# zbMATH — the first resource for mathematics

Landau-Lifshitz equation: solitons, quasi-periodic solutions and infinite-dimensional Lie algebras. (English) Zbl 0571.35104
In previous papers, the authors have developed an important viewpoint of soliton equations wherein the solution spaces in Hirota’s bilinear form can be regarded as orbits of infinite-dimensional groups. Here, the method is extended to include the Laudau-Lifshitz equation earlier rendered in bilinear form by Hirota. The hierarchy of the Landau-Lifshitz equations with full anisotropy is formulated in terms of a free fermion on an elliptic curve. Explicit forms of $$N$$ solitons and the corresponding wave functions are given. Infinitesimal Bäcklund transformations are presented and are shown to form an infinite-dimensional Lie algebra.
Reviewer: C.Rogers

##### MSC:
 35Q51 Soliton equations 17B65 Infinite-dimensional Lie (super)algebras 17B80 Applications of Lie algebras and superalgebras to integrable systems 37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures 37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
Full Text: