary: WKB ANALYSIS FOR THE GROSSPITAEVSKII EQUATION
WITH NONTRIVIAL BOUNDARY CONDITIONS AT INFINITY
THOMAS ALAZARD AND R´EMI CARLES
Abstract. We consider the semiclassical limit for the GrossPitaevskii equa
tion. In order to consider nontrivial boundary conditions at infinity, we work
in Zhidkov spaces rather than in Sobolev spaces. For the usual cubic nonlin
earity, we obtain a pointwise description of the wave function as the Planck
constant goes to zero, so long as no singularity appears in the limit system.
For a cubicquintic nonlinearity, we show that working with analytic data may
be necessary and sufficient to obtain a similar result.
1. Introduction
We study the semiclassical limit 0 for the GrossPitaevskii equation
i tu +
2
2m
u = V u + f u2
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
