  The 1-d Schrodinger flow on 2-sphere, the Gauss-Codazzi equation for flat
Lagrangian submanifolds in C^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, Backlund transformations, and
solutions to Cauchy problems for these equations.
