## Abstract

We consider a superfluid described by the Gross-Pitaevskii equation passing an obstacle ∂u ε^{2}∆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.

Original language | English (US) |
---|---|

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