Differential Geometry, group actions, and soliton equations

孤子方程是完全可积哈密顿系统的非线性波动方程，它具有：Lax对、无穷多个显式孤子解族、一个代数叠加公式、从给定解产生PDE新解的变换和一个非常大的对称群。最著名的这类方程是KdV方程、非线性薛定谔方程（NLS）和正弦戈登方程（SGE）。KdV和NLS是由几何流自然产生的，SGE是$3$-空间中具有恒定负高斯曲率的曲面的高斯-科达齐方程。在几何中出现的许多其他偏微分方程现在也被认为是孤子方程。Terng将研究对称空间和仿射平面三维空间中控制方程为孤子方程或椭圆可积系统的子流形，构造黎曼几何、伪黎曼几何和仿射几何中的曲线流，使曲线不变量的对应方程为孤子方程。Terng还将研究反自对偶Yang-Mills方程的模空间。Terng与B. Dai和Uhlenbeck共同解决了初始数据较小的Cauchy问题，构造了规范群为酉群时的全孤子时空单极子。她提出在规范群是任意紧李群、单李群或特殊线性群时研究这个方程。孤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方程模拟了波在浅通道中的运动，NLS方程模拟了光纤中包络波的运动，而等离子体物理中出现了SGE方程。NLS的类粒子孤子解的存在有助于加快互联网的速度。孤子方程也出现在许多重要几何问题的方程中。Terng计划研究孤子方程的几何方面，利用孤子理论中的技术构造新的有趣的几何对象，并利用几何构造新的孤子方程。

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.