Together with strange attractors and chaos, solitons play a key role in nonlinear science. Solitons are stable, particle-like, nonlinear pulses which result from a critical balance between nonlinearity and dispersion. My research is concerned with nonlinear partial differential equations (PDEs) as well as discretized equations, that are completely integrable and admit exact (soliton) solutions. Such PDEs model shallow water waves, nonlinear optical pulses, currents in electrical networks, nerve pulses, waves in the atmosphere, etc.
Powerful symbolic manipulation programs such as Mathematica, Maple, Macsyma, and REDUCE offer virtually unlimited potential to do mathematics on a computer. The "symbol-crunching" capabilities of such software packages allow one to investigate properties of nonlinear equations without having to do the tedious algebra and calculus with pen and paper.
My research is currently focused on designing symbolic tools that help establish the integrability of certain nonlinear PDEs, differential-difference equations, and fully-discretized lattices. I have designed symbolic programs for the Painlevé test, and codes to calculate Lie-point and generalized symmetries, conservation laws, Lax pairs, recursion operators, and exact solutions, including solitary wave and soliton solutions. Since 1991, my research projects have been supported by the National Science Foundation.
From my student days in Belgium, my curiosity was kindled by real-world applications of mathematics; I was happy to see what "all that stuff was finally used for." My doctoral dissertation dealt with the mathematics of acousto-optics: the interaction of ultrasound and laser light. Engineering applications of acousto-optics include radar signal processing, nondestructive testing, optical computing, laser shows, and medical scanners. These research projects were supported in part by the North Atlantic Treaty Organization (NATO). I have returned to scattering problems later in my career. For example, I investigated the scattering of waves from rough surfaces, as part of a project sponsored by the Air Force Office of Scientific Research (AFOSR). With colleagues at the University of Colorado in Boulder, I also worked on the theory and applications of wavelets.
I like to be surrounded by a group of undergraduate and graduate students and work together on symbolic programs and applied mathematics projects. I was also involved in solving problems from industry, including the modeling of acousto-optical materials, the design of speakers, and the positioning of equipment in an open-pit coal mine based on trilateration.
Last updated: Sunday, June 23, 2013