Mathematical Sciences: Nonlinear Dispersive Waves

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DMS-9600128 McLaughlin The proposer continues his research in the theory and application of nonlinear dispersive waves, in the specific areas of chaotic nonlinear waves, waves which occur in the interaction of the atmosphere with the ocean, and topics in liquid crystal nonlinear optics. His work on the mathematical theory of chaotic nonlinear waves will focus upon the persistence and consequences of homoclinic orbits for partial differ- ential equations. In the project he develops geometric singular perturbation methods for partial differential equations, including including invariant manifold fibrations and normal forms. He then combines these methods from dynamical systems theory with Melnikov analysis to prove the persistence of homoclinic orbits. Consequences of these orbits for nonlinear wave systems are then investigated numerically. Such modern methods in the mathematical theory of nonlinear dispersive waves are applied to topics in atmosphere ocean physics and to nonlinear optics. %%% Atmosphere ocean interactions and laser propagation in nonlinear optics are two of the most natural areas in science for the occurence of nonlinear dispersive waves. The proposer will develop modern mathe- matical theory of these waves, and apply it to these areas of science. Chaotic and irregular behavior, which occurs in the temporal evolution of these nonlinear wave systems, is just becoming understood mathematically. The proposer will study the effects of such nonlinear and irregular behavior on oscillations of the sea surface temperature, on the prediction and evolution of storm tracks, and on the propagation of laser light through liquid crystal materials. Methods for this scientific study include mathematical theory, formal perturbation theory, and scientific computation. ***