大振幅带涡旋的重力孤立波
Vortex-carrying solitary gravity waves of large amplitude
AI总结：
该研究在数学上证明了超临界弗劳德数下存在与水下点涡平衡的大振幅重力孤立波族，允许自由表面悬伸，还通过数值研究发现周期波会翻卷，并证明点涡可去奇异化为带旋转气泡的空心涡孤立波。
摘要：
本文研究仅受重力作用的有限深度水中的二维行波。我们证明，对于任意超临界弗劳德数（无量纲波速），存在一个连续单参数族$\u27e8\b{\u2102}\u27e9$的孤立波，与水下点涡保持平衡。该波族从无旋均匀流分叉而来，且至少在大弗劳德数下，会一直延伸至表面奇点形成。这是首次严格构造出无表面张力的重力波载点涡，值得注意的是，我们的公式允许自由表面出现悬伸形态。我们还对带水下点涡的周期重力行波进行了数值分叉研究，有力表明其中部分波确实会发生翻卷。最后，我们证明在$\b{\u2102}$上的通有解处——包括大振幅甚至悬伸的解——点涡可被去奇异化，从而得到带水下空心涡的孤立波。从物理上讲，这些波可被视为携带旋转空气泡的行波。
英文摘要：
In this paper, we study two-dimensional traveling waves in finite-depth water that are acted upon solely by gravity. We prove that, for any supercritical Froude number (non-dimensionalized wave speed), there exists a continuous one-parameter family $\mathcal{C}$ of solitary waves in equilibrium with a submerged point vortex. This family bifurcates from an irrotational uniform flow, and, at least for large Froude numbers, extends up to the development of a surface singularity. These are the first rigorously constructed gravity wave-borne point vortices without surface tension, and notably our formulation allows the free surface to be overhanging. We also provide a numerical bifurcation study of traveling periodic gravity waves with submerged point vortices, which strongly suggests that some of these waves indeed overturn. Finally, we prove that at generic solutions on $\mathcal{C}$ $\unicode{x2013}$ including those that are large amplitude or even overhanging $\unicode{x2013}$ the point vortex can be desingularized to obtain solitary waves with a submerged hollow vortex. Physically, these can be thought of as traveling waves carrying spinning bubbles of air.
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