Breaking, Peaking, and Disintegration

水波表现为丰富的现象，从涟漪到海啸再到巨浪，它们的研究提出了深刻而微妙的数学挑战。本计画将结合联合收割机严格的渐近分析、数值计算与模型化，以解决水波数学方面的基本问题。重点是真正的非线性现象和高度复杂的过程，如突破和峰值，敏锐的理解将依赖于从广泛不同的数学和其他学科领域的观点和技术的融合。预计该项目的进展将加深我们对表面水波数学的理解，并导致在物理海洋学和地球物理流体动力学方面的应用。该项目还将为研究生提供参与研究的机会。PI将寻求严格的分析，以解释最近的-显着的和意想不到的-关于峰值和接触波，暴跌，和白浪过程的数值结果。她将开发高度精确的计算方法，以揭示新的波动现象，刺激进一步的调查。她还致力于对斯托克斯波进行严格的不稳定性分析：Benjamin-Feir，高频和非线性。这需要对现有方法进行重大修改和重要扩展，以及开发新方法。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

Water waves manifest in a wealth of phenomena, from ripples to tsunamis to rogue waves, and their study presents profound and subtle mathematical challenges. This project will combine rigorous and asymptotic analysis, numerical computation, and modeling to address fundamental issues in the mathematical aspects of water waves. Emphasis is on genuinely nonlinear phenomena and highly complex processes, such as breaking and peaking, an acute understanding of which will rely on the convergence of perspectives and techniques from widely diverse areas of mathematics and other disciplines. Progress in the project is expected to deepen our understanding of the mathematics of surface water waves and lead to applications in physical oceanography to geophysical fluid dynamics. The project will also provide opportunities for involvement of graduate students in the research.The PI will seek rigorous analysis to explain recent -- remarkable and unexpected -- numerical results concerning peaking and touching waves, plunging, and whitecapping processes. She will develop highly accurate computational methods to reveal novel wave phenomena, stimulating further investigations. She also aims at rigorous instability analysis for a Stokes wave: Benjamin-Feir, high-frequency, and nonlinear. This requires significant modifications of and nontrivial extensions to the existing methods, as well as development of new ones. The main thrust is the use of the Euler equations rather than simplified approximate models.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.