Non-Perturbative Interfacial Waves

界面波是沿着分隔两种物质或区域的边界传播的波。它们在自然界中随处可见，从海洋中的内部波到生物学中的传播传染病和材料科学中的缺陷模式。现在人们对某种意义上的微扰波已有了很多了解。例如，如果界面没有受到很大的干扰，它的运动可以用更简单的线性或弱非线性模型来很好地预测。然而，许多重要的现象远远超出了这一范畴。例如，在2022年汤加火山喷发之后，观察到类似海啸的波在浅水中的移动速度比线性理论预测的要快得多，甚至在深水中似乎会放大。这个项目旨在加深对这些和类似的非微扰波的数学理解，对于这些非微扰波，潜在系统表现出内在的非线性动力学、表面奇异性或共振现象。这方面的进展将使更大的数学和科学界以及更广泛的社会受益，因为它们将带来预测能力的改善，并最终帮助减轻未来的灾害。该项目将为本科生和研究生提供研究培训机会。内部钻孔是在层状水体中沿着跃层传播的前沿。它们在混合、能量传输和海洋环流中起着重要的地球物理作用。利用整体分叉理论和自由边界椭圆正则性理论方法，证明了悬垂重力内波的存在，并验证了von Karman关于精确重力流的存在和形式的猜想。另一个目标是关于界面水动力波的时间演化。作为该项目的一部分，研究人员将使用无限维哈密顿系统的技术来表征多峰内毛细重力孤立波的频谱稳定性，并证明具有强表面张力的孤立波相对于横向扰动是轨道不稳定的。了解汤加火山喷发造成的大气海啸-是另一个中心目标。对它们形成的一个重要解释是基于两相轻可压缩欧拉系统中的三波共振。这个项目将首次对这一理论进行严格的处理，通过构建稳定的轴对称三维解作为初始大气激波的模型，并构建精确的三维双周期稳定解来展示共振三分量。该项目的最终目标是由波浪携带的局部化涡旋结构。中空涡旋是被涡片包围并悬浮在理想流体中的恒定压力的有界区域；自19世纪以来，人们一直在研究它们在平面上的存在和稳定性。研究人员将利用水下中空漩涡构造精确的水波，并确定它们的轨道稳定性，从而深入了解波浪与漩涡的相互作用。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Interfacial waves are waves that propagate along a boundary separating two substances or regions. They are found throughout nature, ranging from internal waves in the ocean to spreading contagions in biology and defect patterns in material science. Much is now known about waves that are perturbative in some sense. For instance, if the interface is not greatly disturbed, its motion can be well predicted using simpler linear or weakly nonlinear models. However, many important phenomena fall far outside this category. As one example, following the 2022 volcanic eruption in Tonga, tsunami-like waves were observed moving much faster than linear theory predicts in shallow water and even seemed to amplify in deep water. This project aims to deepen the mathematical understanding of these and similar non-perturbative waves for which the underlying systems exhibit intrinsically nonlinear dynamics, surface singularities, or resonance phenomena. Progress in this direction will be benefit the larger mathematics and science communities, and society more broadly as they will bring predictive capabilities improvements and ultimately help mitigate future disasters. The project will provide research training opportunities for undergraduate and graduate students.Internal bores are fronts that propagate along pycnoclines in stratified bodies of water. They play a geophysically significant role in mixing, energy transport, and oceanic circulation. Using global bifurcation theory and free boundary elliptic regularity theory methods, this project seeks to prove the existence of overhanging internal gravity waves and verify a conjecture of von Karman regarding the existence and form of exact gravity currents. Another objective concerns the time evolution of interfacial hydrodynamic waves. As part of this project, the investigator will use techniques from infinite-dimensional Hamiltonian systems to characterize the spectral stability of multimodal internal capillary-gravity solitary waves, and prove that solitary waves with strong surface tension are orbitally unstable with respect to transverse perturbations. Understanding meteotsunamis --- atmospherically generated tsunamis --- created by the Tonga eruption is another central objective. One prominent explanation for their formation is based on three-wave resonance in a two-phase lightly compressible Euler system. This project will give the first rigorous treatment to this theory by constructing steady axisymmetric three-dimensional solutions as models for the initial atmospheric shockwaves and exact three-dimensional doubly-periodic steady solutions for exhibiting the resonant triad. The final aim of the project concerns localized vortical structures carried by waves. A hollow vortex is a bounded region of constant pressure encircled by a vortex sheet and suspended inside a perfect fluid; their existence and stability in the plane have been studied since the 19th century. The investigator will construct exact water waves with a submerged hollow vortex and ascertain their orbital stability, giving insight into wave-vortex interaction.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.