Applications of loop group factorization to geometric soliton equations

Abstract

The 1-d Schrödinger flow on S_2, the Gauss–Codazzi equation for flat Lagrangian submanifolds in ℝ2_n, and the space-time monopole equation are all examples of geometric soliton equations. The linear systems with a spectral parameter (Lax pair) associated to these equations satisfy the reality condition associated to SU(n). In this article, we explain the method developed jointly with K. Uhlenbeck that uses various loop group factorizations to construct inverse scattering transforms, Bäcklund transformations, and solutions to Cauchy problems for these equations.