Singularity analysis of a new discrete nonlinear Schr\"{odinger equation}
\textsc{S.~Yu.~Sakovich}\medskip
We apply the Painlev\'{e} test for integrability to the new discrete nonlinear Schr\"{o}dinger equation introduced by Leon and Manna. Since the singular expansions of solutions of this equation turn out to contain non-dominant logarithmic terms, we conclude that the studied equation is nonintegrable. This result supports the recent observation of Levi and Yamilov that the Leon--Manna equation does not admit high-order generalized symmetries.\medskip PACS numbers: 02.30.Ik, 02.30.Hq
nlin0111060.tex
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\title{\textbf{Singularity analysis of a new discrete nonlinear Schr\"{o}dinger equation}}
\author{\textsc{S.~Yu.~Sakovich}\medskip\\{\small Institute of Physics, National Academy of Sciences, 220072 Minsk, Belarus}\\{\small E-mail: saks@pisem.net}}
\date{}
\maketitle

\begin{abstract}

We apply the Painlev\'{e} test for integrability to the new discrete nonlinear
Schr\"{o}dinger equation introduced by Leon and Manna. Since the singular
expansions of solutions of this equation turn out to contain non-dominant
logarithmic terms, we conclude that the studied equation is nonintegrable.
This result supports the recent observation of Levi and Yamilov that the
Leon--Manna equation does not admit high-order generalized symmetries.\medskip

PACS numbers: 02.30.Ik, 02.30.Hq

\end{abstract}

Recently, Leon and Manna \cite{LM} introduced the following new discrete
nonlinear Schr\"{o}dinger equation:%
\begin{equation}
\alpha\psi_{m,tt}=\mathrm{i}\beta\left(  \psi_{m+1}-\psi_{m-1}\right)
+2\left|  \psi_{m}\right|  ^{2}\psi_{m} \label{psi}%
\end{equation}
where $\alpha$ and $\beta$ are nonzero real parameters, $m=0,\pm1,\pm2,\ldots
$. In \cite{LM}, the equation (\ref{psi}) was derived from the integrable Toda
lattice equation through the reductive perturbation analysis. More recently,
Levi and Yamilov \cite{LY} studied the integrability of (\ref{psi}) by means
of the generalized symmetry analysis, and found that the equation (\ref{psi})
does not admit local generalized symmetries of order higher than three and,
hence, does not possess the same integrability properties as the Toda lattice
equation, from which it has been derived.

In the present short note, we study the integrability of the Leon--Manna
equation (\ref{psi}) by means of the Painlev\'{e} test, following the
Ablowitz--Ramani--Segur algorithm of singularity analysis \cite{ARS} (see also
\cite{RGB}). We find the presence of non-dominant logarithmic terms in the
singular expansions of solutions of~(\ref{psi}). This brings us to the same
conclusion as made in \cite{LY}: the new discrete nonlinear Schr\"{o}dinger
equation must be nonintegrable.

Following \cite{LY}, we study (\ref{psi}) in its equivalent form:%
\begin{equation}%
\begin{array}
[c]{l}%
u_{m,tt}-u_{m}^{2}v_{m}-u_{m+1}+u_{m-1}=0\medskip\\
v_{m,tt}-v_{m}^{2}u_{m}+v_{m+1}-v_{m-1}=0
\end{array}
\label{uv}%
\end{equation}
$m=0,\pm1,\pm2,\ldots$, where $u_{m}$ and $v_{m}$ stand for $\psi_{m}$ and
$\bar{\psi}_{m}$, and a complex-valued scale transformation of variables has
been made. From the standpoint of singularity analysis, the equation
(\ref{uv}) is an infinite system of second-order nonlinear ordinary
differential equations, which admits infinitely many branches of the dominant
behavior of solutions, i.e.\ the choices of constants $\alpha_{m},\beta
_{m},u_{m,0},v_{m,0}$ in $u_{m}=u_{m,0}\phi^{\alpha_{m}}+\cdots$ and
$v_{m}=v_{m,0}\phi^{\beta_{m}}+\cdots$, where $m=0,\pm1,\pm2,\ldots$ and
$\phi_{t}=1$. We will analyze only one of those branches, namely,%
\begin{equation}
\alpha_{m}=\beta_{m}=-1\qquad u_{m,0}v_{m,0}=2\qquad m=0,\pm1,\pm2,\ldots.
\label{br}%
\end{equation}

Using the expansions%
\begin{equation}%
\begin{array}
[c]{l}%
u_{m}=u_{m,0}\phi^{-1}+\cdots+u_{m,r_{m}}\phi^{r_{m}-1}+\cdots\medskip\\
v_{m}=v_{m,0}\phi^{-1}+\cdots+v_{m,r_{m}}\phi^{r_{m}-1}+\cdots
\end{array}
\label{re}%
\end{equation}
with $u_{m,0}v_{m,0}=2$, we find from (\ref{uv}) that the positions of
resonances are $r_{m}=-1,0,3,4$, for all $m$, $m=0,\pm1,\pm2,\ldots$. This
means that the branch (\ref{br}) is non-generic: since the resonance $-1$
corresponds to the arbitrariness of the constant in $\phi=t+\mathrm{constant}%
$, the expansions (\ref{re}) contain at most $3N+1$ arbitrary constants for a
subsystem of $2N$ equations (say, with $m=1,\ldots,N$) of the infinite system
(\ref{uv}), not $4N$ arbitrary constants as should be for the general solution
due to the Cauchy theorem. Though the branch (\ref{br}) represents only a
class of special solutions, its analysis is sufficient for reaching the
conclusion that the system (\ref{uv}) does not pass the Painlev\'{e} test for integrability.

Assuming that the solutions of (\ref{uv}) are represented in the case of
(\ref{br}) by the Laurent type expansions%
\begin{equation}
u_{m}=\sum_{i=0}^{\infty}u_{m,i}\phi^{i-1}\qquad v_{m}=\sum_{i=0}^{\infty
}v_{m,i}\phi^{i-1}\label{ex}%
\end{equation}
we find the following recursion relations for the constant coefficients
$u_{m,n}$ and $v_{m,n}$, $m=0,\pm1,\pm2,\ldots$, $n=0,1,2,\ldots$:%
\begin{equation}%
\begin{array}
[c]{l}%
\left(  n-1\right)  \left(  n-2\right)  u_{m,n}-%
{\displaystyle\sum\limits_{i=0}^{n}}
{\displaystyle\sum\limits_{j=0}^{n-i}}
u_{m,i}u_{m,j}v_{m,n-i-j}+u_{m-1,n-2}\\
\multicolumn{1}{r}{-u_{m+1,n-2}=0\medskip}\\
\left(  n-1\right)  \left(  n-2\right)  v_{m,n}-%
{\displaystyle\sum\limits_{i=0}^{n}}
{\displaystyle\sum\limits_{j=0}^{n-i}}
v_{m,i}v_{m,j}u_{m,n-i-j}+v_{m+1,n-2}\\
\multicolumn{1}{r}{-v_{m-1,n-2}=0}%
\end{array}
\label{rr}%
\end{equation}
where $u_{m,-2}=v_{m,-2}=u_{m,-1}=v_{m,-1}=0$ formally.

Now we have to check whether the recursion relations (\ref{rr}) are consistent
at the resonances or not. At $n=0$, we have $u_{m,0}=2/v_{m,0}$ with arbitrary
nonzero constants $v_{m,0}$, $m=0,\pm1,\pm2,\ldots$, and no compatibility
conditions arise. Then we find%
\begin{equation}
u_{m,1}=v_{m,1}=0\label{n1}%
\end{equation}
and%
\begin{equation}%
\begin{array}
[c]{l}%
u_{m,2}=\dfrac{2}{3v_{m-1,0}}-\dfrac{2}{3v_{m+1,0}}+\dfrac{v_{m-1,0}%
-v_{m+1,0}}{3v_{m,0}^{2}}\medskip\\
v_{m,2}=\dfrac{v_{m,0}^{2}}{6v_{m+1,0}}-\dfrac{v_{m,0}^{2}}{6v_{m-1,0}}%
+\dfrac{v_{m+1,0}-v_{m-1,0}}{3}%
\end{array}
\label{n2}%
\end{equation}
from (\ref{rr}) at $n=1$ and $n=2$, respectively; $m=0,\pm1,\pm2,\ldots$. At
the resonance $n=3$, we obtain%
\begin{equation}
v_{m,3}=-\frac{1}{2}v_{m,0}^{2}u_{m,3}\label{n3}%
\end{equation}
where $u_{m,3}$ are arbitrary constants, $m=0,\pm1,\pm2,\ldots$, and no
compatibility conditions arise again. At the highest resonance, $n=4$,
however, the recursion relations (\ref{rr}) fail to be consistent, and the
infinite set of compatibility conditions%
\begin{equation}%
\begin{array}
[c]{l}%
\dfrac{v_{m-1,0}^{2}}{v_{m,0}v_{m-2,0}}+\dfrac{2v_{m,0}}{v_{m-2,0}}%
+\dfrac{2v_{m-2,0}}{v_{m,0}}+\dfrac{v_{m,0}v_{m-2,0}}{v_{m-1,0}^{2}}\medskip\\
+\dfrac{v_{m+1,0}^{2}}{v_{m,0}v_{m+2,0}}+\dfrac{2v_{m,0}}{v_{m+2,0}}%
+\dfrac{2v_{m+2,0}}{v_{m,0}}+\dfrac{v_{m,0}v_{m+2,0}}{v_{m+1,0}^{2}}\medskip\\
-\dfrac{2v_{m,0}^{2}}{v_{m-1,0}v_{m+1,0}}-\dfrac{4v_{m-1,0}}{v_{m+1,0}}%
-\dfrac{4v_{m+1,0}}{v_{m-1,0}}-\dfrac{2v_{m-1,0}v_{m+1,0}}{v_{m,0}^{2}}=0
\end{array}
\label{cc}%
\end{equation}
arises there for the infinite set of constants $v_{m,0}$, $m=0,\pm
1,\pm2,\ldots$.

Since we obtained the nontrivial compatibility conditions (\ref{cc}), we have
to modify the singular expansions (\ref{ex}) by introducing appropriate
logarithmic terms, starting from the terms proportional to $\phi^{3}\log\phi$
\cite{RGB}. The appearance of such logarithmic terms, however, is generally
believed to be a clear symptom of nonintegrability \cite{RGB}, \cite{AC}. For
this reason, we conclude that the Leon--Manna equation (\ref{psi}) should not
be expected to be integrable.

\begin{thebibliography}{9}                                                                                                %

\bibitem {LM}Leon~J and Manna~M 1999 Multiscale analysis of discrete nonlinear
evolution equations \textit{J.~Phys.\ A: Math.\ Gen.}\ \textbf{32} 2845--69

\bibitem {LY}Levi~D and Yamilov~R 2001 On the integrability of a new discrete
nonlinear Schr\"{o}dinger equation \textit{J.~Phys.\ A: Math.\ Gen.}%
\ \textbf{34} L553--62

\bibitem {ARS}Ablowitz~M~J, Ramani~A and Segur~H 1980 A connection between
nonlinear evolution equations and ordinary differential equations of P-type.~I
\textit{J.~Math.\ Phys.}\ \textbf{21} 715--21

\bibitem {RGB}Ramani~A, Grammaticos~B and Bountis~T 1989 The Painlev\'{e}
property and singularity analysis of integrable and non-integrable systems
\textit{Phys.\ Rep.}\ \textbf{180} 159--245

\bibitem {AC}Ablowitz~M~J and Clarkson~P~A 1991 \textit{Solitons, Nonlinear
Evolution Equations and Inverse Scattering} (Cambridge: Cambridge University Press)

\end{thebibliography}

\end{document}
nlin0111060.extracted_bib
\begin{thebibliography}{9}                                                                                                %

\bibitem {LM}Leon~J and Manna~M 1999 Multiscale analysis of discrete nonlinear
evolution equations \textit{J.~Phys.\ A: Math.\ Gen.}\ \textbf{32} 2845--69

\bibitem {LY}Levi~D and Yamilov~R 2001 On the integrability of a new discrete
nonlinear Schr\"{o}dinger equation \textit{J.~Phys.\ A: Math.\ Gen.}%
\ \textbf{34} L553--62

\bibitem {ARS}Ablowitz~M~J, Ramani~A and Segur~H 1980 A connection between
nonlinear evolution equations and ordinary differential equations of P-type.~I
\textit{J.~Math.\ Phys.}\ \textbf{21} 715--21

\bibitem {RGB}Ramani~A, Grammaticos~B and Bountis~T 1989 The Painlev\'{e}
property and singularity analysis of integrable and non-integrable systems
\textit{Phys.\ Rep.}\ \textbf{180} 159--245

\bibitem {AC}Ablowitz~M~J and Clarkson~P~A 1991 \textit{Solitons, Nonlinear
Evolution Equations and Inverse Scattering} (Cambridge: Cambridge University Press)

\end{thebibliography}