The Heisenberg magnet equation and the Birkhoff factorization

摘要

A geometrical description of the Heisenberg magnet (HM) with classical spins is given in terms of flows on the homogeneous space G/H ₊ where G is a Banach Lie group and G ₊ is a subgroup of G. The flows are induced by an action of the abelian group on G/H ₊, and the solutions of the HM equation can be found by solving a Birkhoff factorization problem for G. The gauge transformation between the HM and nonlinear Schrödigner (NLS) equations is interpreted as a transformation between a canonical pair of Birkhoff factorizations for G. It is shown that for the HM flows which are Laurent polynomials in the spectral variable this transformation gives rise to a map between the HM and NLS solutions.