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

TL;DR

This paper develops a WKB analysis for the Gross-Pitaevskii equation in the semi-classical limit with non-trivial boundary conditions, using Zhidkov spaces and analytic data to describe the wave function as Planck's constant approaches zero.

Contribution

It extends WKB analysis to the Gross-Pitaevskii equation with non-trivial boundary conditions by employing Zhidkov spaces and analytic data for cubic-quintic nonlinearities.

Findings

Point-wise description of wave function in the semi-classical limit

Necessity of analytic data for cubic-quintic nonlinearities

Applicability of Zhidkov spaces for non-trivial boundary conditions

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.