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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 which 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: ultra-short pulse
propagation in mode locked lasers; optical communications; nonlinear
waves in periodic and complex lattice wave guides; 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.
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