Many physical systems admit nonlinear waves. These systems often exhibit dissipation that is considered to be negligible or weak. However, recent work by the investigators and recently published observations of dissipation rates of ocean swell, show that the models using weak dissipation provide an inadequate description of some aspects of the evolution of nonlinear surface water waves. Thus, a main emphasis of this research is to develop a theory from first principles for the propagation and evolution of nonlinear surface water waves on a viscous fluid, with no approximation on the size of dissipation. The research addresses several projects that build on this notion and/or are intended to apply the mathematical results to ocean applications. The projects fall into two categories: A) surface waves propagating on a viscous fluid and B) waves in shallow or finite-depth water. The investigators will approach these problems by combining their individual strengths in modeling and analysis, numerical simulations, and physical experiments to obtain a fundamental mathematical description of the physical phenomena. They will then use publicly available data from ocean observations to apply their results to ocean settings.
The primary goal of this research is to understand at a fundamental, mathematical level the propagation of waves on a surface of water, and to apply this understanding to observations of waves measured in the laboratory and in the oceans. Particular applications include the effects of dissipation on the generation of waves by wind, the subsequent propagation of ocean swell, and the development of rogue waves in shallow and deep water; predicting dangerous versus benign tsunamis given the Richter-scale measure of the magnitude of the responsible earthquake and some readily available information about the geology of its location; and modeling large-amplitude waves on beaches with variable bathymetry using the new mathematical approaches investigated here. In addition, the mathematical results are expected to describe nonlinear waves in several settings, including light waves in an optical fiber, spin waves in a thin magnetic film and others. Graduate and undergraduate students will be mentored and involved in experimental studies and theoretical analysis; through this project they will receive training in interdisciplinary research.