Breaking, Peaking, and Disintegration

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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.