Differential Geometry and Integrable Systems

微分几何和可积系统

• 批准号: 9972172
• 负责人: Chuu-lian Terng
• 金额: $ 12.14万
• 依托单位: Northeastern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-07-01 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9972172&HistoricalAwards=false
• 关键词: Differential; Geometry; Integrable; Systems

AbstractAward: DMS-9972172Principal Investigator: Chuu-Lian TerngThe principal investigator proposes jointly with Karen Uhlenbeckto work on the following projects concerning geometric integrablesystems: (1) to understand the symmetries and geometric aspectsof soliton equations, (2) to study the relations among Lax pairs,bi-Hamiltonian structures, infinitely many commutingHamiltonians, Backlund and Darboux transformations, and theinverse scattering transform, (3) to apply the techniques ofsoliton theory to geometric problems, in particular, submanifoldsin symmetric spaces, harmonic maps and self-dual Yang Millsfields.The three most well-known soliton equations are KdV, SGE and NLSequations. The KdV equation describes wave motion in a shallowcanal; the SGE equation arose from plasma physics; and the NLSequation describes the propogation of light in a fiber opticcable and is currently used in telecommunication. Theseequations also arose in the 19 century differential geometry.The geometric connection gives interesting methods to constructmore soliton equations and solutions. Many interesting problemsin mathematical physics and geometry are now known to be relatedto soliton equations. The main goal of this proposal is to usegeometric method to study soliton equations and apply the theoryof soliton equations to problems in differential geometry.

AbstractAward：DMS-9972172主要研究者：Chuu-Lian Tern主要研究者建议与Karen Uhlenbeck共同研究以下关于几何积分系统的项目：（1）理解孤子方程的对称性和几何性质;（2）研究Lax对、双Hamilton结构、无穷多个双Hamilton、Backlund变换和Darboux变换以及逆散射变换之间的关系;（3）将孤子理论应用于几何问题，特别是对称空间中的子流形、调和映射和自对偶杨米尔斯场，最著名的三个孤子方程是KdV方程、SGE方程和NLS方程。 KdV方程描述了浅管道中的波动; SGE方程源于等离子体物理学; NLS方程描述了光在光纤电缆中的传播，目前用于电信。 这些方程也出现在19世纪的微分几何中，几何上的联系为构造更多的孤子方程及其解提供了有趣的方法。数学、物理和几何中许多有趣的问题都与孤子方程有关. 本文的主要目的是用几何方法研究孤子方程，并将孤子方程的理论应用于微分几何问题。