On the linear wave regime of the Gross-Pitaevskii equation

TL;DR

This paper investigates the long wave-length behavior of the Gross-Pitaevskii equation, establishing bounds on vortex formation and comparing nonlinear solutions to linear wave models using hydrodynamical transformations.

Contribution

It introduces a novel analysis of the wave regime of the Gross-Pitaevskii equation through the Madelung transform and bounds vortex emergence in this context.

Findings

Lower bounds on vortex formation in the wave regime

Comparison between nonlinear Gross-Pitaevskii solutions and linear wave solutions

Application of hydrodynamical form via Madelung transform

Abstract

We study a long wave-length asymptotics for the Gross-Pitaevskii equation corresponding to perturbation of a constant state of modulus one. We exhibit lower bounds on the first occurence of possible zeros (vortices) and compare the solutions with the corresponding solutions to the linear wave equation or variants. The results rely on the use of the Madelung transform, which yields the hydrodynamical form of the Gross-Pitaevskii equation, as well as of an augmented system.