Landau-Lifshitz hierarchy and infinite dimensional Grassmann variety

Abstract

The Landau-Lifshitz equation is an example of soliton equations with a zero-curvature 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 $2 \times 2$ matrices of Laurent series of the spectral parameter $z$. A special base point $W_0$, 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.