\protect\vspace*{-15mm}Zero curvature representation\\ for a new fifth-order integrable system
In this brief note we present a zero-curvature representation for one of the new integrable systems found by Mikhailov, Novikov and Wang in nlin.SI/0601046.
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%\jl{1}



\title{\protect\vspace*{-15mm}\bf Zero curvature representation\\ for a new fifth-order integrable system}


\author{A. Sergyeyev\\
%\address{
Silesian University in Opava, Mathematical Institute,\\ Na
Rybn\'\i{}\v{c}ku 1, 746\,01 Opava, Czech Republic,\\
%\ead
E-mail: \tt{Artur.Sergyeyev@math.slu.cz}}
\date{}
\maketitle
\begin{abstract}
In this brief note we present a zero-curvature representation
for one of the new integrable systems found by Mikhailov, Novikov and Wang
in nlin.SI/0601046.
\end{abstract}

\section*{Introduction}

The zero curvature representations (ZCRs) are well known to be of paramount importance
in the theory of integrable systems. Indeed, knowing a ZCR involving
an essential (spectral) parameter enables one to use
the inverse scattering transform (and/or find and use Darboux and B\"acklund transformations)
and construct plenty of exact solutions
for the system in question, including multisoliton and finite-gap solutions,
see e.g.\ the classical monographs \cite{as, tf, ne, ts} and references therein.
%%Moreover, one can construct new solutions

%On the other hand,
%finding  a ZCR
However, for integrable systems discovered
using the symmetry approach, see e.g.\
\cite{miknov, mnw, mss, msy, mikyam} and references therein,
there seems to be no straightforward procedure leading to
a ZCR suitable for the inverse scattering transform
(the ZCRs arising from the recursion operators often lead to pseudodifferential
spectral problems that are difficult to handle), so finding
``good" ZCRs for such systems
can be rather challenging.

In this paper we present a ZCR
for a new integrable system of the form
\be\label{e}
\ba{l}
u_t = u_{xxxxx}-20 u u_{xxx} -50 u_x u_{xx}+80 u^2 u_x-w_x,\\
w_t = -6 w u_{xxx}-2 u_{xx} w_x+96 w u u_x+16 w_x u^2
\ea
\ee
found by Mikhailov, Novikov and Wang \cite{mnw} using the symmetry approach.
This system is bihamiltonian and possesses a recursion operator \cite{mnw}.
More precisely, Eq.(\ref{e}) can
be obtained  \cite{mnw} from another integrable system, Eq.(101) of \cite{mnw},
through a simple invertible change
of variables, and hence the recursion operator, Hamiltonian and symplectic structure
found for Eq.(101) in \cite{mnw} can be readily transferred to (\ref{e}).

%Mikhailov, Novikov and Wang \cite{mnw} noted that
Setting $w=0$
reduces \cite{mnw} Eq.(\ref{e}) to the well-known Kaup--Kupershmidt equation
\be\label{kk}
u_t = u_{xxxxx}-20 u u_{xxx} -50 u_x u_{xx}+80 u^2 u_x.
\ee
The ZCR for (\ref{kk}) is readily extracted from the corresponding Lax pair
\cite{kaup} and reads
\be\label{zcrgen}
A_t-B_x+[A,B]=0,
\ee
where
\begin{equation}\label{zcr0}
\ba{l}
A=\left(\begin{array}{@{}ccc@{}} 0 & -2 & 0 \\ -u & 0 & -2 \\ \lambda &
-u & 0 \end{array}\right),\\[10mm]
B=\left(\begin{array}{@{}ccc@{}} 2 u_{xxx} - 32 u u_x + 24 \lambda u &
4 u_{xx} - 32 u^2 & -144 \lambda \\[3mm]
-u_{xxxx} + 18 u u_{xx} + 16 u_x^2 - 12 \lambda u_x - 16 u^3 + 72 \lambda^2 &
-48 \lambda u & 4 u_{xx} - 32 u^2 \\[3mm]
4\lambda u_{xx} - 20 \lambda u^2 &
b_{32} &
-2 u_{xxx} + 32 u u_x + 24 \lambda u \end{array}\right),
\ea
\end{equation}
and
\[
b_{32}=-u_{xxxx} + 18 u u_{xx} + 16 u_x^2 + 12 \lambda u_x - 16 u^3 + 72 \lambda^2.
\]

We succeeded in modifying the above $A$ and $B$ in such a way
that the resulting matrices go into (\ref{zcr0}) if $w=0$
{\em and} yield a ZCR for (\ref{e}) if $w\neq 0$:

\begin{prop}
The matrices \begin{equation}\label{zcr}
\ba{l}
A=\left(\begin{array}{@{}ccc@{}} 0 & -2 & 0 \\ -u & 0 & -2 \\
\ds\frac{1}{72} \frac{w}{\lambda} + \lambda & -u & 0 \end{array}\right),\\[9mm]
B=\left(\begin{array}{@{}ccc@{}} 2 u_{xxx} - 32 u u_x + 24 \lambda u &
4 u_{xx} - 32 u^2 & -144 \lambda \\[3mm]
-u_{xxxx} + 18 u u_{xx} + 16 u_x^2 - 12 \lambda u_x - 16 u^3 + 72 \lambda^2  + w&
-48 \lambda u & 4 u_{xx} - 32 u^2 \\[3mm]
4\lambda u_{xx} - 20 \lambda u^2-\ds\frac{w (u_{xx} - 8 u^2)}{36\lambda} &
b_{32} &
-2 u_{xxx} + 32 u u_x + 24 \lambda u \end{array}\right),
\ea
\end{equation} where
\[
b_{32}=-u_{xxxx} + 18 u u_{xx} + 16 u_x^2 + 12 \lambda u_x - 16 u^3 + 72 \lambda^2+w,
\]
provide a ZCR for (\ref{e}) with a spectral parameter $\lambda$, that is, the matrix equation
\[
A_t-B_x+[A,B]=0
\]
is equivalent to (\ref{e}) for all $\lambda\neq 0$.
\end{prop}

%As a final remark, note that the parameter $\lambda$ in (\ref{zcr})
%is essential, i.e., it cannot be removed by a suitably chosen gauge transformation
%(if it could be removed, then the same would hold for (\ref{zcr0}),
%so we would arrive at a contradiction).
%%This can be seen e.g. from the fact that the Jordan normal form of the matrices characteristic element
%%of (\ref{zcr})
\section*{Acknowledgements}
%\ack
%I am sincerely grateful to Prof M B\l aszak and
%Dr M Marvan for stimulating discussions.
%on results presented here.
%I would also like to thank Dr M V Pavlov for pointing me out
%the reference \cite{rubts}.

This research was supported in part by
the Czech Grant Agency (GA\v CR) under grant No.\ 201/04/0538,
%DFG via Graduiertenkolleg
%``Geometrie und Nichtlineare Analysis'', where the author now holds
%a postdoctoral fellowship.
%I also acknowledge the partial support
%from
by the Ministry of Education, Youth and Sports of Czech Republic
under grant MSM 4781305904.
%I am pleased to thank the anonymous referee for useful suggestions.
\looseness=-1
%,
%and from the Czech Grant Agency under Grant No. 201/00/0724.

%\newpage




\begin{thebibliography}{99}
\bibitem{as}Ablowitz M.J., Segur H. {\it Solitons and the inverse scattering transform}.
%SIAM Studies in Applied Mathematics, 4.
Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1981.
%x+425 pp. ISBN: 0-89871-174-6
\vspace{-2mm}


\bibitem{tf} Faddeev L.D. and Takhtajan L.A.
{\it Hamiltonian methods in the theory of solitons.}
%Translated from the Russian by A. G. Reyman [A. G. Re\u\i man].
%Springer Series in Soviet Mathematics.
Springer-Verlag, Berlin, 1987.
\looseness=-1
%%x+592 pp. ISBN: 3-540-15579-1
\vspace{-2mm}

\bibitem{kaup}Kaup D.J., On the inverse scattering problem
for cubic eigenvalue problems of the class
$\psi_{xxx} + 6q\psi_x + 6r\psi = \lambda\psi$,
{\it Stud. Appl. Math.} {\bf 62} (1980), 189--216.\vspace{-2mm}

\bibitem{miknov} Mikhailov A.V., Novikov V.S.
Perturbative symmetry approach. {\it J. Phys. A} {\bf 35} (2002),
no. 22, 4775--4790.
\vspace{-2mm}

\bibitem{mnw}Mikhailov A.V., Novikov V.S., Wang J.P.
On classification of integrable non-evolutionary equations,
Preprint nlin.SI/0601046, 22 p.
\vspace{-2mm}

\bibitem{mss}
Mikhailov A.V., Shabat A.B. and Sokolov V.V.
%{\em
The symmetry approach to classification of integrable equations, in
{\em What is Integrability?}, ed. V.E.~Zakharov, Springer, New York, 1991,
p.~115--184.\looseness=-2
\vspace{-2mm}

\bibitem{msy}
Mikhailov A.V., Shabat A.B. and Yamilov R.I.
%\newblock
%{\it
The symmetry approach to classification of
nonlinear equations. Complete lists of integrable systems,
{\em Russ.\ Math.\ Surv.} {\bf 42} (1987), no.4, 1--63.
\looseness=-1
\vspace{-2mm}


%\bibitem{sok}Sokolov V V  1988 {\em Russ.\ Math.\ Surv.\/}
%{\bf 43}(5) 165--204



\bibitem{mikyam} Mikhailov A.V., Yamilov R.I.
Towards classification of $(2+1)$-dimensional integrable equations. Integrability conditions. I.
{\it J. Phys. A: Math. Gen.} {\bf 31} (1998), no. 31, 6707--6715.
\vspace{-2mm}

\bibitem{ne}Newell A.C. {\it Solitons in mathematics and physics.}
%CBMS-NSF Regional Conference Series in Applied Mathematics, 48.
Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1985.
%xvi+244 pp. ISBN: 0-89871-196-7
\vspace{-2mm}



\bibitem{ts}Novikov S.P., Manakov S.V., Pitaevskii L.P., Zakharov V.E.
{\it Theory of solitons. The inverse scattering method.}
%Translated from the Russian. Contemporary Soviet Mathematics.
Consultants Bureau (Plenum), New York, 1984.
%xi+276 pp. ISBN: 0-306-10977-8
\vspace{-2mm}



\end{thebibliography}

\end{document}

nlin0604064.extracted_bib
\begin{thebibliography}{99}
\bibitem{as}Ablowitz M.J., Segur H. {\it Solitons and the inverse scattering transform}.
%SIAM Studies in Applied Mathematics, 4.
Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1981.
%x+425 pp. ISBN: 0-89871-174-6
\vspace{-2mm}


\bibitem{tf} Faddeev L.D. and Takhtajan L.A.
{\it Hamiltonian methods in the theory of solitons.}
%Translated from the Russian by A. G. Reyman [A. G. Re\u\i man].
%Springer Series in Soviet Mathematics.
Springer-Verlag, Berlin, 1987.
\looseness=-1
%%x+592 pp. ISBN: 3-540-15579-1
\vspace{-2mm}

\bibitem{kaup}Kaup D.J., On the inverse scattering problem
for cubic eigenvalue problems of the class
$\psi_{xxx} + 6q\psi_x + 6r\psi = \lambda\psi$,
{\it Stud. Appl. Math.} {\bf 62} (1980), 189--216.\vspace{-2mm}

\bibitem{miknov} Mikhailov A.V., Novikov V.S.
Perturbative symmetry approach. {\it J. Phys. A} {\bf 35} (2002),
no. 22, 4775--4790.
\vspace{-2mm}

\bibitem{mnw}Mikhailov A.V., Novikov V.S., Wang J.P.
On classification of integrable non-evolutionary equations,
Preprint nlin.SI/0601046, 22 p.
\vspace{-2mm}

\bibitem{mss}
Mikhailov A.V., Shabat A.B. and Sokolov V.V.
%{\em
The symmetry approach to classification of integrable equations, in
{\em What is Integrability?}, ed. V.E.~Zakharov, Springer, New York, 1991,
p.~115--184.\looseness=-2
\vspace{-2mm}

\bibitem{msy}
Mikhailov A.V., Shabat A.B. and Yamilov R.I.
%\newblock
%{\it
The symmetry approach to classification of
nonlinear equations. Complete lists of integrable systems,
{\em Russ.\ Math.\ Surv.} {\bf 42} (1987), no.4, 1--63.
\looseness=-1
\vspace{-2mm}


%\bibitem{sok}Sokolov V V  1988 {\em Russ.\ Math.\ Surv.\/}
%{\bf 43}(5) 165--204



\bibitem{mikyam} Mikhailov A.V., Yamilov R.I.
Towards classification of $(2+1)$-dimensional integrable equations. Integrability conditions. I.
{\it J. Phys. A: Math. Gen.} {\bf 31} (1998), no. 31, 6707--6715.
\vspace{-2mm}

\bibitem{ne}Newell A.C. {\it Solitons in mathematics and physics.}
%CBMS-NSF Regional Conference Series in Applied Mathematics, 48.
Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1985.
%xvi+244 pp. ISBN: 0-89871-196-7
\vspace{-2mm}



\bibitem{ts}Novikov S.P., Manakov S.V., Pitaevskii L.P., Zakharov V.E.
{\it Theory of solitons. The inverse scattering method.}
%Translated from the Russian. Contemporary Soviet Mathematics.
Consultants Bureau (Plenum), New York, 1984.
%xi+276 pp. ISBN: 0-306-10977-8
\vspace{-2mm}



\end{thebibliography}