This collaborative proposal concerns the investigation of nonlinear, dispersive wave phenomena described by the Kadomtsev-Petviashvili (KP) equation and its physical applications, particularly in two-dimensional shallow water waves. The KP equation admits a particular solution called line-soliton, which is a steady propagating wave with high amplitude, like a beach wave. Broadly speaking, the project has two main goals that are interrelated, namely, (i) to study combinatorial and geometric aspects of the solution space of the KP solitons, and (ii) to develop an asymptotic theory where the KP equation as the leading order equation for real applications in two-dimensional wave phenomena. Detailed analytical and numerical studies of the interactions, stability and initial value problem of the KP solitons will be carried out in this project. The theoretical results will be carefully compared with experimental measurements.
Preliminary work suggests that some of these newly discovered solutions of the KP equation may have important physical applications such as the Mach reflection of an oblique incidence wave onto a vertical wall and in the generation of large amplitude "rogue" waves in shallow water near a beach. An objective of this research is to apply the results of this project in order to investigate possible mechanisms generating waves of extremely high elevations frequently observed in open seas and along coastlines, for example, tsunamis. Understanding the nature and dynamics of such extreme waves, and ultimately predicting such wave phenomena in oceans near highly populated coastal areas are significant and urgent tasks. The proposed research activities will involve several undergraduate and graduate students who will gain first-hand research experience in applied mathematics. Since the theory is shared by various other physical systems, it is anticipated that the results from the proposed work would provide insights into areas such as light waves in nonlinear optics, and spin waves in magnetic thin films.