Differential Geometry and Integrable Systems

项目编号：dms -9972172项目负责人：张楚莲项目负责人拟与Karen uhlenbeck共同开展以下关于几何可积系统的研究项目：(1)了解孤子方程的对称性和几何方面；(2)研究Lax对、双哈密顿结构、无穷多交换哈密顿、Backlund和Darboux变换以及逆散射变换之间的关系；(3)将孤子理论技术应用于几何问题，特别是对称空间中的子流形、调和映射和自对偶杨密尔场。三个最著名的孤子方程是KdV， SGE和NLSequations。KdV方程描述了浅渠中的波浪运动；SGE方程起源于等离子体物理学；NLSequation描述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。几何联系为构造更多的孤子方程和解提供了有趣的方法。现在已知数学、物理和几何中的许多有趣的问题都与孤子方程有关。本文的主要目的是利用几何方法研究孤子方程，并将孤子方程理论应用于微分几何问题。

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.