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Symmetry 2015, 7(3), 1352-1375; https://doi.org/10.3390/sym7031352

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

1
Eurasian International Center for Theoretical Physics and Department of General, Theoretical Physics, Eurasian National University, Astana 010008, Kazakhstan
2
Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli 620 024, India
*
Author to whom correspondence should be addressed.
Academic Editor: Antonio Palacios
Received: 9 April 2015 / Accepted: 22 July 2015 / Published: 3 August 2015
(This article belongs to the Special Issue Symmetry Breaking)
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Abstract

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 Schrödinger 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 Schrödinger family of equations, including the (2 + 1)-dimensional version of nonlinear Schrödinger–Hirota–Maxwell–Bloch equations, along with their Lax pairs. View Full-Text
Keywords: Heisenberg ferromagnet equation; integrable systems; solitons; nonlinear Schrodinger equations; gauge equivalence; nonlinear Schrodinger-Maxwell-Bloch equations; Lax epresentations; spin systems Heisenberg ferromagnet equation; integrable systems; solitons; nonlinear Schrodinger equations; gauge equivalence; nonlinear Schrodinger-Maxwell-Bloch equations; Lax epresentations; spin systems
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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MDPI and ACS Style

Myrzakulov, R.; Mamyrbekova, G.; Nugmanova, G.; Lakshmanan, M. Integrable (2 + 1)-Dimensional Spin Models with Self-Consistent Potentials. Symmetry 2015, 7, 1352-1375.

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