TY - JOUR
T1 - Superfluids passing an obstacle and vortex nucleation
AU - Lin, Fanghua
AU - Wei, Juncheng
N1 - Funding Information:
Acknowledgments. The research of the first author is partially supported by the NSF grant DMS-1501000. The research of the second author is partially supported by NSERC of Canada.
Publisher Copyright:
© 2019 American Institute of Mathematical Sciences. All rights reserved.
PY - 2019
Y1 - 2019
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
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U2 - 10.3934/dcds.2019232
DO - 10.3934/dcds.2019232
M3 - Article
AN - SCOPUS:85072936917
SN - 1078-0947
VL - 39
SP - 6801
EP - 6824
JO - Discrete and Continuous Dynamical Systems- Series A
JF - Discrete and Continuous Dynamical Systems- Series A
IS - 12
ER -