Superfluids passing an obstacle and vortex nucleation

Fanghua Lin, Juncheng Wei

Research output: Contribution to journalArticlepeer-review


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 languageEnglish (US)
Pages (from-to)6801-6824
Number of pages24
JournalDiscrete and Continuous Dynamical Systems- Series A
Issue number12
StatePublished - 2019


  • And phrases. Traveling waves
  • Gross-Pitaevskii equations
  • Singular perturbation
  • Vortices

ASJC Scopus subject areas

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


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