Superfluids passing an obstacle and vortex nucleation

We consider a superfluid described by the Gross-Pitaevskii equation passing an obstacle ∂u ε²∆u + u(1 − |u|²) = 0 in R^d\Ω, _∂ν = 0 on ∂Ω where Ω is a smooth bounded domain in R^d (d ≥ 2), which is referred as the obstacle and ε > 0 is sufficiently small. We first construct a vortex free solution of the form u = ρε(x)e^{i Φ} ε^ε with ρε(x) → 1 − |∇Φ^δ(x)|², Φε(x) → Φ^δ(x) where Φ^δ(x) is the unique solution for the subsonic irrotational flow equation ∇((1 − |∇Φ|²)∇Φ) = 0 in R^d\Ω, ^∂ _∂ν ^Φ = 0 on ∂Ω, ∇Φ(x) → δ~e_d as |x| → +∞ and |δ| < δ∗ (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 |∇Φ^δ(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.