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\title{Riemann-Hilbert Problems and Soliton equations.
The Reduction problem and Hamiltonian properties.}

\author{ V. S. Gerdjikov \\
Institute of Nuclear Research and Nuclear Energy\\ Bulgarian Academy of Sciences,
Sofia 1784, Bulgaria}
\date{}

\maketitle

Our main tool is the Riemann-Hilbert problem (RHP) with canonical normalization
\begin{equation}\label{eq:rhp}\begin{split}
\xi^+(\vec{x},t,\lambda) &= \xi^-(\vec{x},t,\lambda) G(\vec{x},t,\lambda) , \qquad \lambda^k\in\mathbb{R}, \qquad
\lim_{\lambda\to\infty} \xi^+(\vec{x},t,\lambda) =\openone ,
\end{split}\end{equation}
where the functions $\xi^\pm(\vec{x},t,\lambda)$ are taking values in a simple Lie group $\mathfrak{G}$ and
allow analytic extension for $\lambda^k\in\mathbb{R}$ respectively, see \cite{VG1,VG2,VYa}. We also assume that the sewing function $G(\vec{x},t,\lambda) $
depends on the auxiliary variables $\vec{x},t$ as follows:
\begin{equation}\label{eq:Gxt}\begin{split}
i \frac{\partial G}{ \partial x_s } -\lambda^k [J_s, G(\vec{x},t,\lambda) ] &=0, \qquad
i \frac{\partial G}{ \partial t } -\lambda^k [K, G(\vec{x},t,\lambda) ] =0.
\end{split}\end{equation}
where $J_s$ belong to the Cartan subalgebra of the relevant simple Lie algebra $ \mathfrak{h}\subset \mathfrak{g}$.

The canonical normalization of the RHP allows us to use the asymptotic expansions:
\begin{equation}\label{eq:xi-as}\begin{split}
\xi^\pm(\vec{x},t,\lambda) = \exp Q(\vec{x},t,\lambda), \qquad Q(\vec{x},t,\lambda)=\sum_{k=1}^{\infty} Q_k(\vec{x},t) \lambda^{-k} .
\end{split}\end{equation}
where all $Q_k(\vec{x},t)\in \mathfrak{g}$.
Next we use Zakharov-Shabat theorem \cite{ZaSha} to construct a
family of commuting operators $L_s$, $M$ provided the coefficients $Q_j(\vec{x},t)$, $j=1,\dots,k$ in (\ref{eq:xi-as}) satisfy a certain set of
soliton equations. New examples of such equations \cite{VG1,VG2} are obtained by applying Mikhailov's $\mathbb{Z}_h$-reduction \cite{Mikh} on
$C\xi^\pm(\vec{x},t,\lambda)C^{-1}=\xi^\pm(\vec{x},t,\lambda\omega)$ where $C^h=\openone$ and $\omega^h=1$. One can also derive
their soliton solutions using the dressing Zakharov-Shabat method \cite{ZaSha}.

It is natural to expect that the Hamiltonian properties of the new types of soliton equations can be obtained from the
the generic Hamiltonian structures of polynomial bundles \cite{KuRe} by considering the reduction conditions as Dirac
constraints.

{\small
\begin{thebibliography}{DD}

\bibitem{VG1} V. S. Gerdjikov. Pliska Stud.\ Math.\ Bulgar.\
{\bf 21}, 201--216 (2012). %subm. March 20, 2012.
{\bf arXiv:1204.2928 }%[nlin.SI]}.

\bibitem{VG2} V. S. Gerdjikov. AIP Conf.
proc. {\bf 1487} pp. 272-279 (2012); %doi:http://dx.doi.org/10.1063/1.4758968 (8 pages).
%Proceedings of AMITANS-4 conference. \quad
{\bf arXiv:1302.1116.}

\bibitem{VYa} V. S. Gerdjikov, A B Yanovski.
Journal of Physics: Conference Series {\bf 482} (2014) 012017. %\quad doi:10.1088/1742-6596/482/1/012017

\bibitem{ZaSha} V. E. Zakharov, A. B. Shabat, \textit{ Functional Annal. \& Appl.} {\bf 8}, 43-53 (1974) (In Russian); \\
V. E. Zakharov, A. B. Shabat, \textit{ Functional Annal. \& Appl.} {\bf 13}, 13-22 (1979) (In Russian).

\bibitem{Mikh} A. V. Mikhailov, \textit{Physica D} \textbf{3D}, 73--117 (1981).

\bibitem{KuRe} P. P. Kulish, A. G. Reyman. %\textit{Hamiltonian structures of polynomial bundles}.
Sci. notes of LOMI seminars, {\bf 123}, 67--76 (1983). (In Russian).
\end{thebibliography}
}
\end{document}