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. ***
