Loop Groups, Grassmannians and KdV Equations

Abstract

This is a report on joint work of Graeme Segal and George Wilson. Its aim is to describe a construction of a large class of solutions of the KdV equations (and, more generally, the KP equations) in terms of the geometry of a certain infinite dimensional Grassmannian Gr. The solutions obtained include all the “quasi-periodic” solutions arising from the algebro-geometric construction of Krichever, and in particular the rational, soliton and multi-soliton solutions.