Twistor Geometry and Field Theory: Gauge theory

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 motion of curves in self-consistent potentials: Relation to spin systems and soliton equations

The Myrzakulov-I equation is a 2 + 1-dimensional generalization of the Heisenberg ferromagnetic equation and has a non-isospectral Lax pair. The Darboux transformation with non-constant spectral parameter is constructed and an extra constraint on the spectral parameter for the existence of the Darboux transformation is derived. Explicit expressions of the solutions of the Myrzakulov-I equation are presented.

Darboux Transformation and Exact Solutions of the Myrzakulov-I Equation

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

A few years ago, some of us devised a method to obtain integrable systems in (2+1)-dimensions from the classical non-Abelian pure Chern–Simons action via the reduction of the gauge connection in Hermitian symmetric spaces. In this article we show that the methods developed in studying classical non-Abelian pure Chern–Simons actions can be naturally implemented by means of a geometrical interpretation of such systems. The Chern–Simons equation of motion turns out to be related to time evolving two-dimensional surfaces in such a way that these deformations are both locally compatible with the Gauss–Mainardi–Codazzi equations and completely integrable. The properties of these relationships are investigated together with the most relevant consequences. Explicit examples of integrable surface deformations are displayed and discussed.

Deformation of surfaces, integrable systems, and Chern–Simons theory

We study the connection between some lattice and continuous Heisenberg spin models. The continuous limits of some generalized compressible Heisenberg spin chains are found. Using the methods of differential geometry of surfaces, the Lakshmanan equivalent counterpart of one of the obtained continuous Heisenberg ferromagnet equation is constructed.

On Continuous Limits of Some Generalized Compressible Heisenberg Spin Chains

Integrable Heisenberg Ferromagnets and Soliton Geometry of Curves and Surfaces

A new class of two-dimensional surfaces generated by formulas which are generalizations of the well known Lelieuvre and Schief formulas is presented. These surfaces are connected with two-dimensional spin systems which are stationary versions of the (2+1)-dimensional classical continuous Heisenberg ferromagnets.

On the Geometry of Stationary Heisenberg Ferromagnets

Integrability of the (2+1)-dimensional Gauss-Codazzi-Mainardi equation is considered. It is shown that this equation is the particular cases of the Yang-Mills-Higgs-Bogomolny and self-dual Yang-Mills equations.

Kur. Myrzakul

Papers

Deformation of surfaces, integrable systems, and Chern–Simons theory

On Continuous Limits of Some Generalized Compressible Heisenberg Spin Chains

Integrable Heisenberg Ferromagnets and Soliton Geometry of Curves and Surfaces

On the Geometry of Stationary Heisenberg Ferromagnets

Integrability of the Gauss-Codazzi-Mainardi equation in (2+1)-dimensions