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 [26,27].

Extensions to Dirichlet boundary conditions, which may be more consistent with the situation in the physical experiments and numerical simulations (see [1] and references therein) for the trapped Bose-Einstein condensates, are also discussed.