Mathematical Sciences: "Dynamics of Superconducting Vortices and Monopoles in Gauge Theories"

这个研究项目的目的是了解超导涡旋在随时间变化的超导方程（金兹堡-朗道理论）中的行为。此外，还将研究类粒子物体在杨-米尔斯希格斯方程和其他系统中的作用。这些方程本质上分别是抛物线型和双曲型方程组。这项工作的目的之一是了解“粒子”的动力学性质如何反映在方程的解析性质中。这很有趣，因为粒子代表了真正的非线性效应。类粒子物体是与时间无关的解，并以拓扑圈数为特征。在有限维系统中对这些“粒子”的动力学进行了简化描述，并对近似的有效性给出了严格的限定。这使得我们能够理解“粒子”之间的作用力以及它们的散射特性。这反过来又导致对原始方程的理解，特别是关于解的大时间行为的猜想和定理。反过来，人们希望对这些方程的大时间行为的研究可以成为分析静态类粒子解本身的有用工具。在对物理世界的数学描述中，“场”和“粒子”是有区别的。前者是连续的媒介，其中信息以波状的方式传播，而后者是离散的对象。然而，对某些物理情况给出“场”描述的某些方程，其解本身就是“类粒子”的。这在两种类型的描述之间提供了一个桥梁。发生这种情况的真实世界的情况是超导：涡旋在超导体中作为连续体描述中的粒子状物体出现。它们在超导体的技术应用中是非常重要的，因为涡流对电流通过材料产生阻力，从而降低了材料在现代技术中的效率。因此，为了充分发挥超导材料在高速计算机设计、电力传输、现代铁路系统等领域的巨大技术潜力，了解涡旋之间以及与材料其他部分的相互作用是极其重要的。＊＊＊

9623463 Stuart The aim of this research project is to understand the behavior of superconducting vortices in the equations of time-dependent superconductivity (Ginzburg-Landau theory). Also the role of particle-like objects in the Yang-Mills Higgs equations and other systems will be studied. These equations are essentially systems of parabolic and hyperbolic type respectively. One of the aims of the work is to understand how the dynamical properties of the ``particles'' are reflected in the analytical properties of the equations. This is interesting since the particles represent a genuinely nonlinear effect. The particle-like objects are solutions which are independent of time and are characterized by a topological winding number. Reduced descriptions of the dynamics of these ``particles'' in terms of finite dimensional systems have been obtained together with rigorous bounds on the validity of the approximations. These allow an understanding of the forces between the ``particles'' and also their scattering properties. This in turn leads to an understanding of the original equations, in particular to conjectures and theorems about the large time behavior of the solutions. In turn it is to be hoped that a study of the large time behavior of these equations may be a useful tool in analyzing the static particle-like solutions themselves. %%% In mathematical descriptions of the physical world there is a division between ``fields'' and ``particles''. The former are continuous media in which information propagates in a wave-like fashion, while the latter are discrete objects. However certain equations which give a ``field'' description of some physical situation have solutions which are themselves ``particle-like''. This provides a bridge between the two types of description. A real world situation in which this occurs is superconductivity: vortices appear in superconductors as particle-like objects within a continuum description. They are very important in technological applications of superconductors because the vortices produce a resistance to passage of current through the material and hence reduce the efficacy of the material in modern technology. It is therefore extremely important to be able to understand the interactions of vortices with each other and with the rest of the material in order to fully realize the great technological potential of superconducting materials in high speed computer design, power transmission, modern rail systems and other areas. ***