Mark J. Ablowitz
Professor of Applied Mathematics

Department of Applied Mathematics
University of Colorado, Box 526
Boulder, CO 80309-0526
 Telephone: 303-492-5502, Fax: 303-492-4066

Research Interests
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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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This site was last updated June, 2007