Superfluids Passing an Obstacle and Vortex Nucleation

TL;DR

This paper constructs vortex-free and vortex nucleating solutions for superfluids passing an obstacle, using the Gross-Pitaevskii equation, and rigorously confirms vortex nucleation phenomena observed in numerical simulations.

Contribution

The paper provides the first rigorous construction of vortex nucleation solutions in superfluids around obstacles within the Gross-Pitaevskii framework, extending previous numerical findings.

Findings

Vortex-free solutions approximating irrotational flow are constructed.

Solutions with single vortices near obstacle boundary are shown to exist.

Vortex solutions converge to traveling wave solutions under scaling.

Abstract

We consider a superfluid described by the Gross-Pitaevskii equation passing an obstacle \[\epsilon^2 \Delta u+ u(1-|u|^2)=0 \ \mbox{in} \ {\mathbb R}^d \backslash \Omega, \ \ \frac{\partial u}{\partial \nu}=0 \ \mbox{on}\ \partial \Omega \] where $ \Omega$ is a smooth bounded domain in $ {\mathbb R}^d$ ($d\geq 2$), which is referred as the obstacle and $ \epsilon>0$ is sufficiently small. We first construct a vortex free solution of the form $ u= \rho_\epsilon (x) e^{i \frac{\Phi_\epsilon}{\epsilon}}$ with $ \rho_\epsilon (x) \to 1-|\nabla \Phi^\delta(x)|^2, \Phi_\epsilon (x) \to \Phi^\delta (x) $ where $\Phi^\delta (x)$ is the unique solution for the subsonic irrotational flow equation \[ \nabla ( (1-|\nabla \Phi|^2)\nabla \Phi )=0 \ \mbox{in} \ {\mathbb R}^d \backslash \Omega, \ \frac{\partial \Phi}{\partial \nu} =0 \ \mbox{on} \ \partial \Omega, \ \nabla \Phi (x) \to \delta \vec{e}_d \ \mbox{as} \ |x| \to +\infty \] and $|\delta | <\delta_{*}$ (the sound speed). In dimension $d=2$, on the background of this vortex free solution we also construct solutions with single vortex close to the maximum or minimum points of the function $|\nabla \Phi^\delta (x)|^2$ (which are on the boundary of the obstacle). The latter verifies the vortex nucleation phenomena (for the steady states) in superfluids described by the Gross-Pitaevskii equations. Moreover, after some proper scalings, the limits of these vortex solutions are traveling wave solution of the Gross-Pitaevskii equation. These results also show rigorously the conclusions drawn from the numerical computations in \cite{huepe1, huepe2}. Extensions to Dirichlet boundary conditions, which may be more consistent with the situation in the physical experiments and numerical simulations (see \cite{ADP} and references therein) for the trapped Bose-Einstein condensates, are also discussed.