A central element in the study of Applied Mathematics is to understand and describe physical phenomena by employing detailed mathematical models. Frequently such models lead to large amplitude or nonlinear systems. Remarkably, in many cases certain prototypical equations are found to be the fundamental underlying systems that can be used to approximate the physical problem.
An important theme in this research program is to understand by approximation, numerical and exact methods, 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, techniques originally developed by physicists and mathematicians in the study of quantum mechanics. IST allows one to construct general solutions to certain initial-boundary value problems 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 are referred to as solitons, which are extremely stable localized waves. Solitons are important in physical applications, including nonlinear optics and fluid dynamics.
Some of the research areas being studied are: nonlinear waves in one and two dimensional periodic and complex lattice wave guides; this includes studies of honeycomb/hexagonal photonic lattices sometimes referred to as `photonic graphene', a photonic analog of material graphene, and their topological insulator properties; ultra-short pulse propagation in mode locked lasers; optical communications; discrete optical solitons; dispersive shock waves and their application to Bose-Einstein condensates and nonlinear optics; water waves; new solutions and analysis of nonlinear equations associated with the inverse scattering transform and novel differential equations arising in integrable systems and number theory.