Geometric Combinatorics and Hypergeometric Functions in Integrable Systems and Their Physical Applications

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This research project is to investigate certain classes of nonlinear differential equations arising as a mathematical description of several physical phenomena. The project includes an investigation of nonlinear waves, particularly, in shallow water. Because of nonlinearity, when one wave is superimposed on the other the waves are not just "additive," as would be true in the linear case, but interact with each other. Such waves form complex wave patterns and have very important physical applications in the generation of large-amplitude waves, for example, in shallow water near a beach, or tsunamis. Understanding the nature and dynamics of such extreme waves, especially near highly populated areas is a significant and urgent task. The project also aims to investigate a new connection between classical special functions and a class of hydrodynamic systems which describe, for example, shock wave phenomena in gas dynamics. The research activities will involve graduate students who will be trained in the fields of both pure and 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, particle physics and plasmas.The PI will continue his work on geometric and combinatorial aspects of certain two-dimensional integrable systems, particularly, the Kadomtsev-Petviashvili (KP) equation. The KP equation admits a large class of solitary wave solutions with complex two-dimensional wave patterns, which are sometimes referred to as the KP web-solitons. This project also aims to give a further investigation of the PI's recent study on a class of integrable hydrodynamic systems in connection with the generalized hypergeometric functions that are defined on the Grassmannian. This project has two main goals, namely, (1) to investigate the detailed structure of the complex two-dimensional patterns of the KP web-solitons using geometric and combinatorial theories, and (2) to construct and classify integrable hydrodynamic systems generated by the generalized hypergeometric functions. Also, the project includes ongoing collaboration with experimentalists in order to compare the obtained theoretical results on the KP solitons with laboratory experiments performed in water wave tanks.