The Landau-Lifshitz equation, elliptic curves and the Ward transform

Abstract

The Landau-Lifshitz (LL) equation is studied from a point of view that is close to that of Segal and Wilson's work on KdV. The LL hierarchy is defined and shown to exist using a dressing transformation that involves parameters λ1, λ2, λ3 that live on an elliptic curve Σ. The crucial role of the groupK ≃ ℤ2 × ℤ2 of translations by the half-periods of Σ and its non-trivial central extension\(\tilde K\) is brought out and an analogue of Birkhoff factorisation for\(\tilde K\)-equivariant loops in Σ is given. This factorisation theorem is given two treatments, one in terms of the geometry of an infinite-dimensional Grassmannian, and the other in terms of the algebraic geometry of bundles over Σ. Further, a Ward-like transform between a class of holomorphic vector bundles on the total spaceZ of a line-bundle over Σ and solutions of LL is constructed. An appendix is devoted to a careful definition of the Grassmannian of the Frechet spaceC ∞(S 1).