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Integrable two-layer spin systems with self-consistent potential
Gulgassyl Nugmanova, Zhanna Sagidullayeva, Ratbay Myrzakulov

Last modified: 2017-04-14


Integrable generalization with self-consistent potential for the Heisenberg ferromagnetic equation reads as
\[iS_{t}+\frac{1}{2}[S, S_{xx}]+\frac{1}{\omega}[S, W]=0, \eqno (1a)\]\[iW_{x}+\omega [S, W]=0, \eqno (1b)\]where $\omega=const$, $S={\Sigma}^{3}_{j=1}S_j(x,y,t)\sigma_j$ is a matrix analogue of the spin vector, $W$ - potential with the matrix form $W={\Sigma}^{3}_{j=1}W_j(x,y,t)\sigma_j$, and $\sigma_j$ are Pauli matrices. In work \cite{key-1} was shown that spin system with self-consistent potentials (1) is gauge equivalent to the Schr\"{o}dinger-Maxwell-Bloch (SMB) equations \cite{key-2}
\[iq_{t}+q_{xx}+2\delta |q|^2q-2ip=0, \]\[p_{x}-2i\omega p -2\eta q=0,\]\[\eta_{x}+\delta(q^{*} p +p^{*} q)=0,\]where $q, p$ are complex functions, $\eta$ is a real function, $\omega, \delta$ are real contants ($\delta=\pm 1$). The Darboux transformation for equation (1) was constructed and some of its exact solutions were found  \cite{key-3}.  In this presentation we show which two-layer spin system is associated with the two-layer SMB equation \cite{key-4}.
\begin{thebibliography}{1}\bibitem[1]{key-1}R. Myrzakulov, G. Mamyrbekova, G. Nugmanova, M. Lakshmanan. Integrable Motion of Curves in Self-Consistent Potentials: Relation to Spin Systems and Soliton Equations. Phys. Lett. A 378 (2014),  2118 \textendash 2123.
\bibitem[2]{key-2}R. Myrzakulov, G. Mamyrbekova, G. Nugmanova, M. Lakshmanan. Integrable (2+1)-dimensional Spin Models with Self-Consistent Potentials.Symmetry. 7 (2015), 1352-1375. 
\bibitem[3]{key-3}Z. Yersultanova, M. Zhassybayeva, G. Mamyrbekova, G. Nugmanova, R. Myrzakulov.  Darboux Transformation and Exact Solutions of the Integrable Heisenberg Ferromagnetic Equation with Self-Consistent Potential. Int. J. Geom. Meth. in Mod. Phys. 12 (2015), 1550134.
\bibitem[4]{key-4}Xiao-Yu Wu, Bo Tian, Hui-Ling Zhen, Wen-Rong Sun and Ya Sun. Solitons for the (2+1)-dimensional nonlinearSchr\"{o}dinger-Maxwell-Bloch equations in an erbium-doped fibre. J. Mod. Opt. 2015,

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