Vera Mikyoung Hur
Breaking, Peaking and Disintegration
The water wave problem, in the simplest form, concerns the wave motion at the surface of an incompressible inviscid fluid, below a body of air, acted on by gravity. Describing what one may see or feel at the beach or in a boat, water waves are a prime example of applied mathematics. They host a wealth of wave phenomena, from ripples to tsunamis and to rogue waves. They have stimulated a considerable part of the historical development of wave motion, such as the discovery of the Korteweg-de Vries soliton. The water wave problem presents profound and subtle challenges. For one thing, the interface between the water and the air is free and to be determined as part of the solution. Free boundaries are mathematically challenging in their own right and they occur in many other situations, such as melting of ice and stretching of a flexible membrane over an obstacle. Moreover, free surface conditions are nonlinear.
During her CAS appointment, Professor Hur will put together rigorous analysis, numerical computation and modeling to address fundamental issues in the mathematical aspects of water waves. Particularly, she will focus on global regularity versus finite time singularities, the existence of traveling waves and their characterization, and the stability and instability of traveling waves. She will emphasize large scale dynamics and genuinely nonlinear behaviors, such as breaking, peaking and touching waves, and the Benjamin-Feir instability. She will expand the use of computational methods to support theoretical advances. Progress in the proposed research will lead to the resolution of several outstanding problems surrounding water waves, and engineering applications.