On the Birkhoff factorization problem for the Heisenberg magnet and nonlinear Schroedinger equation (CROSBI ID 141298)
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Saša Krešić-Jurić
engleski
On the Birkhoff factorization problem for the Heisenberg magnet and nonlinear Schroedinger equation
A geometrical description of the Heisenberg magnet (HM) equation with classical spins is given in terms of flows on the quotient space $G/H_+$ where $G$ is an infinite dimensional Lie group and $H_+$ is a subgroup of $G$. It is shown that the HM flows are induced by an action of $\R^2$ on $G/H_+$, and that the HM equation can be integrated by solving a Birkhoff factorization problem for $G$. For the HM flows which are Laurent polynomials in the spectral variable we derive an algebraic transformation between solutions of the nonlinear Schr\"{; ; ; o}; ; ; dinger (NLS) and Heisenberg magnet equation. The Birkhoff factorization problem for $G$ is treated in terms of the geoemetry of the Segal-Wilson Grassmannian $Gr(H)$. The solution of the problem is given in terms of a pair of Baker functions for special subspaces in $Gr(H)$. The Baker functions are constructed explicitly for subspaces which yield multisoliton solutions of NLS and HM equations.
Birkhoff factorization; Heisenberg magnet; Segal-Wilson Grassmannian
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