Differential Geometry, group actions, and soliton equations

微分几何、群作用和孤子方程

• 批准号: 1109342
• 负责人: Chuu-lian Terng
• 金额: $ 26.05万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1109342&HistoricalAwards=false
• 关键词: Differential; Geometry; group; actions; soliton

Soliton equations are non-linear wave equation that are completely integrable Hamiltonian systems and have: Lax pairs, infinitely many families of explicit soliton solutions, an algebraic superposition formula, transformations that produce new solutions of the PDE from a given one, and a very large symmetry group. The most well-known such equations are the KdV equation, the non-linear Schrodinger (NLS) equation, and the sine-Gordon equation (SGE). The KdV and NLS arise naturally from geometric flows and the SGE is the Gauss-Codazzi equation for surfaces in $3$-space with constant negative Gaussian curvature. Many other partial differential equations arising in geometry are also now known to be soliton equations. Terng will study submanifolds in symmetric spaces and in affine flat 3-space whose governing equations are soliton equations or elliptic integrable systems, construct curve flows in Riemannian, pseudo-Riemannian, and affine geometries so that the corresponding equations for curve invariants are soliton equations. Terng will also study the moduli space of the anti-self-dual Yang-Mills equation. Jointly with B. Dai, and Uhlenbeck, Terng solved the Cauchy problem with small initial data and constructed all soliton space-time monopoles when the gauge group is the unitary group. She proposes to investigate this equation when the gauge group is an arbitrary compact, simple Lie group or the special linear group. Soliton theory is an enticingly elegant part of modern mathematics. It has a multitude of interpretations in geometry, analysis, algebra, and applied mathematics. The KdV equation models the motion of waves in shallow canal, the NLS equation models the motion of a wave envelope in an optical fiber, and the SGE equation arises in Plasma physics. The existence of particle like soliton solutions of the NLS has helped speed up the internet. Soliton equations also arise as equations for many important geometric problems. Terng plans to study geometric aspects of soliton equations, use techniques in soliton theory to construct new and interesting geometric objects, and use geometry to construct new soliton equations.

孤子方程是完全可积的哈密顿系统的非线性波动方程，它有：Lax对，无穷多个显式孤子解，一个代数叠加公式，从给定的偏微分方程组产生新解的变换，以及一个非常大的对称群。最著名的此类方程是KdV方程、非线性薛定谔(NLS)方程和Sine-Gordon方程(SGE)。KdV和NLS是由几何流自然产生的，而SGE是$3空间中具有常负高斯曲率的曲面的Gauss-Codazzi方程。几何中出现的许多其他偏微分方程现也被称为孤子方程。Terng将研究对称空间和仿射平坦三维空间中的子流形，这些空间的控制方程是孤子方程或椭圆可积系统，构造黎曼几何、伪黎曼几何和仿射几何中的曲线流，使得曲线不变量的相应方程是孤子方程。Terng还将研究反自对偶杨-米尔斯方程的模空间。Terng与B.Dai和Uhlenbeck共同解决了初值较小的柯西问题，并构造了当规范群为酉群时的所有孤子时空单极子。当规范群是任意紧的单李群或特殊的线性群时，她建议研究这个方程。孤立子理论是现代数学中一个迷人而优雅的部分。它在几何、分析、代数和应用数学方面有多种解释。KdV方程模拟浅水管道中波的运动，NLS方程模拟光纤中波包的运动，SGE方程产生于等离子体物理。NLS类粒子孤子解的存在帮助加快了互联网的速度。孤子方程也作为方程出现在许多重要的几何问题中。Terng计划研究孤子方程的几何方面，使用孤子理论中的技术来构建新的和有趣的几何对象，并使用几何来构建新的孤子方程。