Integrable spin systems possess interesting geometrical and gauge invariance properties and have important applications in applied magnetism and nanophysics. They are also intimately connected to the nonlinear Schr\"odinger family of equations. In this paper, we identify three different integrable spin systems in (2 + 1) dimensions by introducing the interaction of the spin field with more than one scalar potential, or vector potential, or both. We also obtain the associated Lax pairs. We discuss various interesting reductions in (2 + 1) and (1 + 1) dimensions. We also deduce the equivalent nonlinear Schr\"odinger family of equations, including the (2 + 1)-dimensional version of nonlinear Schr\"odinger--Hirota--Maxwell--Bloch equations, along with their Lax pairs.

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Integrable (2+1)-Dimensional Spin Models with Self-Consistent Potentials

Integrable spin systems possess interesting geometrical and gauge invariance properties and have important applications in applied magnetism and nanophysics. They are also intimately connected to the nonlinear Schrodinger family of equations. In this paper, we identify three different integrable spin systems in (2 + 1) dimensions by introducing the interaction of the spin field with more than one scalar potential, or vector potential, or both. We also obtain the associated Lax pairs. We discuss various interesting reductions in (2 + 1) and (1 + 1) dimensions. We also deduce the equivalent nonlinear Schrodinger family of equations, including the (2 + 1)-dimensional version of nonlinear Schrodinger–Hirota–Maxwell–Bloch equations, along with their Lax pairs.

Integrable (2 + 1)-Dimensional Spin Models with Self-Consistent Potentials

Integrable Heisenberg ferromagnetic equations are an important subclass of integrable systems. The M-XCIX equation is one of a generalizations of the Heisenberg ferromagnetic equation and are integrable. In this paper, the Darboux transformation of the M-XCIX equation is constructed. Using the DT, a 1-soliton solution of the M-XCIX equation is presented.

Darboux transformation and exact solutions of the integrable Heisenberg ferromagnetic equation with self-consistent potentials

Soliton excitations in the anisotropic Heisenberg chain are studied in the classical limit by means of the Holstein-Primakoff boson representation, taking into account the complete series. The results obtained are equivalent to those obtained by the classical treatment of spins. The connection with different calculations using limited numbers of terms is established, as well as the relation to the generalised coherent-states method.

The classical limit for the Holstein-Primakoff representation in the soliton theory of Heisenberg chains

In this paper, we study integrable multilayer spin systems, namely, the multilayer M-LIII equation. We investigate their relation with the geometric flows of interacting curves and surfaces in some...

Integrable geometric flows of interacting curves/surfaces, multilayer spin systems and the vector nonlinear Schrödinger equation

On the basis of the method of generalized coherent states (GCS) of the SU(2) group we formulate the reduction procedure of the quantum lattice Heisenberg XYZ model to the continuum Landau-Lifshitz model. The choice of the GCS representation is determined (i) by their proximity to the corresponding classical states and (ii) by similarity of the geometric structure of the homogeneous spaces on which the SU(2) GCS and the vector of the magnetic moment of the corresponding Landau-Lifshitz model are defined. The present reduction procedure is allowed only in the case of weak anisotropy. Accounting for a weak one-ion anisotropy leads to the Landau-Lifshitz model with a renormalized anisotropy tensor. The contribution of such an anisotropy vanishes for the s = 1/2 case.

https://iopscience.iop.org/article/10.1088/0031-8949/35/3/003/pdf

R Myrzakulov

Papers

Generalized Coherent States and the Continuous Heisenberg XYZ Model With One-Ion Anisotropy