Mathematical Sciences: Nonlinear Dispersive Waves

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9302988 McLaughlin The investigator continues his research in the theory of nonlinear waves, using a combination of mathematical theory, formal approximation methods, and scientific computation. In addition, some of the projects in nonlinear optics will be done in collaboration with members of the Department of Physics at Princeton University who will provide a direct experimental component to the research. McLaughlin will work in three general areas of nonlinear waves -- (i) temporally chaotic waves, (ii) nonlinear optics, and (iii) spatially and temporally chaotic waves. In the project on temporally chaotic nonlinear waves, mathematical methods from dynamical systems theory will be adapted to the partial differential setting of waves. The project on spatially and temporally chaotic waves will develop both stochastic and dynamical mathematical methods for the partial differential equations of waves. The projects in nonlinear optics will focus upon laser light interacting with liquid crystals. Because liquid crystal media are extremely nonlinear, they provide excellent materials in which to study ``light-matter''interactions. All nonlinear waves in nature, from the light waves of laser physics to water waves on the surface of the ocean, are modeled mathematically by partial differential equations known as nonlinear wave equations. While linear wave equations have been well understood for some time, scientists and mathematicians knew little about the behavior of solutions of nonlinear wave equations until the recent explosion in scientific computing power. While scientific computation can suggest and discover new and unexpected behavior in solutions of nonlinear wave equations, one can never be certain of the outcome of numerical experiments. Mathematical studies, often of further idealized models, are required -- not only to ensure the validity of the numerical experiments, but also to establish qualitative behavior in the model. With results certain for the model nonlinear partial differential equations, physical experiments can then be used to validate the wave equations as a description of real waves in nature. This project, taken together with the experiments in nonlinear optics on the ``light-liquid crystal'' interaction, addresses each of these related areas (scientific computation, mathematical studies, and physical experiments) in the mathematical studies of nonlinear waves. ***