### 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) |
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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 - Jan 1 2019 |

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### Keywords

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

### ASJC Scopus subject areas

- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Discrete and Continuous Dynamical Systems- Series A*,

*39*(12), 6801-6824. https://doi.org/10.3934/dcds.2019232

**Superfluids passing an obstacle and vortex nucleation.** / Lin, Fang-Hua; Wei, Juncheng.

Research output: Contribution to journal › Article

*Discrete and Continuous Dynamical Systems- Series A*, vol. 39, no. 12, pp. 6801-6824. https://doi.org/10.3934/dcds.2019232

}

TY - JOUR

T1 - Superfluids passing an obstacle and vortex nucleation

AU - Lin, Fang-Hua

AU - Wei, Juncheng

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

KW - And phrases. Traveling waves

KW - Gross-Pitaevskii equations

KW - Singular perturbation

KW - Vortices

UR - http://www.scopus.com/inward/record.url?scp=85072936917&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85072936917&partnerID=8YFLogxK

U2 - 10.3934/dcds.2019232

DO - 10.3934/dcds.2019232

M3 - Article

AN - SCOPUS:85072936917

VL - 39

SP - 6801

EP - 6824

JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

SN - 1078-0947

IS - 12

ER -