Differential Geometry, group actions, and soliton equations

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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.