Integrable generalizations of Schrodinger maps and Heisenberg spin models from Hamiltonian flows of curves and surfaces

JOURNAL OF GEOMETRY AND PHYSICS

卷:60

Article

Anco, Stephen C.¹  Myrzakulov, R.² 

[1] Brock Univ, Dept Math, St Catharines, ON L2S 3A1, Canada

[2] Eurasian Natl Univ, Dept Gen & Theoret Phys, Astana 010008, Kazakhstan

关键词: Integrable vector model;    Curve flow;    Schrodinger map;    Heisenberg model;    bi-Hamiltonian;   

DOI  :  10.1016/j.geomphys.2010.05.013

来源: Elsevier

【 摘 要 】

A moving frame formulation of non-stretching geometric curve flows in Euclidean space is used to derive a 1 + 1 dimensional hierarchy of integrable SO(3)-invariant vector models containing the Heisenberg ferromagnetic spin model as well as a model given by a spin vector version of the mKdV equation. These models describe a geometric realization of the NLS hierarchy of soliton equations whose bi-Hamiltonian structure is shown to be encoded in the Frenet equations of the moving frame This derivation yields an explicit bi-Hamiltonian structure, recursion operator, and constants of motion for each model in the hierarchy A generalization of these results to geometric surface flows is presented, where the surfaces are non-stretching in one direction while stretching in all transverse directions. Through the Frenet equations of a moving frame, such surface flows are shown to encode a hierarchy of 2 + 1 dimensional integrable SO(3)-invariant vector models, along with their bi-Hamiltonian structure, recursion operator, and constants of motion, describing a geometric realization of 2 + 1 dimensional bi-Hamiltonian NLS and mKdV soliton equations. Based on the well-known equivalence between the Heisenberg model and the Schrodinger map equation in 1 + 1 dimensions, a geometrical formulation of these hierarchies of 1 + 1 and 2 + 1 vector models is given in terms of dynamical maps Into the 2-sphere. In particular, this formulation yields a new integrable generalization of the Schrodinger map equation in 2 + 1 dimensions as well as a mKdV analog of this map equation corresponding to the mKdV spin model in 1 + 1 and 2 + 1 dimensions. (C) 2010 Elsevier B V All rights reserved

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