Title: Gravity wave resonances in closed basins

Abstract: Nonlinear surface gravity waves sloshing in a container of rectangular cross-section can behave very differently than those with other cross sections. Why? Wave resonance is a mechanism by which energy is continuously exchanged between a small number of wave modes and is common to many nonlinear dispersive wave systems. They have been studied extensively over the past 60-years, almost always on domains that are large (or infinite) compared to the characteristic wavelength. In this case, the dispersion relation for gravity waves dictates that only quartic (4-wave) resonances can occur. In contrast, wave resonances in confined three-dimensional geometries have received relatively little attention, where, perhaps surprisingly, stronger 3-wave resonances of gravity waves can occur (except for rectangular domains!). We will present the results characterizing the configuration and dynamics of resonant triads of gravity waves in cylindrical basins of arbitrary cross sections. Extensions to internal waves and other geometries will also be discussed.

Bio Sketch

Dr. Paul Milewski is a Professor and Department Head in the Department of Mathematics at Penn State University. He received his B.Sc. and M.Sc. in Aerospace Engineering at Boston University and his Ph.D. (1993) from Mathematics at M.I.T. Prior to joining Mathematics at Penn State in 2023, he had positions in the Departments of Mathematical Sciences at the University of Bath (UK, 2011-2023), and in Mathematics at Wisconsin-Madison (1995-2011) and at Stanford. He has held visiting positions, among others, at IMPA (Brazil) and ENS (France). He was recipient of a Royal Society Wolfson Fellowship and a Sloan Fellowship. His research is in applied and computational mathematics, mainly in nonlinear waves in fluids, but also with interests across mathematical modeling of physical and biological systems, and data science.