Nonlinear Wave Motion

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A class of fundamental nonlinear wave equations and related nonlinear systems that have important physical applications is being investigated. New solutions and properties of both one dimensional and multi-dimensional equations will be obtained. The physically significant systems that will be investigated include the Kadomtsev-Petviashvili (KP) equation, which is a two space - one time dimensional extension of the Korteweg-deVries equation and the one space - one time dimensional vector nonlinear Schroedinger equation. New ordinary differential equation reductions of the four-dimensional self-dual equations will be pursued and properties of their solutions analyzed. Experimental and theoretical research in water waves has shown that modulation of periodic waves in deep water yield nonrepeatible chaotic dynamics, whereas solitons do not exhibit this behavior. Applications to other physical problems, such as fiber optics, will be investigated. It appears that this phenomenon is universal in character.An essential element in the study of Applied Mathematics is to explain physical phenomena by mathematical models. Frequently such models lead to nonlinear systems and in a surprisingly large number of cases certain prototypical equations. This research aims to understand by exact and approximate methods of an analysis and computational techniques solutions to these underlying equations and their properties. An important method used to solve certain nonlinear wave equations is the so-called Inverse Scattering Transform (IST). The IST is conceptually analogous to the Fourier Transform; IST employs methods of direct and inverse scattering that are techniques originally developed by physicists and mathematicians studying quantum mechanics. The IST allows one to construct general solutions to equations that arise in a variety of physical problems such as nonlinear optics, water waves, plasma physics, lattice vibrations, and relativity. A special class of solutions referred to as solitons are extremely stable localized pulse-like waves. Solitons are important in many physical applications. The PI's research has been extensively referenced worldwide; the Web of Science lists the PI as one of the most highly cited people in the field of mathematics.