Analysis of Topological Singularities and Their Dynamics

点击翻译按钮获取中文摘要

Fang-Hua Lin, NYU Courant Institute. DMS-0201443Abstract: The first part of this proposal is to study the dynamics and thestability of topological singularities in several concrete cases including:magnetic vortices in the Landau-Lefschitz equations(Schrodinger and wavemaps coupled with magnetic fields),vortices and vortex rings in the trappeddilute Bose-Einstein condensates(certain nonlinear Schrodinger systemscoupled with potentials).The second part of the proposal is concerned withtopological singularities of Sobolev mappings.We shall examine the effectsof singularities on the denisity of smooth maps,the topology of mappingspaces,the regularity and singularity of various energy minimizers.Ofparticular interest is the global structures of singular sets and asymptotic behavior of energy minimizers near such topological singularities. Many interesting natural phenomena contain some sort singular behaviorand they are often manifiested through energy concentrations.Singularitiesof solutions of partial differential equations which describe thesephenomena are,therefore,an important part of facets.We propose here toestablish a rigourous mathematical theory concerning probabilly a mostimportant class of such singularities ,topological singularity.Topologicalsingularities arise in many physical problems and have been an importantsubject of much study over past many years.Among known examples aremagnetic bubbles in a fereomagnetic(or anti-ferromagnetic)continuum,vortices in Bose-Einstein condensates and superconductivity,topological defects inliquid crystals,as well as skyrminors,monopoles and instantons which areparticle like solutions in generic models of high energy physics.Thesesingularities not only carry definite topological informations but alsoquantified amount of energies.Because of this they are often moreobservable and stable energitically and dynamically.Thus a rigourousmathematical theory for such singularity would be not only mathematicallychallenging and interesting but also of fundamental importance in sciences.