On the Birkhoff factorization problem for the Heisenberg magnet and nonlinear Schrödinger equations

A geometrical description of the Heisenberg magnet (HM) equation with classical spins is given in terms of flows on the quotient space G/H+ where G is an infinite dimensional Lie group and H+ is a subgroup of G. It is shown that the HM flows are induced by an action of R2 on G/H+, and that the HM equation can be integrated by solving a Birkhoff factorization problem for G. For the HM flows which are Laurent polynomials in the spectral variable we derive an algebraic transformation between solutions of the nonlinear Schrödinger (NLS) and Heisenberg magnet equation. The Birkhoff factorization problem for G is treated in terms of the geoemetry of the Segal-Wilson Grassmannian Gr(H). The solution of the problem is given in terms of a pair of Baker functions for special subspaces in Gr(H). The Baker functions are constructed explicitly for subspaces which yield multisoliton solutions of NLS and HM equations.