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
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
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; 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.