Landau-Lifshitz hierarchy and infinite dimensional Grassmann variety

The Landau-Lifshitz equation is an example of soliton equations with a zerocurvature representation defined on an elliptic curve. This equation can be embedded into an integrable hierarchy of evolution equations called the Landau-Lifshitz hierarchy. This paper elucidates its status in Sato, Segal and Wilson’s universal description of soliton equations in the language of an infinite dimensional Grassmann variety. To this end, a Grassmann variety is constructed from a vector space of 2×2 matrices of Laurent series of the spectral parameter z. A special base point W0, called “vacuum,” of this Grassmann variety is chosen. This vacuum is “dressed” by a Laurent series φ(z) to become a point of the Grassmann variety that corresponds to a general solution of the Landau-Lifshitz hierarchy. The Landau-Lifshitz hierarchy is thereby mapped to a simple dynamical system on the set of these dressed vacua. A higher dimensional analogue of this hierarchy (an elliptic analogue of the Bogomolny hierarchy) is also presented. Mathematics Subject Classification: 35Q58, 37K10, 58F07