Mathematical Sciences: Evolution of Structures in Nonlinear PDE's

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9404374 Muraki The theme of proposed work centers on the evolution of spatial structure in nonlinear 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) flow structures in complex fluids. In particular, the mathematical objective is to establish relationships between coherent structures, instability and dynamics as pertains to the development of weakly-turbulent behaviors in simple nonlinear systems. The basic strategy involves both the identification of the primary structural modes, and the determination of their dynamics and interaction properties. A combination of asymptotic analysis, dynamical systems methods and numerical computation are employed to obtain clear illustrations of the mechanisms by which spatial structures are created and sustained by nonlinearity. This, in turn, provides an analytical basis leading to a physical understanding for the underlying nonlinear phenomena. A familiar example of a spatio-temporal evolution is the video record of satellite weather maps as often shown on the evening news. While the imagery of spatial features that evolve over time is a common theme for many physical and biological phenomena, our quantitative understanding of these processes is actually quite limited. The proposed research investigates the model equations for unusually complex patterns -- the break-up of optical laser beams, and the spreading of chemicals in surface reactions -- which have only recently been produced in the laboratory, but hold future promise for the design and control of new fine-scale processes.