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

小行星9623463 本研究项目的目的是了解 在含时超导方程中的超导涡旋（金-朗道理论）。此外，类粒子物体在杨-米尔斯希格斯方程和其他系统中的作用 将被研究。这些方程本质上分别是抛物型和双曲型方程组。这项工作的目的之一是 理解“粒子”的动力学性质如何反映在方程的分析性质中。这是 有趣的是，粒子代表了真正的非线性效应。类粒子对象是与时间无关的解，其特征在于具有拓扑缠绕数。 减少对这些“粒子”动力学的描述 在有限维系统方面已经获得了严格的边界上的有效性的近似。这使得人们能够理解“粒子”之间的作用力及其散射特性。 这反过来又导致了一个 理解原始方程，特别是关于解的大时间行为的图解和定理。反过来，人们希望，这些方程的大时间行为的研究可能是一个有用的工具，在分析静态粒子的解决方案本身。 %%% 在物理世界的数学描述中，有“场”和“粒子”之分。前者是连续的介质，其中信息以波状方式传播，而后者是离散的对象。然而，某些给出某种物理情形的“场”描述的方程，其解本身就是“类粒子”的。这在两种类型的描述之间架起了一座桥梁。这种现象发生在真实的世界中，那就是超导性：涡旋在超导体中表现为连续统描述中的粒子状物体。他们非常 在超导体的技术应用中是重要的，因为涡流对电流通过材料产生阻力，因此降低了材料在现代技术中的功效。因此，能够 了解涡旋之间以及涡旋与其他材料之间的相互作用，以充分实现超导材料在高速计算机设计、电力传输、现代铁路系统和其他领域的巨大技术潜力。 ***

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. ***