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

这个项目是研究非线性常微分方程和偏微分方程领域的三个重要物理问题。第一个问题是最近提出的一个演化方程的模式形成动力学，作为各种结晶固体中微观结构形成的模型，其目的是对演化进行分析研究和解释，并将其与已获得的各种数值结果联系起来。二是研究超导中与金兹堡-朗道方程相关的涡旋，即当某一参数变小时，金兹堡-朗道泛函变简并的最小化问题；McLeod将研究当参数趋于零时最小化器的行为。最后，他将研究具有最近邻相互作用的晶格上行波的存在性和性质。每一个问题都有其自身的意义。第一个与各种晶体固体有关，在这些固体中，观察到微观结构随着施加的应力、温度变化或施加的电场或磁场而出现或消失。因此，了解这些微观结构的形成对于这些材料的工业应用至关重要。第二个问题是关于超导这一非常重要的领域的相变，并试图将数学理论与物理实验和观察结合起来。

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.