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\begin{document}

\title{Symmetric spaces and new 2-component integrable NLS equations}

\author{{\bf V. S. Gerdjikov} \\[5pt]
\sl Institute of Mathematics and Informatics \\
Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria.
%Acad. Georgi Bonchev Str., Block 8, \\
%Institute for Advanced Physical Studies, New Bulgarian University \\
%21 Montevideo Street, Sofia 1618, Bulgaria
}
\date{}
\maketitle

The fundamental properties of the multi-component nonlinear
Schr\"o\-dinger (NLS) type models related to the D.III symmetric spaces are
analyzed. Their Lax operator in the case $SO^*(2r)/U(r)$ take the form:
\begin{equation*}\label{eq:}\begin{split}
L\psi & \equiv \left( i \frac{\partial }{ \partial x } +
\left(\begin{array}{ccc} 0 & \q \\ \p & 0 \end{array}\right) - \lambda
\left(\begin{array}{ccc} \openone & 0 \\ 0 & -\openone \end{array}\right) \right) \psi(x,t,\lambda)=0,
\end{split}\end{equation*}
New types of $\mathbb{Z}_r\otimes \mathbb{Z}_2$ Mikhailov reductions of these systems are
constructed, after which $Q(x,t)$ with $r=5$ depends on only two complex functions as follows:
\[ \q(x,t) = \frac{1}{\sqrt{6}} \left(\begin{array}{ccccc} q_1 & q_2 & q_2 & q_1 & 0 \\
q_2 & -q_2 & q_1 & 0 & q_1 \\ q_2 & q_1 & 0 & q_1 & -q_2 \\ q_1 & 0 & q_1 & q_2 & q_2 \\
0 & q_1 & -q_2 & q_2 & -q_1 \end{array}\right), \qquad \p = \q^\dag . \]

As a result we obtain a 2-component NLS type system with Hamiltonian \cite{1}:
\begin{equation*}\label{eq:}\begin{split}
H= \frac{1}{2}\left| \frac{\partial q_1}{ \partial x } \right|^2 +
\frac{1}{2}\left| \frac{\partial q_2}{ \partial x } \right|^2 - \left( |q_1|^2 + |q_2|^2\right)^2
- \frac{1}{2} \left(|q_1|^2 + G\right)^2 - \frac{1}{2} \left(|q_2|^2 - G\right)^2,
\end{split}\end{equation*}
where $G=q_1 q_2^* + q_1^*q_2$.
The spectral properties of the reduced Lax operator $L $
and the fundamental properties of the relevant class
of nonlinear evolution equations are described. Special attention is paid to the
recursion operators and the bi-Hamiltonian properties of the relevant NLEE.

Other examples of new 2-component NLS equations related to other types of
symmetric spaces are presented in \cite{2}.

\begin{thebibliography}{1}
\bibitem{1} V.S. Gerdjikov, A. A. Stefanov. New types of two component NLS-type equations.
Pliska Studia Mathematica {\bf 26}, 53--66 (2016).

\bibitem{2} V. S. Gerdjikov. Kulish-Sklyanin type models: integrability and reductions.
Submitted to TMF.
\end{thebibliography}


\end{document}