The Landau-Lifshitz equation,elliptic curves and the ward transform |
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Authors: | A L Carey K C Hannabuss L J Mason M A Singer |
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Institution: | (1) Department of Pure Mathematics, University of Adelaide, 5001, South Australia;(2) Balliol College, Oxford, UK;(3) Mathematical Institute, Oxford, UK;(4) Lincoln College, Oxford, UK |
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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 1, 2, 3 that live on an elliptic curve . The crucial role of the groupK 2 × 2 of translations by the half-periods of and its non-trivial central extension
is brought out and an analogue of Birkhoff factorisation for
-equivariant loops in 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 . Further, a Ward-like transform between a class of holomorphic vector bundles on the total spaceZ of a line-bundle over and solutions of LL is constructed. An appendix is devoted to a careful definition of the Grassmannian of the Frechet spaceC
(S
1). |
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Keywords: | |
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