Mathematical Sciences: Study of Nonlinear Problems by the IST Method

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This project supports the collaboration between a group of U.S. scientists at Clarkson University and a group of Russian scientists from University of Leningrad. The research concerns the study of nonlinear partial differential equations by the Inverse Scattering Transform (IST) Method. The IST method has been a successful and powerful method of solution in the 1+1 (one space and one time dimension) setting. The power of the IST method comes from the fact that the solution of certain nonlinear partial differential equations can be reduced to the solution of the inverse problem for linear ordinary differential equations, and such an inverse problem can usually be formulated as a Riemann- Hilbert problem. As a result, the solution of certain nonlinear partial differential equations have been obtained simply by using the methods to solve linear differential equations. The generalization of the IST method to N+1, where N is greater than 1, has not been as successful and so far some results have only been obtained for N=2. The Russian team at Leningrad is noted as one of the top teams in the world in this area of science. A collaboration would have significant scientific advantage to U.S. scientists. The science supported in this US-USSR collaboration is in the general area of partial differential equations. This research topic concerns the equations of motion, fluid flow, physics, etc. that model our physical universe.