Proposal # DMS-9731097 PI: Harvey Segur Title: Nonlinear Wave Motion The Kadomtsev-Petviashvili equation is a nonlinear partial differential equation in two spatial dimensions plus time. It is one of very few nonlinear equations that: (i) is completely integrable and admits "soliton" solutions; (ii) involves two spatial dimensions plus time; and (iii) arises naturally in physics. Specifically, the equation describes approximately the motion of ocean waves in shallow water. (Ocean waves require two spatial dimensions because they occur on the water's two-dimensional surface.) For this application, the most interesting solutions are either exactly or approximately periodic, just as typical ocean waves are approximately periodic. Based on recent discoveries of the mathematical structure of this equation, we can now solve this problem as an initial-value problem for initial data that are either exactly or approximately periodic in space. The work to be carried out in this study will exploit this structure to build an accurate model of waves in shallow water. The waves in the model will be as complicated as waves in the ocean usually are: steady or unsteady, with either exact or approximate periodicity in time or in space or both, with arbitrarily large amplitudes, and with fully two-dimensional spatial patterns. No other functioning model of water waves includes all of these features simultaneously, so this model should vastly improve our ability to predict waves in shallow water. Nonlinear phenomena occur when a perturbation to a system triggers a response that is not proportional to the perturbation. Many of the dramatic events in nature are nonlinear, including the "big bang" in cosmology, sonic booms in aerodynamics, hurricanes and breaking ocean waves in geophysics. The mathematical models used to describe these phenomena are intrinsically nonlinear, and typically we cannot solve them in any general sense. The set of models that admit "solitons" and are "completely integrable" are exceptions to this rule: usually these models can be solved in complete detail, and they have provided great insight into the nature of nonlinear wave phenomena. The work to be carried out in this study will exploit recently a discovered mathematical structure of one of these completely integrable models to build an accurate and realistic model of ocean waves in shallow water. Practical implications of more accurate predictions of shallow water waves include better control of beach erosion, fewer problems with shoreline pollution from dispersal of man-made waste, and better design of offshore structures like oil platforms.
