Mathematical Sciences: Investigations into Development of Microstructure, Ginzburg-Landau Vortices, and Travelling Waves on Lattics

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DMS-9501395 McLeod The project is to study three physically important problems in the area of nonlinear ordinary and partial differential equations. The first problem is the dynamics of pattern formation for an evolution equation recently proposed as a model for the formation of microstructures in various crystalline solids, and the object is to give an analytical study and explanation of the evolution, and link this to various numerical results which have been obtained. The second problem is the study of vortices associated with the Ginzburg-Landau equation in superconductivity, i.e. minimizers of the Ginzburg-Landau functional which becomes degenerate as a certain parameter becomes small; McLeod will study the behavior of the minimizers as the parameter tends to zero. Finally, he will study the existence and properties of traveling waves on lattices with nearest-neighbor interactions. Each of these problems has its own significance. The first has relevance for a variety of crystalline solids where microstructures are observed to appear or disappear in response to imposed stresses, changes of temperature or applied electric or magnetic fields. An understanding of the formation of these microstructures is therefore crucial for the industrial use of these materials. The second problem is relevant to phase transitions in the highly important area of superconductivity, and attempts to marry the mathematical theory to physical experiment and observation.