Nonlinear Wave Motion

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Nonlinear wave motion is manifested in many natural phenomena and technological processes. Nonlinearity effects are most prominent for the high-power signals and large-magnitude waves: when one wave is superimposed on the other the waves are not just adding as would be true in the linear case but interact with each other. Such waves form complex wave patterns and have very important applications, including among others propagation of electromagnetic waves for fiber optics and lasers, shock waves in aerodynamics, and rogue waves in the ocean. Mathematical analysis of these phenomena is hampered by the nonlinearity of the governing equations. For nonlinear equations formulating a general theory is often unfeasible, and obtaining solutions requires a case-by-case study. This research effort focuses on solving a number of open problems that will substantially extend the ability of mathematicians to develop deeper understanding and explain the behavior of large-amplitude wave phenomena which arise widely in applications. The Inverse Scattering Transform (IST) method is one of the few analytical techniques that allows one to analyze and obtain solutions to a number of important equations in mathematical physics. IST will be used to obtain solutions and understand properties of new classes of nonlinear wave equations which exhibit PT (parity-time) symmetry properties.  IST can also be used to describe certain classes of localized solutions, termed lump solutions, which decay in all directions. The key properties of these solutions will be understood and they will be connected to the non-stationary Schrödinger equation. The research effort will also be directed to investigating a class of shock wave phenomena termed dispersive shock waves (DSWs). DSWs are shock waves which are regularized by dispersion, in contrast with standard shock waves which are regularized by dissipation. DSWs arise in many applications including water waves, Bose- Einstein condensates, nonlinear optics, etc. The theory of DSWs will be extended in order to obtain improved approximations to the underlying equations in one dimension, and a detailed multidimensional analysis will be developed.