Analysis of Topological Singularities and Their Dynamics

New York University, New York NY

Investigators

Abstract

Fang-Hua Lin, NYU Courant Institute. DMS-0201443 Abstract: The first part of this proposal is to study the dynamics and the stability of topological singularities in several concrete cases including :magnetic vortices in the Landau-Lefschitz equations(Schrodinger and wave maps coupled with magnetic fields),vortices and vortex rings in the trapped dilute Bose-Einstein condensates(certain nonlinear Schrodinger systems coupled with potentials).The second part of the proposal is concerned with topological singularities of Sobolev mappings.We shall examine the effects of singularities on the denisity of smooth maps,the topology of mapping spaces,the regularity and singularity of various energy minimizers.Of particular 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 behavior and they are often manifiested through energy concentrations.Singularities of solutions of partial differential equations which describe these phenomena are,therefore,an important part of facets.We propose here to establish a rigourous mathematical theory concerning probabilly a most important class of such singularities ,topological singularity.Topological singularities arise in many physical problems and have been an important subject of much study over past many years.Among known examples are magnetic bubbles in a fereomagnetic(or anti-ferromagnetic)continuum,vortices in Bose-Einstein condensates and superconductivity,topological defects in liquid crystals,as well as skyrminors,monopoles and instantons which are particle like solutions in generic models of high energy physics.These singularities not only carry definite topological informations but also quantified amount of energies.Because of this they are often more observable and stable energitically and dynamically.Thus a rigourous mathematical theory for such singularity would be not only mathematically challenging and interesting but also of fundamental importance in sciences.