Differential Geometry, group actions, and soliton equations

孤立子方程是一种非线性波动方程，是完全可积的哈密顿系统，具有：Lax对，无穷多个显式孤立子解族，代数叠加公式，从给定解产生PDE新解的变换，以及非常大的对称群。最著名的这类方程是KdV方程、非线性薛定谔（NLS）方程和正弦-戈登方程（SGE）。KdV和NLS自然地产生于几何流，SGE是3 $-空间中具有恒定负高斯曲率的曲面的Gauss-Codazzi方程。许多其他的偏微分方程在几何中出现，现在也知道是孤立子方程。Terng将研究对称空间和仿射平坦3空间中的子流形，其控制方程是孤子方程或椭圆可积系统，在黎曼，伪黎曼和仿射几何中构造曲线流，以便相应的曲线不变量方程是孤子方程。 Terng还将研究反自对偶杨-米尔斯方程的模空间。 和B一起。Dai和Uhlenbeck，Terng在小初值条件下解决了Cauchy问题，构造了规范群为幺正群时的所有孤子时空单极子。 她建议当规范群是一个任意紧的简单李群或特殊线性群时研究这个方程。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、代数和应用数学中有多种解释。 KdV方程模拟了波在浅管道中的运动，NLS方程模拟了光纤中波包络的运动，SGE方程起源于等离子体物理学。NLS的类粒子孤子解的存在有助于加快互联网的速度。 孤子方程也作为许多重要几何问题的方程出现。 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有趣的几何对象，并使用几何来构建新的孤子方程。

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.