This is a collaborative research project to investigate certain types of nonlinear wave phenomena. Due to their inherent nonlinear behavior, such waves have very important physical applications in the generation of large amplitude waves, for example in shallow water near a beach. A primary goal of this research is to provide further insights into possible mechanisms that generate extremely high waves frequently observed in open seas and along coastlines. Important examples of such phenomena include tsunamis. Understanding the nature and dynamics of such extreme waves, particularly in oceans near highly populated coastal areas, is a significant and urgent task. The research activities will involve undergraduate and graduate students who will be trained in the field of applied mathematics and will gain first-hand research experience. It is further anticipated that the results from the proposed work would be useful in the study of similar nonlinear wave phenomena that occur in other physical problems including nonlinear optics, plasmas, and spin waves in magnetic thin films.

The nonlinear waves indicated above are closely described by the weakly dispersive, quasi-two-dimensional Kadomtsev-Petviashvili (KP) equation which admits a class of solitary wave solutions with complex two-dimensional web patterns. These solutions are sometimes referred to as the KP web-solitons.The purpose of this collaborative research project is to continue investigations into the physical as well as the mathematical aspects of the KP equation. Broadly speaking, this project has two main goals that are interrelated, namely, (1) to study solitary wave interactions observed in certain physical situations such as shallow water where the KP equation describes the leading order model, and (2) to investigate the detailed structure of the complex two-dimensional patterns of the KP web-solitons using geometric and combinatorial theories, which bring new techniques to the area of nonlinear waves. In particular, detailed analytical and numerical studies of the interactions, stability and initial value problem of the KP web-solitons will be conducted in this project. Furthermore, the project envisions collaboration with experimentalists in order to compare the obtained theoretical results with laboratory experiments performed in water wave tanks.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Victor Roytburd
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