Mathematical Models of Nonlinear Wave Processes

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9704724 Muraki The theme of proposed work centers on the generation and propagation of nonlinear waves in PDE models which derive from physical systems of current interest: a) self-focussing of light in nematic liquid crystals, b) interfacial pattern formation in chemical reactions, c) wave behaviors in atmospheric weather systems. The mathematical objective is to quantify relationships between spatial structure, instability, and dynamics as pertains to wave behavior in nonlinear PDE systems. In all of the proposed projects, the model equations involve coupled PDE systems where the wave behavior is strongly coupled to other background influences. The complexity of these models requires the development of new methods for nonlinear waves which combine the techniques of asymptotic analysis, modern dynamical systems theory, and numerical simulation. These projects focus on the understanding of nonlinear waves through the identification of solutions representing the primary wave modes, along with their dynamics and interaction properties -- while remaining relevant within the underlying physical context. Change in nature is often mediated by waves -- a process in which spatial features evolve over time. Familiar examples are outward spreading of ripples in a pond and waves in the form of atmospheric turbulence, which result in the "bumpy" flights experienced by aircraft. The science of waves (as with science in general) is becoming increasingly quantitative, due to higher-precision instrumentation and larger-scale computing. The interpretation and understanding of greater amounts of refined data demands an equal enhancement of our ability to analyze increasingly sophisticated mathematical models of wave generation and propagation. The proposed research seeks to develop analytical and computational techniques to quantify wave behavior in model equations for optical, chemical, and weather systems and to re-interpret the mathem atical implications within the original scientific context.