Mathematical Sciences: Geometry of Integrable Systems

数学科学：可积系统的几何

• 批准号: 8901607
• 负责人: David Sattinger
• 金额: $ 8.6万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-06-15 至 1992-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8901607&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Geometry; Integrable; Systems

8901607 Sattinger This project concerns the investigation of Hamiltonian hierachies based on semi-simple Lie algebras viewed as gauge theories of a flat connection. Flat connections arise in a multitude of problems, such as the self-dual Yang-Mills equations, and embedding problems in differential geometry. An approach to extending Sato's infinite dimensional Grassmannian picture of the Kadomtsev-Petviashvili hierachy to the nxn hierarchies is proposed, based on the solution of the inverse scattering problem for nxn first order systems on the line. The co-adjoint action for the hierachies is to be investigated from the point of view gauge transformations and the Riemann-Hilbert factorization associated with the solution of the inverse scattering problem for nxn systems. The approach is based on the formulation of the results of scattering theory in terms of principal bundles, and the "dressing method" of Zakharov and Shabat.

8901607 Sattinger 该项目涉及基于被视为平面连接的规范理论的半简单李代数的哈密尔顿层次结构的研究。 平面连接出现在许多问题中，例如自对偶杨-米尔斯方程以及微分几何中的嵌入问题。 基于在线 nxn 一阶系统的逆散射问题的求解，提出了一种将 Sato 的 Kadomtsev-Petviashvili 层次结构的无限维格拉斯曼图扩展到 nxn 层次结构的方法。 层次结构的共伴随作用将从规范变换和与 nxn 系统的逆散射问题的解相关的黎曼-希尔伯特分解的角度进行研究。 该方法基于用主束表示的散射理论结果以及扎哈罗夫和沙巴特的“修整方法”。