Abstract
We consider a superfluid described by the Gross-Pitaevskii equation passing an obstacle ∂u ε2∆u + u(1 − |u|2) = 0 in Rd\Ω, ∂ν = 0 on ∂Ω where Ω is a smooth bounded domain in Rd (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)ei Φ εε with ρε(x) → 1 − |∇Φδ(x)|2, Φε(x) → Φδ(x) where Φδ(x) is the unique solution for the subsonic irrotational flow equation ∇((1 − |∇Φ|2)∇Φ) = 0 in Rd\Ω, ∂ ∂ν Φ = 0 on ∂Ω, ∇Φ(x) → δ~ed 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 |∇Φδ(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.
Original language | English (US) |
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Pages (from-to) | 6801-6824 |
Number of pages | 24 |
Journal | Discrete and Continuous Dynamical Systems- Series A |
Volume | 39 |
Issue number | 12 |
DOIs | |
State | Published - 2019 |
Keywords
- And phrases. Traveling waves
- Gross-Pitaevskii equations
- Singular perturbation
- Vortices
ASJC Scopus subject areas
- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics