On the (2+1)-Dimensional Integrable Inhomogeneous Heisenberg Ferromagnet Equation

Zhang, Zhen-Huan; Deng, Ming; Wang, Zhao; Wu, Ke

doi:10.1143/jpsj.75.104002

Cited by 10 publications

(6 citation statements)

References 11 publications

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“…This flow equation describes the motion of a sheet of non-stretching filaments in Euclidean space, in analogy with the form of the vortex filament equations (2.8). Some properties of the model (7.1) have been studied recently in [9,30].…”

Section: +1 Vector Models and Dynamical Mapsmentioning

confidence: 99%

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

Anco

Myrzakulov

2010

Journal of Geometry and Physics

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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 Schrödinger 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 Schrödinger 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. * Electronic address: sanco@brocku.ca † Electronic address: cnlpmyra@mail.ru

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Section: +1 Vector Models and Dynamical Mapsmentioning

confidence: 99%

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

Anco

Myrzakulov

2010

Journal of Geometry and Physics

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“…here we use the property is proves that the -HF is geometrical equivalent to the following -NLSE. where By choosing the coefficients of for the identity (13), (13) leads to the -NLS equation (12). ☐ e Lax pair of (12) can be represented as where and where is a identity matrix.…”

Section: Theorem 1 E Following Equation Holdsmentioning

confidence: 99%

“…Many (2+1)-dimensional integrable inhomogeneous Heisenberg ferromagnet equations have been of interest, for instance, Inhomogeneous M-I equation [13] and the Ishimori equation [11]. e Ishimori equation [11] is a well-known (2+1)-dimensional integrable extension of the HF model, which involves an infinite dimensional symmetry algebra with a loop algebra structure and is solved by the inverse scattering transform approach.…”

Section: Generalized -Heisenberg Ferromagnet Model In (2+1)-dimensionsmentioning

confidence: 99%

“…(1+1)-dimensional generalized HF models involving inhomogeneous and higher order deformed HF models have been analyzed [6,7]. e deformed HF models in (2+1)-dimensions also have been investigated, such as the higher order HF models [8,9], the HF models with self-consistent potentials [10], the Ishimori equation [11], and inhomogeneous HF models [12,13].…”

Section: Introductionmentioning

confidence: 99%

“…. + 퐵 1(푛−1) ,where When 푗 = 0, the -NLS equation (45) degrades into the (2+1)-dimensional focusing nonlinear Schrödinger equation equation[13]. e Lax representation of the -HLS equation can be expressed as where where 휆 0 ≤ 푗 ≤ 푛 − 1 are spectral parameters.…”

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confidence: 99%

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Three Types Generalized Zn-Heisenberg Ferromagnet Models

Zhou

Wan

Bai

et al. 2020

Advances in Mathematical Physics

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By taking values in a commutative subalgebra g푙(푛, ℂ), we construct a new generalized -Heisenberg ferromagnet model in (1+1)-dimensions. e corresponding geometrical equivalence between the generalized -Heisenberg ferromagnet model and -mixed derivative nonlinear Schrödinger equation has been investigated. e Lax pairs associated with the generalized systems have been derived. In addition, we construct the generalized -inhomogeneous Heisenberg ferromagnet model and -Ishimori equation in (2+1)-dimensions. We also discuss the integrable properties of the multi-component systems. Meanwhile, the generalized Z n -nonlinear Schrödinger equation, Z n -Davey-Stewartson equation and their Lax representation have been well studied.

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The Prolongation Structure of a Coupled Kdv Equation

Jia

Duan

2018

Acta Math. Appl. Sin. Engl. Ser.

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On the (2+1)-Dimensional Integrable Inhomogeneous Heisenberg Ferromagnet Equation

Cited by 10 publications

References 11 publications

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

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

Three Types Generalized Zn-Heisenberg Ferromagnet Models

The Prolongation Structure of a Coupled Kdv Equation

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