关于Gross-Pitaevskii方程的涡丝猜想
On the vortex filament conjecture for the Gross-Pitaevskii equation
- University of Bath（巴斯大学）
- Chinese University of Hong Kong（香港中文大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文在三维Gross-Pitaevskii方程中验证了涡丝猜想的一种形式，构造了收敛到任意光滑闭嵌入副法线流的小核极限精确解族，并推导了涡丝的精细调制律。
AI中文摘要：
我们建立了三维Gross-Pitaevskii方程涡丝猜想的一种形式。给定紧时间区间上任意光滑闭嵌入的曲线副法线流，我们在小核极限ε→0下构造一族精确解，其度一涡丝一致收敛到指定的流。在涡丝附近，解具有标准平面涡旋剖面；而在远离涡丝处，其相位梯度收敛到相关的Biot-Savart场。我们还推导了涡丝的精细调制律。
英文摘要：
We establish one form of the vortex filament conjecture for the three-dimensional Gross-Pitaevskii equation. Given any smooth closed embedded binormal flow of curves on a compact time interval, we construct, in the small-core limit $\varepsilon\to0$, a family of exact solutions whose degree-one vortex filaments converge uniformly to the prescribed flow. Near the filament the solutions have the standard planar vortex profile, while away from it their phase gradients converge to the associated Biot-Savart field. We also derive a refined modulation law for the vortex filament.
