Superfluids passing an obstacle and vortex nucleation
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Bibliographic record
Abstract
We consider a superfluid described by the Gross-Pitaevskii equation passing an obstacle \begin{document}$ \epsilon^2 \Delta u+ u(1-|u|^2) = 0 \ \mbox{in} \ {\mathbb R}^d \backslash \Omega, \ \ \frac{\partial u}{\partial \nu} = 0 \ \mbox{on}\ \partial \Omega $\end{document} where $ \Omega $ is a smooth bounded domain in $ {\mathbb R}^d $ ($ d\geq 2 $), which is referred as the obstacle and $ \epsilon>0 $ is sufficiently small. We first construct a vortex free solution of the form $ u = \rho_\epsilon (x) e^{i \frac{\Phi_\epsilon}{\epsilon}} $ with $ \rho_\epsilon (x) \to 1-|\nabla \Phi^\delta(x)|^2, \Phi_\epsilon (x) \to \Phi^\delta (x) $ where $ \Phi^\delta (x) $ is the unique solution for the subsonic irrotational flow equation \begin{document}$ \nabla ( (1-|\nabla \Phi|^2)\nabla \Phi ) = 0 \ \mbox{in} \ {\mathbb R}^d \backslash \Omega, \ \frac{\partial \Phi}{\partial \nu} = 0 \ \mbox{on} \ \partial \Omega, \ \nabla \Phi (x) \to \delta \vec{e}_d \ \mbox{as} \ |x| \to +\infty $\end{document} and $ |\delta | <\delta_{*} $ (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 $ |\nabla \Phi^\delta (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.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it