A note on the Painlev\'{e property\\ of coupled KdV equations}
\textsc{S.~Yu.~Sakovich}\bigskip
National Academy of Sciences
[-4pt] 220072 Minsk
We prove that one system of coupled KdV equations, recently claimed by Hirota, Hu and Tang [J.~Math.\ Anal.\ Appl.\ 288:326--348 (2003)] to pass the Painlev\'{e} test for integrability, actually fails the test at the highest resonance of the generic branch.
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\title{\textbf{A note on the Painlev\'{e} property\\
of coupled KdV equations}}

\author{\textsc{S.~Yu.~Sakovich}\bigskip \\
\small Institute of Physics, National Academy of Sciences,\\[-4pt]
\small 220072 Minsk, Belarus. E-mail: saks@pisem.net}

\date{}

\maketitle

\begin{abstract}
We prove that one system of coupled KdV equations, recently
claimed by Hirota, Hu and Tang [J.~Math.\ Anal.\ Appl.\
288:326--348 (2003)] to pass the Painlev\'{e} test for
integrability, actually fails the test at the highest resonance of
the generic branch.
\end{abstract}

\section*{Introduction}

In Section~6 of their recent work \cite{HHT}, Hirota, Hu and Tang
reported that the system of coupled KdV equations
\begin{equation}
\label{e1}
\begin{split}
\frac{\partial u_i}{\partial t} & + 6a \left( \sum_{k=1}^N u_k
\right) \frac{\partial u_i}{\partial x} + 6(1-a) \left(
\sum_{k=1}^N \frac{\partial u_k}{\partial x} \right) u_i +
\frac{\partial^3 u_i}{\partial x^3} = 0,\\ i & = 1,2, \dotsc ,N
\qquad (N \geq 2)
\end{split}
\end{equation}
passes the Painlev\'{e} test for integrability iff the parameter
$a$ is equal to $1$ or $1/2$ or $3/2$. The authors of \cite{HHT}
pointed out that the cases $a=1$ and $a=1/2$ of \eqref{e1}
correspond to integrable systems of coupled KdV equations, whereas
the problem of integrability of \eqref{e1} with $a=3/2$ remains
open. In the present short note, we show that the system
\eqref{e1} with $a=3/2$ actually does not pass the Painlev\'{e}
test, and its integrability should not be expected therefore.

\section*{Singularity analysis}

First of all, let us notice that the $N$-component system
\eqref{e1} can be studied in the following form of two coupled KdV
equations:
\begin{equation}
\label{e2} v_t + 6 v v_x + v_{xxx} = 0, \qquad w_t + 6a v w_x +
6(1-a) w v_x + w_{xxx} = 0,
\end{equation}
where $v = \sum_{k=1}^N u_k$, $w$ is any one of the $N$ components
$u_1 , \dotsc , u_N$, and the subscripts $x$ and $t$ denote
partial derivatives. This equivalence between \eqref{e1} and
\eqref{e2} means that any solution $v$ of the first equation of
\eqref{e2} and any $N-1$ solutions $w$ (not necessarily different,
but corresponding to this $v$) of the second equation of
\eqref{e2} determine a solution $u_1 , \dotsc , u_N$ of the system
\eqref{e1}, and vice versa.

Now, setting $a=3/2$ in \eqref{e2} and starting the Weiss--Kruskal
algorithm of singularity analysis \cite{WTC,JKM}, we use the
expansions $v = v_0 (t) \phi^{\alpha} + \dotsb + v_r (t) \phi^{r +
\alpha} + \dotsb$ and $w = w_0 (t) \phi^{\beta} + \dotsb + w_r (t)
\phi^{r + \beta} + \dotsb$ with $\phi_x (x,t) = 1$, and determine
branches (i.e.\ admissible choices of $\alpha$ and $\beta$)
together with corresponding positions $r$ of resonances (where
arbitrary functions of $t$ can enter the expansions). The
exponents $\alpha$ and $\beta$ and positions of resonances turn
out to be integer in all branches. In what follows, we only
consider the generic singular branch, where $\alpha = \beta = -2$,
$v_0 = -2$, $w_0 (t)$ is arbitrary, and $r=-1,0,1,4,6,8$; this
branch describes the singular behavior of the general solution.

Substituting the expansions
\begin{equation}
\label{e3} v = \sum_{n=0}^{\infty} v_n (t) \phi^{n-2} , \qquad w =
\sum_{n=0}^{\infty} w_n (t)  \phi^{n-2}
\end{equation}
into the system \eqref{e2} with $a=3/2$, we obtain recursion
relations for the coefficients $v_n$ and $w_n$ of \eqref{e3}, and
then analyze the compatibility of those recursion relations at the
resonances. We find that the recursion relations are compatible at
the resonances $0$, $1$, $4$ and $6$, where the arbitrary
functions $w_0 (t)$, $w_1 (t)$, $v_4 (t)$ and $v_6 (t)$ appear in
the expansions \eqref{e3}, respectively; the resonance $-1$, as
usual, corresponds to the arbitrariness of $\psi$ in $\phi = x +
\psi (t)$. However, at the highest resonance, $r=8$, where the
arbitrary function $w_8 (t)$ enters the general solution, we
encounter the following compatibility condition which restricts
the arbitrary functions appeared at lower resonances:
\begin{equation}
\label{e4} 300 w_1 {v_4}' - 7 w_1 \phi' \phi'' + 6 {w_0}' \phi'' =
0,
\end{equation}
where the prime denotes $\partial / \partial t$.

The appearance of the compatibility condition \eqref{e4} means
that the Laurent type expansions \eqref{e3} do not represent the
general solution of the studied system, and we have to modify the
expansion for $w$ by introducing logarithmic terms, starting from
the term proportional to $\phi^6 \log \phi$. This non-dominant
logarithmic branching of solutions is a clear symptom of
non-integrability. Thus, the case $a=3/2$ of the system \eqref{e2}
(and of the system \eqref{e1}, equivalently) fails the
Painlev\'{e} test.

\section*{Conclusion}

We have shown that, contrary to what was claimed by Hirota, Hu and
Tang in \cite{HHT}, the system of coupled KdV equations \eqref{e1}
with $a=3/2$ does not pass the Painlev\'{e} test for
integrability. We have to note, moreover, that the singularity
analysis of coupled KdV equations has been done in the papers
\cite{Kar} and \cite{Sak}, published prior to \cite{HHT}. For
instance, the integrable cases $a=1$ and $a=1/2$ of the system
\eqref{e1} can be found in \cite{Sak} as the systems (vi) and
(vii) which passed the Painlev\'{e} test well, whereas the case
$r_1 = 1$ in Section~2.1.3 of \cite{Sak} tells that the system
\eqref{e1} with $a=3/2$ must fail the Painlev\'{e} test for
integrability.

\begin{thebibliography}{9}
\bibitem{HHT} R.~Hirota, X.-B.~Hu, X.-Y.~Tang. A vector potential
KdV equation and vector Ito equation: soliton solutions, bilinear
B\"{a}cklund transformations and Lax pairs. J.~Math.\ Anal.\
Appl.\ 288:326--348 (2003).

\bibitem{WTC} J.~Weiss, M.~Tabor, G.~Carnevale. The Painlev\'{e}
property for partial differential equations. J.~Math.\ Phys.\
24:522--526 (1983).

\bibitem{JKM} M.~Jimbo, M.~D.~Kruskal, T.~Miwa. Painlev\'{e} test
for the self-dual Yang--Mills equation. Phys.\ Lett.\ A 92:59--60
(1982).

\bibitem{Kar} A.~(Kalkanli)~Karasu. Painlev\'{e} classification of
coupled Korteweg--de~Vries systems. J.~Math.\ Phys.\ 38:3616--3622
(1997).

\bibitem{Sak} S.~Yu.~Sakovich. Coupled KdV equations of
Hirota--Satsuma type. J.~Nonlinear Math.\ Phys.\ 6:255--262
(1999); arXiv solv-int/9901005.
\end{thebibliography}

\end{document}

nlin0402004.extracted_bib
\begin{thebibliography}{9}
\bibitem{HHT} R.~Hirota, X.-B.~Hu, X.-Y.~Tang. A vector potential
KdV equation and vector Ito equation: soliton solutions, bilinear
B\"{a}cklund transformations and Lax pairs. J.~Math.\ Anal.\
Appl.\ 288:326--348 (2003).

\bibitem{WTC} J.~Weiss, M.~Tabor, G.~Carnevale. The Painlev\'{e}
property for partial differential equations. J.~Math.\ Phys.\
24:522--526 (1983).

\bibitem{JKM} M.~Jimbo, M.~D.~Kruskal, T.~Miwa. Painlev\'{e} test
for the self-dual Yang--Mills equation. Phys.\ Lett.\ A 92:59--60
(1982).

\bibitem{Kar} A.~(Kalkanli)~Karasu. Painlev\'{e} classification of
coupled Korteweg--de~Vries systems. J.~Math.\ Phys.\ 38:3616--3622
(1997).

\bibitem{Sak} S.~Yu.~Sakovich. Coupled KdV equations of
Hirota--Satsuma type. J.~Nonlinear Math.\ Phys.\ 6:255--262
(1999); arXiv solv-int/9901005.
\end{thebibliography}