Mathematical Sciences: Geometry of Integrable Systems

点击翻译按钮获取中文摘要

8901607 Sattinger This project concerns the investigation of Hamiltonian hierachies based on semi-simple Lie algebras viewed as gauge theories of a flat connection. Flat connections arise in a multitude of problems, such as the self-dual Yang-Mills equations, and embedding problems in differential geometry. An approach to extending Sato's infinite dimensional Grassmannian picture of the Kadomtsev-Petviashvili hierachy to the nxn hierarchies is proposed, based on the solution of the inverse scattering problem for nxn first order systems on the line. The co-adjoint action for the hierachies is to be investigated from the point of view gauge transformations and the Riemann-Hilbert factorization associated with the solution of the inverse scattering problem for nxn systems. The approach is based on the formulation of the results of scattering theory in terms of principal bundles, and the "dressing method" of Zakharov and Shabat.