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\markright{Physica Scripta \textbf{55}, 129 (1997). }
\title{A comparison of two discrete mKdV equations}
\author{C. Chandre}
\address{Laboratoire de Physique, CNRS, Universit\'e de Bourgogne,
BP 400, F-21011 Dijon, France}
\address{e-mail: cchandre@jupiter.u-bourgogne.fr}
\maketitle
\begin{abstract}
{\em Abstract}--- We consider here two discrete versions of the modified KdV
equation. In one
case, some solitary wave solutions, B\"acklund transformations and integrals
of motion are known. In the other one, only solitary wave solutions were given,
and we supply the corresponding results for this equation. We also derive the
integrability of the second equation and give a transformation which links the
two models.
\end{abstract}
\vspace*{.2cm}
{\em Introduction}--- In a recent paper, Takeno~\cite{take} considered
two nonlinear excitation
transfer models in a $d$-dimensional version of the simple cubic lattice
with nearest neighbour interactions. These models highlight two discrete
versions of the mKdV equation
\begin{eqnarray}
\dot{\phi_n}=(\phi_{n+1}-\phi_{n-1})(1+ \phi_n ^2) ,\\
\dot{\phi_n}=(\phi_{n+1}-\phi_{n-1})(1+\phi_n )^2 .
\end{eqnarray}
The first equation which is integrable has been widely studied
(see \cite{xiao,yang,ablo}) and is known to be a discrete mKdV equation.
But, the second equation is also related to the continuous mKdV equation
(see \cite{take,homm}). Indeed, in the continuous limit (the lattice spacing
$h$ becomes zero and $x=nh$), $(1)$ and $(2)$ become
\begin{eqnarray}
&& \frac{\partial \Phi}{\partial t}=2(1+\Phi^2 )\frac{\partial \Phi}{\partial x}
+\frac{1}{3}\frac{\partial ^3 \Phi}{\partial x^3} ,\\
&&\frac{\partial \Phi}{\partial t}=2(1+\Phi )^2 \frac{\partial \Phi}{\partial x}
+\frac{1}{3}\frac{\partial ^3 \Phi}{\partial x^3} ,
\end{eqnarray}
respectively. Applying the transformation $\Phi(x,t)=\Psi(\xi,t)-1$ where
$\xi=x-2t$, $(4)$ becomes $(3)$. Therefore, the continuous versions of $(1)$
and $(2)$ are linked by a simple Galilean transformation. But, this
transformation can not be applied in the discrete case. One result of this
paper is to find the transformation which links $(1)$ and $(2)$.\\
\\
%-------------------------------------------------------------------------------
{\em Travelling wave solutions}--- For completeness, we first review
the known travelling wave solutions of $(1)$ and $(2)$ of the form
$ \phi_n(t)=\Phi(n-c t )$. Some solutions are given by
Takeno~\cite{take,homm} for both equations and Xiao~\cite{xiao} for eq.$(2)$.
Both equations have bright solitary wave solutions with a localised shape;
for $(1)$ and $(2)$ respectively we have~\cite{take}
\begin{eqnarray}
&&\phi_n=\pm \,\sinh (C) \, \mathrm{sech} [\it C(n-n_0)-\omega t],\\
&& \omega=-2\sinh C , \\
&&\phi_n= \,\sinh ^2 (C) \, \mathrm{sech} ^2 [\it C(n-n_0)-\omega t],\\
&& \omega=-\sinh 2C .
\end{eqnarray}
Besides, $(2)$ has the singular solution~\cite{take}
$$ \phi_n= \, -\sinh ^2 (C) \, \mathrm{cosech} ^2 [\it C(n-n_0)-\omega t] .$$
We notice that the solitary wave solutions of $(1)$ are closer to the
continuous case which has sech-shaped solitary wave solutions. But, $(2)$ is
interesting because it has non vanishing waves with a sech$^2$ shape.
Xiao~\cite{xiao} also gives solitary wave solutions with nonzero boundary
conditions of $(2)$ by using the real exponential approach
\begin{eqnarray*}
&&\phi_n= a_0 + (1+a_0) \,\sinh ^2 (C) \, \mathrm{sech} ^2
[\it C(n-n_0)-\omega t], \\
&&\phi_n= a_0 - (1+a_0) \,\sinh ^2 (C) \, \mathrm{cosech} ^2
[\it C(n-n_0)-\omega t] ,
\end{eqnarray*}
where $\omega=-(1+a_0) ^2 \sinh 2C$.
For $(1)$, there is no known corresponding solutions with a sech shape with
nonzero boundary conditions.\\
%-------------------------------------------
{\em Lax pairs}---
We now consider the Lax pairs for $(1)$ and $(2)$. Recall that for discrete
equation $ \partial \phi_n/\partial t=F(\phi_n,\phi_{n-1},\phi_{n+1})$,
a Lax pair is a set of two matrices $( M_n , N_n )$ which satisfy
\begin{eqnarray*}
&& V_{n+1} = M_n V_n, \\
&& \partial V_n/\partial t = N_n V_n,
\end{eqnarray*}
where $ N_n =\displaystyle \left( \begin{array}{cc}
A_n & B_n \\ C_n & D_n \end{array} \right)$ and $ V_n=
\displaystyle\left(
\begin{array}{ll} V_{n,1} \\ V_{n,2} \end{array} \right) ,$
and also, the compatibility condition
$$ \frac{\partial}{\partial t} \left( S V_n \right) = S
\frac{\partial V_n}{\partial t},$$
where $S$ is the shift operator $ S V_n = V_{n+1}$.
In this case, this condition is $
\partial M_n/\partial t= N_{n+1} M_n - M_n N_n $ and gives the
dynamical equation for $\phi_n$.
For Equation $(1)$, the Lax pair is given by Ablowitz and Segur~\cite{ablo2}:
$$M_n =\left( \begin{array}{cc} \eta & \phi_n \\ -\phi_n & 1/\eta
\end{array} \right) \indent N_n =\left( \begin{array}{cc} A_n & B_n \\
C_n & D_n \end{array} \right),$$
\begin{eqnarray*}
\mbox{ where } && A_n= \eta^2 +\phi_{n-1} \phi_n, \indent
B_n=\phi_n \eta + \phi_{n-1}/\eta, \\
&& C_n=-\phi_{n-1} \eta - \phi_n/\eta, \indent
D_n= \eta^{-2} +\phi_{n-1} \phi_n,
\end{eqnarray*}
and $\eta$ is the spectral parameter associated with the problem.
It is straightforward to show that the Lax pair for $(2)$ is
$$ M_n =\left( \begin{array}{cc} \eta & 1+\phi_n \\ -(1+\phi_n) & 0
\end{array} \right), \indent N_n =\left( \begin{array}{cc} A_n & B_n \\
C_n & D_n \end{array} \right), $$
\begin{eqnarray*}
\mbox{ where } && A_n=(1+\phi_n) (1+\phi_{n-1}) +\eta^2, \indent
B_n=\eta (1+\phi_n). \\
&& C_n=-\eta(1+\phi_{n-1} ), \indent D_n= (1+\phi_n ) (1+\phi_{n-1}).
\end{eqnarray*}
This proves the complete integrability of both differential-difference
equations.
{\em Integrals of motion}---
We now consider integrals of motion, which are easy to obtain
(see Ablowitz and Ladik~\cite{ablo}, Homma and Takeno~\cite{homm}, Scharf
and Bishop~\cite{scha}) for an infinite 1D-lattice for $(1)$
\begin{equation} N_1=\sum_{n=-\infty}^{+\infty}
\ln (1+\phi_n ^2) ,\indent N_1 '=\sum_{n=-\infty}^{+\infty}
\arctan \phi_n .\end{equation}
In a similar way for eq.$(2)$, we have
\begin{equation} N_2=\displaystyle \sum_{n=-\infty}^{+\infty}
\ln (1+\phi_n ) ,\indent N_2 '=\displaystyle \sum_{n=-\infty}^{+\infty}
\frac{\phi_n}{1+\phi_n} .\end{equation}
Other conserved quantities of $(1)$ are given by Ablowitz and
Ladik~\cite{ablo}: for instance,
\begin{eqnarray}
&& C_1 = \sum_{n=-\infty}^{+\infty} \phi_n \phi_{n-1} ,\\
&& C_2 = \sum_{n=-\infty}^{+\infty} \left\{ \phi_n \phi_{n-2}
( 1+\phi_{n-1}^2) + \phi_n^2 \phi_{n-1}^2/2 \right\}.
\end{eqnarray}
{\em B\"acklund transformations}---
The B\"acklund transformation for $(1)$ was derived by
Yang and Schmid~\cite{yang}, who give the two following Miura transformations
\begin{eqnarray}
&& \phi_n=\frac{u_{n+1}/\eta -u_n \eta}{1+u_n u_{n+1}}
= \Theta_1 (u_n,u_{n+1}) ,\\
&& \phi_{n+1} '=\displaystyle \frac{u_{n}/\eta
-u_{n+1} \eta}{1+u_n u_{n+1}} = \Theta_2 (u_n,u_{n+1}).
\end{eqnarray}
where $u_n=V_{n,1}/V_{n,2}$ is a solution of the following
differential-difference equation
\begin{eqnarray*}
\frac{\partial u_n}{\partial t} =&& \displaystyle \frac{u_{n+1} -u_n
\left( \eta^2 +\eta^{-2} \right) -u_n ^3}{1+u_n u_{n+1}}\\
&& - \frac{u_{n-1} -u_n \left( \eta^2 + \eta^{-2} \right)
-u_n ^3}{1+u_n u_{n-1}}
\end{eqnarray*}
Therefore, $ \Phi ' = (\Theta_2 \circ \Theta_1 ^{-1} ) \Phi $ defines a
B\"acklund transformation for $(1)$.
%-------------------------------------------
Applying the same method to $(2)$, we find two Miura transformations which
define a B\"acklund transformation for this equation
\begin{eqnarray}
&& \phi_n= \displaystyle -1 -\eta
\frac{v_n}{1+v_n v_{n+1}}=\tilde{\Theta_1} (v_n,v_{n+1}) ,\\
&& \phi_{n+1} '=\displaystyle -1 -\eta \frac{v_{n+1}}{1+v_n v_{n+1}} =
\tilde{\Theta_2} (v_n,v_{n+1}).
\end{eqnarray}
where $v_n=V_{n,1}/V_{n,2}$ is a solution of the following
differential-difference equation:
$$
\frac{\partial v_n}{\partial t}= \eta ^2 v_n ^2
\left( \frac{v_{n+1}}{1+v_n v_{n+1}} -\frac{v_{n-1}}{1+v_n v_{n-1}} \right)
$$
Therefore, $ \Phi ' = (\tilde{\Theta_2} \circ \tilde{\Theta_1} ^{-1} ) \Phi $
defines a B\"acklund transformation for $(2)$.\\
These B\"acklund transformations link two solutions of the same
differential-difference equation.
%-------------------------------------------
{\em Miura-type transformation}---
Finally, we derive a transformation which links the two discrete mKdV
equations under consideration, in the same way as their continuous versions
are linked by a Galilean transformation. Unfortunately, some features of $(2)$
are not similar to those of $(1)$ (for instance, the travelling wave solutions
$(5,6)$ or the integrals of motion $(7,8)$). So, the transformation which
links both discrete equations is more difficult to obtain than the continuous
case.\\
The idea is to link both equations with a discrete version of the KdV
equation by two Miura~transformations. \\
If we assume that
\begin{equation} u_n = (1+i \phi_n) (1-i \phi_{n+1}), \end{equation} $(1)$
becomes by a straightforward calculation
\begin{equation}
\dot{u_n} = u_n (u_{n+1} -u_{n-1} ) .
\end{equation}
This defines a Miura transformation $\hat{\Theta_1}$ which maps a solution of
$(13)$ into a solution of $(1)$. \\
On the other hand, if we define
\begin{equation} u_n = (1+ \phi_n) (1+ \phi_{n+1}) , \end{equation}
$(2)$ becomes $(13)$. This also defines a Miura transformation
$\hat{\Theta_2}$ which maps a solution of $(13)$ into a solution of $(2)$.
Therefore, the transformation
$\Phi ' = (\hat{\Theta_2} \circ \hat{\Theta_1} ^{-1} ) \Phi $
maps a solution of $(1)$ into a solution of $(2)$. In theory, this
transformation could be used to derive the results above, but such
calculations do not appear to be straightforward.
{\em Acknowledgements}--- I am grateful to J.C. Eilbeck
for suggesting this problem to me, and for the hospitality of the
Department of Mathematics, Heriot-Watt University, where much of this
work was carried out. The visit was supported by funds from the ENS.
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% system.
\newblock { Phys. Lett. A} {\bf 193}, 419 (1994).
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X.~Yang and R.~Schmid.
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\newblock { Phys. Lett. A} {\bf 195}, 63 (1994).
\bibitem{ablo}
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\bibitem{homm}
S.~Homma and S.~Takeno.
%\newblock Soliton scattering by an impurity on a nonlinear lattice.
\newblock { Phys. Lett. A} {\bf 169}, 355 (1992).
\bibitem{ablo2}
M.J. Ablowitz and H.~Segur.
\newblock { Solitons and the inverse scattering transform}.
\newblock (SIAM, Philadelphia, 1981).
\bibitem{scha}
R.~Scharf and A.R. Bishop.
%\newblock Properties of the nonlinear {S}chr{\"{o}}dinger equation on a
% lattice.
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\end{references}
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