Mathematical Sciences: Nonlinear Oscillations/Completely Integrable Systems

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Professor Venakides intends to study mathematical problems of oscillation and wave propagation. He wants to determine the nature of oscillations in waveforms which are governed by a nonlinear partial differential equation of Schrodinger type. The interesting part of this work consists of studying the passage to the limit when the dispersive term becomes very small. Problems of this type were studied for the Korteweg-DeVries equation by Peter Lax, Henry McKean and Eugene Trubowitz of Courant Institute and by Herman Flaschka and others from the University of Arizona. Venakides brings new techniques to his study and wants to derive an integral equation which would characterize the solution in the limit case. He also wants to perform a similar study for a system of nonlinear ordinary differential equations known as Toda Lattice. Other projects described by him include an inverse problem for a Schrodinger equation, study of equations of granular flow and the work on the understanding of certain procedure for stochastic equations with a nonlinear operator. This research falls into the general area of studies in nonlinear partial differential equations which describe wave propagation phenomena such as solitary waves, acoustic waves, shocks and wavefronts, optical communications and laser technology. This particular research is intended to add to the knowledge base underlying such physical phenomena. The nonlinear partial differential equations of the type investigated in this project are important mathematical tools for engineering problems encountered in wave propogation.