Superfluids Passing an Obstacle and Vortex Nucleation

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.