Analysis of Topological Singularities and Their Dynamics

拓扑奇点及其动力学分析

• 批准号: 0201443
• 负责人: Fang-Hua Lin
• 金额: --
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-01 至 2008-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0201443&HistoricalAwards=false
• 关键词: Analysis; Topological; Singularities; Their; Dynamics

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.

林芳华，纽约大学考兰特学院。DMS-0201443摘要：本方案的第一部分是研究几种具体情况下拓扑奇点的动力学和稳定性，包括：Landau-Lefschitz方程(与磁场耦合的薛定谔方程和波映射)中的磁涡旋，陷阱稀疏玻色-爱因斯坦凝聚(某些与势耦合的非线性薛定谔系统)中的涡旋和涡环。第二部分是关于Sobolev映射的拓扑奇点。我们将研究奇点对光滑映射的致密性、映射空间的拓扑、各种能量最小化的正则性和奇异性的影响。特别感兴趣的是奇异集的整体结构和拓扑奇点附近能量极小子的渐近行为。许多有趣的自然现象都包含着某种奇异行为，它们通常是通过能量集中来描述的。因此，描述这些现象的偏微分方程解的奇异性是各方面的一个重要部分。我们建议建立关于这类奇点中最重要的一类--拓扑奇性的严格的数学理论。拓扑奇性存在于许多物理问题中，在过去的许多年里一直是许多研究的重要课题。已知的例子有铁磁(或反铁磁)连续介质中的磁泡，玻色-爱因斯坦凝聚体中的涡旋和超导，液晶中的拓扑缺陷，以及天空小单极子和瞬子在高能物理的一般模型中是类似粒子的解。这些奇点不仅携带确定的拓扑信息，而且还量化了能量的量。正因为如此，它们通常在能量和动力学上更可观察和稳定。因此，对于这种奇点，一个严格的数学理论不仅在数学上具有挑战性和趣味性，而且在科学上也具有基础性的重要性。