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\begin{document}

\begin{titlepage}
\null
\begin{flushright}
UT-02-56
\\
hep-th/0301213
\\
January, 2003
\end{flushright}

\vskip 1.5cm
\begin{center}

  {\LARGE \bf Noncommutative Burgers Equation}

\lineskip .75em
\vskip 2cm
\normalsize

  {\large Masashi Hamanaka\footnote{
e-mail: hamanaka@hep-th.phys.s.u-tokyo.ac.jp}
and
Kouichi Toda\footnote{e-mail: kouichi@yukawa.kyoto-u.ac.jp}
}

\vskip 2cm

  {\it Department of Physics, University of Tokyo,\\
               Tokyo 113-0033, Japan}

\vskip 0.5cm

  {\it Department of Mathematical Physics,
Toyama Prefectural University,\\
%Kurokawa 5180, Kosugi, Imizu,
Toyama, 939-0398, Japan}

\vskip 1.5cm

{\bf Abstract}

\end{center}

We discuss the Burgers equation and the exact solutions
on $(1+1)$-dimensional noncommutative spaces.
We obtain the noncommutative version of the Burgers equation
which possesses the Lax representation
by the Lax-pair generating technique
and succeed in the linearization of it
by noncommutative version of the Cole-Hopf transformations.
We find the simple solutions of the linearized equation exactly
and discuss more general solutions by the Weyl transformation.
We also present the noncommutative Burgers hierarchy.
%which contains integrable evolution equations.

\end{titlepage}
\clearpage
\baselineskip 6.7mm

%\tableofcontents \newpage \setcounter{section}{-1}

\section{Introduction}

The extensions of ordinary integrable systems
to noncommutative spaces
have grown to be one of hot topics
in the recent studies 
and various aspects of the integrabilities
have been studied intensively \cite{DiMH}-\cite{CFZ}.
Noncommutative theories can be described 
as deformed theories from commutative ones.
The deformation is essentially unique and, in gauge theories,
it means physically the presence 
of the background magnetic fields \cite{NC}.
Especially noncommutative (anti-)self-dual Yang-Mills equations
have been proved to be important \cite{YM} since
the great success of noncommutative deformation \cite{NeSc}
of ADHM construction \cite{ADHM} of instantons.
They are expected to yield various
lower-dimensional noncommutative integrable equations by reductions,
which is the noncommutative version \cite{Toda, HaTo} 
of Richard Ward conjecture \cite{Ward}.

Noncommutative spaces are defined
by the noncommutativity of the coordinates:
\begin{eqnarray}
\label{nc_coord}
[x^i,x^j]=i\theta^{ij},
\end{eqnarray}
where $\theta^{ij}$ are real constants.
This relation looks like the canonical commutation
relation in quantum mechanics
and leads to ``space-space uncertainty relation.''
Hence the singularity which exists on commutative spaces
could resolve on noncommutative spaces.
This is one of the prominent features of noncommutative theories
and yields various new physical objects.

In \cite{Toda, HaTo}, we have presented a powerful method to
generate noncommutative Lax pairs and obtain various Lax pairs
on noncommutative $(2+1)$ and $(1+1)$ dimensional spaces.
Though the integrability of noncommutative Lax equations
is non-trivial unlike commutative ones, 
the generated noncommutative Lax equations 
actually contain noncommutative integrable equations
which possess infinite number of conserved quantities and so on.
The noncommutative Lax equations are expected to
have good properties in the integrable viewpoints
because the noncommutative deformations preserve
many good properties of the commutative equations.
In order to confirm it more perfectly,
it is worth investigating the various integrable aspects,
such as, the construction of multi-soliton solutions and so on.

The Burgers equation \cite{Burgers}
can be derived from the Navier-Stokes equation
and describes real phenomena, such as the turbulence and shock waves.
In this point, the Burgers equation draws much attention
amongst many integrable equations.
Moreover the Burgers equation
can be linearized by the Cole-Hopf transformation \cite{CoHo},
which shows that the equation can be solved and integrable.

At the first step of establishing NC-deformed world of integrable systems,
it is worth studying the noncommutative extension of
the Burgers equation because of both the importance
and the linearizability.

In this article, we present the noncommutative version 
of the Burgers equation
by the Lax-pair generating technique.
The generated equation has the noncommutative Lax representation
and is naively expected to be integrable.
Then by using the noncommutative Cole-Hopf transformation,
we can linearize the noncommutative Burgers equation.
This shows that the noncommutative Burgers equation is really integrable
and noncommutative Lax equations would be expected to have good properties.
We also find the condition that the noncommutative Burgers
equation should be linearized and discuss 
some solutions of the linearized equation in various ways.
Finally the Burgers hierarchy is presented
up to the third order by using the Lax-pair generating technique.
All results are brand-new.

\section{The Noncommutative Burgers Equation}

First let us construct the noncommutative version of the Burgers equation
by using the Lax-pair generating technique.
The Lax pair is the pair of operators $(L,T)$
which satisfies the Lax equation:
\begin{eqnarray}
\label{lax}
[L,T+\del_t]=0,
\end{eqnarray}
where $\del_t=\del/\del t$.
The technique is a method to
find a corresponding $T$-operator for a given $L$-operator.
This is based on the following ansatz 
for the $T$-operator
\begin{eqnarray}
\label{ansatz}
T=\del_i^n L +T^\prime.
\end{eqnarray}
Then the problem reduces to that for the $T^\prime$-operator
and becomes enough easy to solve in many cases.
This procedure is called, 
the {\it Lax-pair generating technique} \cite{ToYu}.

This technique can be applied to noncommutative cases \cite{Toda, HaTo}.
The noncommutative version of the Lax equation (\ref{lax}),
the {\it noncommutative Lax equation}, is easily defined
just by replacing the product of $L$ and $T$ with the star product.
The star product is defined by
\begin{eqnarray}
f\star g(x):=
\exp{\left(\frac{i}{2}\theta^{ij}\partial^{(x^\prime)}_i
\partial^{(x^{\prime\prime})}_j\right)}
f(x^\prime)g(x^{\prime\prime})\Big{\vert}_{x^{\prime}
=x^{\prime\prime}=x.}%\nonumber\\
%&=&f(x)g(x)+\frac{i}{2}\theta^{ij}\partial_if(x)\partial_jg(x)
%+{\cal O}(\theta^2).
\end{eqnarray}
The star product has associativity: $f\star (g\star h)=(f\star g)\star h$,
and reduces to the ordinary product 
in the {\it commutative limit} $\theta^{ij}\rightarrow 0$.
The modification of the product  makes the ordinary
coordinate ``noncommutative,''
which means : $[x^i,x^j]_\star :=x^i\star x^j-x^j\star x^i=i\theta^{ij}$.
In this article, the noncommutativity is introduced as
\begin{eqnarray}
[t,x]_\star=i\theta,~~~\theta>0.
\end{eqnarray}

Let us apply this technique to 
the noncommutative extension of the Burgers equation.
The $L$-operator of the Burgers equation is given by
\begin{eqnarray}
L=\del_x+u(t,x).
\end{eqnarray}
The ansatz for the $T$-operator is now taken as
\begin{eqnarray}
T=\del_x L+T^\prime,
\end{eqnarray}
which is the case for $n=1$ in the general ansatz (\ref{ansatz}).
The ansatz for $n=2,3,\ldots$ gives rise to
the Burgers hierarchy, which is discussed in section 4.

Then the noncommutative Lax equation becomes
\begin{eqnarray}
[\del_x+u,T^\prime]_\star= u_x \del_x+u_t+u_x \star u.
\label{tpr}
\end{eqnarray}
Here the term $u_x \del_x$ in the RHS of Eq. (\ref{tpr})
is troublesome because the Lax equation should be a
differential equation without bare derivatives $\del_i$.
Hence we have to delete this term to find an appropriate
$T^\prime$-operator so that the bare derivative term in
the LHS of Eq. (\ref{tpr})
should cancel out the troublesome term in the RHS.
In order to do this, we can take the $T^\prime$-operator as
the following form:
\begin{eqnarray}
T^\prime=A\del_x+B,
\end{eqnarray}
where $A$ and $B$ are polynomials of $u, u_x, u_t,$ etc.
Then the Lax equation becomes $f\del_x+g=0$ and
the condition $f=0$ determines some part of $A,B$ and
finally a differential equation $g=0$ is expected to be
the Burgers equation which possesses the Lax representation.

The condition $f=0$ is
\begin{eqnarray}
A_x+[u,A]_\star=u_x.    
\end{eqnarray}
The solution is $A=u$.
Then the differential equation $g=0$ becomes
\begin{eqnarray}
B_x+[u,B]_\star=u_t+u\star u_x+u_x\star u.
\end{eqnarray}
Taking the dimensions of the variables into account,
that is, $[x]=-1, [u]=1, [t]=-2$, hence, $[B]=2$,
we can take the unknown $B$ as\footnote{Here we don't consider
fractional terms such as $u_{xx}\star u^{-1}$ and so on.}
\begin{eqnarray}
B=a u_x+b u^2,
\end{eqnarray}
where $a$ and $b$ are constants.\footnote{
We note that $u\star u\equiv u^2$.}

And now finally we get the noncommutative version of the Burgers equation:
\begin{eqnarray}
\label{burgers}
u_t-a u_{xx}+(1+a-b)u_x\star u+(1-a-b)u\star u_x=0.
\end{eqnarray}
In the commutative limit, it reduces to
\begin{eqnarray}
u_t-a u_{xx}+2(1-b)u u_x=0.
\end{eqnarray}

In general, the appearance of arbitrary constants 
in the Lax equations means that the equations 
are not always integrable.
Here let us seek for the condition on the constants 
that the Burgers equations should be linearizable 
in the next section.

\section{The Noncommutative Cole-Hopf Transformation}

In commutative case, it is well known that
the Burgers equation is linearized by the
Cole-Hopf transformation
\begin{eqnarray}
\label{com_ch}
u=\fr{1}{c}\del_x \log\psi=\fr{1}{c}\fr{\psi_x}{\psi}.
\end{eqnarray}
Taking the transformation for the Burgers equation (\ref{burgers}),
we get\footnote{Here we can set the constant $C(t)$ zero  
in Eq. (\ref{com_linear}) after the linearization 
without loss of generality
because it can be absorbed 
by the scale transformation $\psi \rar \psi \exp
\left\{\pm\int^t C(t^\prime)dt^\prime\right\}$.}
\begin{eqnarray}
\label{com_linear}
\psi_t=a\psi_{xx}
-\left(a-\fr{b-1}{c}\right)\fr{\psi_x^2}{\psi}.
\end{eqnarray}
Hence we can see that only when $ac=b-1$, 
the Burgers equation
reduces to the linear equation $\psi_t=a\psi_{xx}$.\footnote{
Without loss of generality, we can take $a>0$.
Then the linear equation is just the diffusion equation 
or the heat equation
where $a$ shows the coefficient of viscosity.}
The linearizable Burgers equation becomes
\begin{eqnarray}
\label{com_lin_burgers}
u_t-a u_{xx}-2ac u u_x=0.
\end{eqnarray}
We note that the scale transformations $t\rar (1/a)t$
and  $u\rar (1/c)u$ absorb the constants $a$ and $c$
in Eqs (\ref{com_ch}) and (\ref{com_lin_burgers}), respectively.

This transformation (\ref{com_ch}) 
still works well in noncommutative case.
Then we have to treat the inverse of $\psi$ carefully.
%with respect to the ordering.
There are typically two possibilities to
define the noncommutative version of the Cole-Hopf transformation:
\begin{eqnarray}
\mbox{(i)}&& u=\dis\fr{1}{c}\psi_x\star \psi^{-1}
\label{ch1}\\
\mbox{(ii)}&& u=\dis\fr{1}{c}\psi^{-1}\star
\psi_x\label{ch2}
\end{eqnarray}

In the case (i),
we can see that when $a+b=1, c=-1$,\footnote{
Here we omit the possibility: $a=0, b=1$ 
because the noncommutative Burgers equation (\ref{burgers}) 
becomes trivial in this case.}
the noncommutative Burgers equation reduces to 
the equation: $(\del_x-\psi_x\star \psi^{-1})\star(\psi_t-a\psi_{xx})=0$.
Hence the solutions of 
the noncommutative diffusion 
equation\footnote{Here we can also put $a>0$ as in commutative case.}
\begin{eqnarray}
\label{diffusion}
\psi_t=a\psi_{xx},
\end{eqnarray}
give rise to the exact solutions of the noncommutative Burgers equation via
the noncommutative Cole-Hopf transformation (\ref{ch1}).
The naive solution of 
the noncommutative diffusion equation (\ref{diffusion}) is
\begin{eqnarray}
\label{naive_sol}
\psi(t,x)=\sum_{i=1}^k h_i e^{a k_i^2t}\star e^{\pm k_i x}
=\sum_{i=1}^k h_i e^{\fr{i}{2}a k_i^3\theta}e^{a k_i^2t\pm k_i x},
\end{eqnarray}
where $h_i, k_i$ are complex constants.
The final form in (\ref{naive_sol})
shows that the naive solution on commutative space
is deformed by $e^{\fr{i}{2}a k_i^3\theta}$ 
due to the noncommutativity.

If we want to know more general solutions,
it would be appropriate to take the Fourier transformation
under some boundary conditions.
The calculation is the same as the commutative case.

In the case (ii),
the same discussion leads us to the similar conclusion
that when $a-b=-1, c=1$,\footnote{
Here we also omit the possibility: $a=0, b=1$ 
for the same reason as in the case (i).}
the solutions of the noncommutative diffusion equation (\ref{diffusion})
yields the exact solutions of the noncommutative Burgers equation via
the noncommutative Cole-Hopf transformation (\ref{ch2}).

The region where the noncommutative Burgers equation (\ref{burgers})
can be linearized is shown in Fig. \ref{region}.

\begin{figure}[htbn]
\epsfxsize=60mm
\hspace{5cm}
\epsffile{region.eps}
\caption{The region where the noncommutative Burgers equation
can be linearized}
\label{region}
\end{figure}

It is interesting that the conditions on $a,b$
say that one part of two coefficients of $u_x\star u$ and $u \star u_x$
in the noncommutative Burgers equation (\ref{burgers}) vanishes.
(See Table 1.)

\begin{center}
Table 1: The Linearizable NC Burgers Equation\\
\begin{tabular}{|c|c|c|} \hline
~&NC Cole-Hopf transformation&NC Burgers Equation\\\hline\hline
(i)&$u=-\psi_x\star \psi^{-1}$
&$u_t-a u_{xx}+2a u_x\star u=0$\\\hline
(ii)&$u=\psi^{-1}\star\psi_x$
&$u_t-a u_{xx}-2a u\star u_x=0$\\\hline
\end{tabular}\\
\end{center}

This is consistent with the condition that
the matrix Burgers equation should be integrable \cite{OlSo},
which is reasonable because the variable $u$
in the noncommutative deformed Burgers equation can be rewritten
as the infinite-size matrix by the Weyl transformation
as we see below.

In the commutative limit, the linearizable NC Burgers equation
reduces to commutative one (\ref{com_lin_burgers})
with $c=\pm 1$.

\vspace{3mm}

There is another way to find more non-trivial solution
for the noncommutative diffusion equation (\ref{diffusion}).
Here let us move to another description
of the noncommutative theory, the {\it operator formalism},
which is equivalent to the deformed description of
the commutative theory presented in the previous discussion.
%where all the products of the fields are the star-products.
The two descriptions are connected one-to-one by the 
{\it Weyl transformation}.

Let us introduce the operator formalism
on noncommutative $(1+1)$-dimensional space briefly.
In this formalism,
we start with the noncommutativity of the spatial coordinates
(\ref{nc_coord}) and define noncommutative gauge theory considering
the coordinates as operators.
{}From now on, we denote the hat on the operators
in order to emphasize that they are operators.

Defining new variables $\ah,\ah^\dagger$ as
\begin{eqnarray}
\ah:=\fr{1}{\sqrt{2\theta}}\zh,~\ah^\dagger:=\fr{1}{\sqrt{2\theta}}\zbh,
\end{eqnarray}
where $\zh=\hat{t}+i\xh,~\zbh=\hat{t}-i\xh$,
we get the Heisenberg's commutation relation:
\begin{eqnarray}
{[\ah,\ah^\dagger]}=1.
\end{eqnarray}
Hence the spatial coordinates can be considered
as the operators acting on a Fock space $\cH$
which is spanned by the occupation number basis $\vert
n\ket:=\left\{(\ah^\dagger)^n/\sqrt{n!}\right\}\vert 0\ket,
~\ah\vert 0\ket=0$:
\begin{eqnarray}
\cH=\oplus_{n=0}^{\infty}\C\vert n\ket.
\end{eqnarray}
The fields on the noncommutative space depend on the spatial coordinates
and are also the operators
acting on the Fock space $\cH$.
They are represented by the occupation number basis as
the infinite-size matrix:
\begin{eqnarray}
\label{field_gen}
\fh(\hat{t},\xh)=\sum_{m,n=0}^{\infty}f_{mn}\vert m\ket\bra n\vert.
\end{eqnarray}
If the fields have the rotational symmetry on the plane,
namely, commute with the number operator $\nh:=\ah^\dagger
\ah\sim (\xh^1)^2+(\xh^2)^2$,
they become diagonal:
\begin{eqnarray}
\label{field_sp}
\fh(\hat{t},\xh)=\sum_{n=0}^{\infty}f_{n}\vert n\ket\bra n\vert.
\end{eqnarray}

The fields (\ref{field_gen}) in the operator formalism
correspond to the fields $f(t,x)$ in the deformation theory
one-to-one by the Weyl transformation as
\begin{eqnarray}
\label{weyl1}
\fh(\hat{t},\xh)&:=&\fr{1}{(2\pi)^2}\int d\omega dk~\widetilde{f}(\omega,k)
e^{-i(\omega\hat{t}+k\xh)},
\end{eqnarray}
where
\begin{eqnarray}
\label{weyl2}
\widetilde{f}(\omega,k)&:=&\int dt dx~f(t,x) e^{i(\omega t+k x)}.
\end{eqnarray}
The Weyl transformation preserves the rule of the products.
Moreover the differentiation and the integration
have the one-to-one correspondences.

Then the noncommutative diffusion equation (\ref{diffusion})
is transformed to
\begin{eqnarray}
[\delh_t, \psih]=[\delh_x,[\delh_x,\psih]].
\label{op_linear}
\end{eqnarray}
In the operator formalism,
the derivative of the operator is defined as
$\del_i \fh := [\delh_i,\fh]:=[-i(\theta^{-1})_{ij} \xh^j,\fh]$.
Hence
\begin{eqnarray}
\delh_t=\fr{i}{\theta}\xh
=\frac{1}{(2\theta)^{\fr{3}{2}}}(\ah-\ah^\dagger),~~~
\delh_x=-\fr{i}{\theta}\hat{t}
=-\frac{i}{(2\theta)^{\fr{3}{2}}}(\ah+\ah^\dagger).
\end{eqnarray}
Substituting the general form of $\psi$ (\ref{field_gen})
into (\ref{op_linear}), we get the following recursion relation:
\begin{eqnarray}
\label{rec_gen}
&&(2\theta)^{\fr{3}{2}}(\sqrt{m}\psi_{m-1,n}+\sqrt{n}\psi_{m,n-1}
-\sqrt{m+1}\psi_{m+1,n}-\sqrt{n+1}\psi_{m,n+1})\nn
&=&\sqrt{(m+2)(m+1)}\psi_{m+2,n}
+\sqrt{(n+2)(n+1)}\psi_{m,n+2}\nn
&&+\sqrt{m(m-1)}\psi_{m-2,n}
+\sqrt{n(n-1)}\psi_{m,n-2}\nn
&&-2\sqrt{(m+1)n}\psi_{m+1,n-1}
-2\sqrt{m(n+1)}\psi_{m-1,n+1}\nn
&&-2\sqrt{(m+1)(n+1)}\psi_{m+1,n+1}
+2(m+n+1)\psi_{m,n}
-2\sqrt{mn}\psi_{m-1,n-1}.
\end{eqnarray}
The solution could be obtained by putting some ansatz
reflecting the boundary conditions for the considered situations.

If we put the ansatz (\ref{field_sp}) on $\psih$,
the recursion relation (\ref{rec_gen}) reduces to
\begin{eqnarray}
&&(2\theta)^{\fr{3}{2}}(\sqrt{n}(\psi_{n-1}-\psi_{n})
+\sqrt{n+1}(\psi_{n}-\psi_{n+1})\nn
&=& +\sqrt{n(n-1)}(\psi_{n}-2\psi_{n-1}+\psi_{n-2})\nn
&&+\sqrt{(n+2)(n+1)}(\psi_{n+2}-2\psi_{n+1}+\psi_n)\nn
&&+2(n+1)(\psi_{n}-\psi_{n+1})
-2n(\psi_{n-1}-\psi_{n}).
\end{eqnarray}
This has a special solution: $\psi_n=\psi_0$ for all $n$
which is a trivial solution.

\section{The Noncommutative Burgers Hierarchy}

So far, we have discussed 
the noncommutative Burgers equation taking $n=1$
in the ansatz (\ref{ansatz}).
Now let us take the other ansatz for the same operator $L=\del_x+u$
\begin{eqnarray}
T_{n\scr\mbox{th-h}}=\del_x^n L +T_{n\scr\mbox{th-h}}^\prime,
\end{eqnarray}
which yields the noncommutative Burgers hierarchy as the following.

\begin{itemize}

\item For $n=1$, the noncommutative Lax equation gives
the (first-order) noncommutative Burgers equation (\ref{burgers}).

\item For $n=2$, the noncommutative Lax equation gives
the second-order noncommutative Burgers equation.

The Lax pair is given by
\begin{eqnarray}
L=\del_x+u(t,x),~~~T_{\scr\mbox{2nd-h}}=\del_x^2 L
+T_{\scr\mbox{2nd-h}}^\prime,
\end{eqnarray}
where
\begin{eqnarray}
\del_x^2 L=\del_x^3+u\del_x^2 +2u_x \del_x+u_{xx}.
\end{eqnarray}
By substituting this ansatz into the noncommutative Lax equation,
then we can take more explicit form on $T_{\scr\mbox{2nd-h}}^\prime$ as
\begin{eqnarray}
T_{\scr\mbox{2nd-h}}^\prime=A\del_x^2+B\del_x+C,
\end{eqnarray}
where $A,B$ and $C$ are polynomials of $u, u_x, u_t,$ etc.
In the similar way to the $n=1$ case,
the unknown polynomials satisfy the following differential equations
\begin{eqnarray}
&&A_x+[u,A]_\star-2u_x=0,\nn
&&B_x+[u,B]_\star-2A\star u_x-u_{xx}-2u_x\star u=0,\nn
&&C_x+[u,C]_\star-A\star u_{xx}-B\star u_x-u_t -2u_x^2-u_{xx}\star u =0,
\label{second}
\end{eqnarray}
and the solutions are:
\begin{eqnarray}
&&A=2u,~~~B=u_x+3u^2,\nn
&&C=au_{xx}+bu_x\star u +cu\star u_x+du^3,
\end{eqnarray}
where the coefficients $a,b,c$ and $d$ are constants.

Then the last equation of (\ref{second})
yields the second-order noncommutative Burgers equation:
\begin{eqnarray}
\label{2_hie}
&&u_t-au_{xxx}+(1+a-b)u_{xx}\star u+(2-a-c)u\star u_{xx}
+(b-d)u_x\star u^2 \nn
&&+(c-b-d)u\star u_x \star u +(3-c-d)u^2\star
u_x+(3-b-c)u_x^2=0.
\end{eqnarray}
The parameter family of this equation 
actually contains four integrable evolution equations
given in Theorem 3.6 in \cite{OlSo}.
However, this time,
the linearizable condition by the noncommutative 
version of Cole-Hopf transformation 
leads to the restricted situation $a=0,b=1,c=2,d=1$
where the second-order noncommutative Burgers equation (\ref{2_hie})
becomes trivial.
The result shows that the linearizable condition
is too strict for 
the second-order noncommutative Burgers equation (\ref{2_hie})
due to the nonlinear effect.

\item For $n=3$, the noncommutative Lax equation gives
the third-order noncommutative Burgers equation. 

The Lax pair is given by
\begin{eqnarray}
L=\del_x+u(t,x),~~~T_{\scr\mbox{3rd-h}}=\del_x^3 L
+T_{\scr\mbox{3rd-h}}^\prime.
\end{eqnarray}
%where
%\begin{eqnarray}
%\del_x^3 L=\del_x^4+u\del_x^3 +3u_x \del_x^2+3u_{xx} \del_x +u_{xxx}.
%\end{eqnarray}
By substituting this ansatz into the Lax equation (\ref{lax}),
then we can take more explicit form on $T_{\scr\mbox{3rd-h}}^\prime$ as
\begin{eqnarray}
T_{\scr\mbox{3rd-h}}^\prime=A\del_x^3+B\del_x^2+C\del_x+D,
\end{eqnarray}
where $A,B,C$ and $D$ are polynomials of $u, u_x, u_t,$ etc
and are determined 
in the similar way to the cases for $n=1,2$ as
%the unknown polynomials satisfy the following differential equations
%\begin{eqnarray}
%&&A_x+[u,A]_\star-3u_x=0,\\
%&&B_x+[u,B]_\star-3A\star u_x-3u_{xx}-3u_x\star u=0,\nn
%&&C_x+[u,C]_\star-3A\star u_{xx}-2B\star u_x -6u_x^2-3u_{xx}\star u
%-u_{xxx}=0,\nn
%&&D_x+[u,D]_\star-A\star u_{xxx}-B\star 
%u_{xx}-C\star u_x-u_t-3u_{xx}\star u_x- 3u_x\star
%u_{xx}=0\nonumber,
%\label{third}
%\end{eqnarray}
\begin{eqnarray}
A&=&3u,~~~B=3u_x+6u^2,\nn
C&=&u_{xx}+4u_x\star u +8u\star u_x+4u^3,\nn
D&=&au_{xxx}+bu_{xx}\star u+cu\star u_{xx}+du_x^2\nn
&&+eu_x\star u^2+fu\star u_x\star u+gu^2\star u_x+hu^4,
\end{eqnarray}
where the coefficients $a,b,\ldots,h$ are constants.
%Then the final equation of (\ref{third})
Then we can get the third-order noncommutative Burgers equation:
\begin{eqnarray}
&&u_t-au_{xxxx}+(1+a-b)u_{xxx}\star u+(3-a-c)u\star u_{xxx}\nn
&&+(4-b-d)u_{xx}\star u_x+(6-c-d)u_{x} \star u_{xx}\nn
&&+(b-e)u_{xx}\star u^2+(c-b-f)u \star u_x \star u+(6-c-g)u^2\star 
u_{xx}\nn
&&+(d-e-f)u_x^2\star u+(4-e-g)u_x\star u\star u_x
+(8-d-f-g)u\star u_x^2\nn
&&+(e-h)u_x\star u^3+(f-e-h)u\star u_x\star u^2\nn
&&+(g-f-h)u^2\star u_x\star u+(4-g-h)u^3\star u_x=0.
\end{eqnarray}
\end{itemize}

In this way, we can generate 
the higher-order noncommutative Burgers equations.
The ansatz for the $n$-th order is more explicitly given by
\begin{eqnarray}
T_{n\scr\mbox{-th}}=\del_x^n L +T_{n\scr\mbox{-th}}^\prime
=\del_x^{n+1}+\sum_{k=0}^{n}%\left(\begin{array}{c}n\\k\end{array}\right)
\fr{n!}{k!(n-k)!}(\del^k_x u )\del_x^{n-k}
+\sum_{l=0}^{n} A_l \del_x^{n-l},
\end{eqnarray}
where $A_l$ are homogeneous polynomials of $u,u_x,u_{xx}$ and so on,
whose degrees are $[A_l]=l+1$. 
%and $\left(\begin{array}{c}n\\k\end{array}\right)$ represents
%the binomial coefficient.
These unknown polynomials are determined one by one as $A_0=nu$ and so on.

\section{Conclusion and Discussion}

In this article,
we reported the noncommutative version of
the Burgers equation
by using the Lax-pair generating technique.
We found the noncommutative version of the Cole-Hopf transformation
and succeeded in linearizing of it.
The linearized equation was found 
to be the noncommutative diffusion equation
and the naive solutions were obtained.
We also discussed another representation of the
noncommutative diffusion equation via the Weyl transformation
and rewrote it as the recursion relation (\ref{rec_gen}).
Finally we gave the Burgers hierarchy explicitly 
up to the third order by using the Lax-pair generating technique.

The results show that
noncommutative Lax equations actually contain
integrable equations in the linearizable sense
and become good news for the study of 
noncommutative extension of integrable equations.
The second-order noncommutative 
Burgers equations also imply
integrable evolution equations
though the linearizable condition
is too strict.

We have mainly discussed formal aspects of the
noncommutative Burgers equation.
Of course, we have to reveal many other things,
the physical meanings, the conserved quantities and so on.
%The noncommutative Burgers equation might
%describe charged fluid in the background external fields
The discretization of the
noncommutative Burgers equation is also
considerably interesting in recent developments
of integrable systems.\footnote{The semi-discretization
of the Burgers equation is discussed in detail,
for example, by T.~Tsuchida \cite{Tsuchida}.}
The further study is worth investigating.

The success of revealing the noncommutative 
Burgers hierarchy suggests the existence of
the fruitful world in noncommutative integrable systems.
In fact, the noncommutative KdV and KP hierarchy are
also obtained by the same technique \cite{Toda},
which might be the dawn of the {\it noncommutative Sato theory}. 


\vskip7mm\noindent
{\bf Acknowledgments}
\vskip2mm

\noindent
We would like to thank the YITP at Kyoto University
during the YITP
workshops YITP-W-02-04 on ``QFT2002''
and
YITP-W-02-16 on ``Development of Superstring Theory''
where this work has been developed.
%our stay as atom-type visitors.
We are also grateful to I.~Kishimoto, T.~Tsuchida and M.~Wadati 
for useful comments.
The work of M.H. was supported in part
by the Japan Securities Scholarship Foundation (\#12-3-0403).
That of K.T. was financially supported by the Sasagawa Scientific
Research Grant (\#13-089K) from The Japan Science Society.

\begin{thebibliography}{99}
%\baselineskip 6.6mm

\bibitem{DiMH}
A.~Dimakis and F.~Muller-Hoissen,
%``Bi-differential calculi and integrable models,''
J.\ Phys.\ A {\bf 33} (2000) 957
[math-ph/9908015];
%%CITATION = MATH-PH 9908015;%%
%``Bicomplexes, integrable models, and noncommutative geometry,''
Int.\ J.\ Mod.\ Phys.\ B {\bf 14} (2000) 2455
[hep-th/0006005];
%%CITATION = HEP-TH 0006005;%%
%``Bicomplexes and integrable models,''
J.\ Phys.\ A {\bf 33} (2000) 6579
[nlin.si/0006029];
%%CITATION = NLIN-SI 0006029;%%
``A noncommutative version of the nonlinear Schroedinger equation,''
hep-th/0007015;
%%CITATION = HEP-TH 0007015;%%
``Noncommutative Korteweg-de-Vries equation,''
hep-th/0007074;
%%CITATION = HEP-TH 0007074;%%
%``Bicomplexes and Backlund transformations,''
J.\ Phys.\ A {\bf 34} (2001) 9163
[nlin.si/0104071].
%%CITATION = NLIN-SI 0104071;%%

\bibitem{KKO}
A.~Kapustin, A.~Kuznetsov and D.~Orlov,
%``Noncommutative instantons and twistor transform,''
Commun.\ Math.\ Phys.\  {\bf 221} (2001) 385
[hep-th/0002193].
%%CITATION = HEP-TH 0002193;%%

\bibitem{Takasaki}
K.~Takasaki,
%``Anti-self-dual Yang-Mills equations on noncommutative spacetime,''
J.\ Geom.\ Phys.\ {\bf 37} (2001) 291
[hep-th/0005194].
%%CITATION = HEP-TH 0005194;%%

\bibitem{GrNe}
D.~J.~Gross and N.~A.~Nekrasov,
%``Monopoles and strings in noncommutative gauge theory,''
JHEP {\bf 0007} (2000) 034
[hep-th/0005204].
%%CITATION = HEP-TH 0005204;%%

\bibitem{Nekrasov2}
N.~A.~Nekrasov,
``Noncommutative instantons revisited,''
hep-th/0010017.
%%CITATION = HEP-TH 0010017;%%

\bibitem{Legare}
M.~Legare,
``Noncommutative generalized NS and super matrix KdV systems from a
noncommutative version of (anti-)self-dual Yang-Mills equations,''
hep-th/0012077;
%%CITATION = HEP-TH 0012077;%%
%``Reduced systems of (2,2) pseudo-Euclidean...,''
J.\ Phys. A {\bf 35} (2002) 5489.

\bibitem{LPS}
O.~Lechtenfeld, A.~D.~Popov and B.~Spendig,
%``Noncommutative solitons in open N = 2 string theory,''
JHEP {\bf 0106} (2001) 011
[hep-th/0103196].
%%CITATION = HEP-TH 0103196;%%

\bibitem{CLMS}
D.~H.~Correa, G.~S.~Lozano, E.~F.~Moreno and F.~A.~Schaposnik,
%``Comments on the U(2) noncommutative instanton,''
Phys.\ Lett.\ B {\bf 515} (2001) 206
[hep-th/0105085].
%%CITATION = HEP-TH 0105085;%%

\bibitem{Paniak}
L.~D.~Paniak,
``Exact noncommutative KP and KdV multi-solitons,''
hep-th/0105185.
%%CITATION = HEP-TH 0105185;%%

\bibitem{LePo}
O.~Lechtenfeld and A.~D.~Popov,
%``Noncommutative multi-solitons in 2+1 dimensions,''
JHEP {\bf 0111} (2001) 040
[hep-th/0106213];
%%CITATION = HEP-TH 0106213;%%
%``Scattering of noncommutative solitons in 2+1 dimensions,''
Phys.\ Lett.\ B {\bf 523} (2001) 178
[hep-th/0108118];
%%CITATION = HEP-TH 0108118;%%
%``Noncommutative 't Hooft instantons,''
JHEP {\bf 0203} (2002) 040
[hep-th/0109209].
%%CITATION = HEP-TH 0109209;%%

\bibitem{Hannabuss}
K.~C.~Hannabuss,
%``Non-commutative twistor space,''
Lett.\ Math.\ Phys.\  {\bf 58} (2001) 153
[hep-th/0108228].
%%CITATION = HEP-TH 0108228;%%

\bibitem{GrPe}
M.~T.~Grisaru and S.~Penati,
``The noncommutative sine-Gordon system,''
hep-th/0112246.
%%CITATION = HEP-TH 0112246;%%

\bibitem{Bieling}
S.~Bieling,
%``Interaction of noncommutative plane waves in 2+1 dimensions,''
J.\ Phys.\ A {\bf 35} (2002) 6281
[hep-th/0203269].
%%CITATION = HEP-TH 0203269;%%

\bibitem{Wolf}
M.~Wolf,
%``Soliton antisoliton scattering configurations 
%in a noncommutative sigma  model in 2+1 dimensions,''
JHEP {\bf 0206} (2002) 055
[hep-th/0204185].
%%CITATION = HEP-TH 0204185;%%

\bibitem{Toda}
K.~Toda,
``Extensions of soliton equations to non-commutative $(2+1)$ dimensions,''
JHEP proceedings of workshop on
integrable theories, solitons and duality, Sao Paulo, Brazil, 1-6 July
2002.

\bibitem{FSIv}
F.~Franco-Sollova and T.~Ivanova,
``On noncommutative merons and instantons,''
hep-th/0209153.
%%CITATION = HEP-TH 0209153;%%

\bibitem{HLW}
Z.~Horv\'ath, O.~Lechtenfeld and M.~Wolf,
%``Noncommutative instantons via dressing and splitting approaches,''
JHEP {\bf 0212} (2002) 060
[hep-th/0211041].
%%CITATION = HEP-TH 02011041;%%

\bibitem{FIY}
K.~Furuta, T.~Inami and M.~Yamamoto,
``Topics in nonlinear sigma models in D = 3,''
hep-th/0211129.
%%CITATION = HEP-TH 0211129;%%

\bibitem{HaTo}
M.~Hamanaka and K.~Toda,
``Towards noncommutative integrable systems,''
hep-th/0211148.
%%CITATION = HEP-TH 0211148;%%

\bibitem{CCMo}
I.~Cabrera-Carnero and M.~Moriconi,
``Noncommutative integrable field theories in 2d,''
hep-th/0211193.
%%CITATION = HEP-TH 0211193;%%

\bibitem{IhUh}
M.~Ihl and S.~Uhlmann,
``Noncommutative extended waves 
and soliton-like configurations in N = 2  string theory,''
hep-th/0211263.
%%CITATION = HEP-TH 0211263;%%

\bibitem{CFZ}
T.~Curtright, D.~Fairlie and C.~K.~Zachos,
%``Integrable symplectic trilinear interaction terms for matrix
%membranes,''
Phys.\ Lett.\ B {\bf 405} (1997) 37
[hep-th/9704037];
%%CITATION = HEP-TH 9704037;%%
C.~K.~Zachos, D.~Fairlie and T.~Curtright,
``Matrix membranes and integrability,''
hep-th/9709042.
%%CITATION = HEP-TH 9709042;%%

\bibitem{NC}
M.~R.~Douglas and N.~A.~Nekrasov,
%``Noncommutative field theory,''
Rev.\ Mod.\ Phys.\  {\bf 73} (2002) 977
[hep-th/0106048];
%%CITATION = HEP-TH 0106048;%%
%\bibitem{Szabo}
R.~J.~Szabo,
%``Quantum field theory on noncommutative spaces,''
to appear in Phys.\ Rept.\
[hep-th/0109162]
%%CITATION = HEP-TH 0109162;%%
and references therein.

\bibitem{YM}
K.~Furuuchi,
``Topological charge of U(1) instantons on noncommutative $\R^4$,''
hep-th/0010006;
%%CITATION = HEP-TH 0010006;%%
N.~A.~Nekrasov,
``Trieste lectures on solitons in noncommutative gauge theories,''
hep-th/0011095;
%%CITATION = HEP-TH 0011095;%%
A.~Konechny and A.~Schwarz,
%``Introduction to M(atrix) theory and noncommutative geometry,''
Phys.\ Rept.\  {\bf 360} (2002) 353
[hep-th/0012145],
%%CITATION = HEP-TH 0012145;%%
%``Introduction to M(atrix) theory and noncommutative geometry. II,''
%Phys.\ Rept.\  {\bf 360} (2002) 353
[hep-th/0107251]
and references therein.

\bibitem{NeSc}
N.~Nekrasov and A.~Schwarz,
%``Instantons on noncommutative ${\bf R}^4$,
%and (2,0) superconformal six dimensional theory,''
Commun.\ Math.\ Phys.\ {\bf 198} (1998) 689
[hep-th/9802068].
%%CITATION = HEP-TH 9802068;%%

\bibitem{ADHM}
M.~F.~Atiyah, N.~J.~Hitchin, V.~G.~Drinfeld and Y.~I.~Manin,
%``Construction of instantons,''
Phys.\ Lett.\ A {\bf 65} (1978) 185.
%%CITATION = PHLTA,A65,185;%%

\bibitem{Ward}
R.~S.~Ward,
%``Integrable And Solvable Systems, And Relations Among Them,''
Phil.\ Trans.\ Roy.\ Soc.\ Lond.\ A {\bf 315} (1985) 451;
%%CITATION = PTRSA,A315,451;%%
%\bibitem{Ward2}
%R.~S.~Ward,
``Multidimensional integrable systems,''
Lect.\ Notes.\ Phys.\ {\bf 280} (Springer, 1986) 106;
%{\sf [ISBN/0-387-17925-9]}.
%\bibitem{Ward3}
%R.~S.~Ward,
``Integrable systems in twistor theory,''
in {\it Twistors in Mathematics and Physics}
(Cambridge UP, 1990) 246.
%{\sf [ISBN/0-521-39783-9]}.

\bibitem{Burgers}
J.~M.~Burgers,
%``,''
Adv.\ Appl.\ Math.\ {\bf 1} (1948) 171.

\bibitem{CoHo}
E. Hopf,
%``The partial differential equation $u_t+uu_x=\mu u_{xx}$,''
Comm.\ Pure Appl.\ Math.\ {\bf 3} (1950) 201;
J. D. Cole,
%``On a quasi-linear parabolic equation occuring in aerodynamics,''
Quart.\ Appl.\ Math.\ {\bf 9} (1951) 225.

\bibitem{ToYu}
K.~Toda and S-J.~Yu,
%``he investigation into the Schwarz Korteweg-de Vries equation and
% the Schwarz derivative in ($2 + 1$) dimensions,''
J.\ Math.\ Phys.\ {\bf 41} (2000) 4747;
%``The investigation into new equations in $(2+1)$ dimensions. 
%Nonlinear evolution equations and dynamical systems,''
J.\ Nonlinear Math.\ Phys.\ Suppl. {\bf 8} (2001) 272;
%``A study of the construction of equations in $(2+1)$ dimensions,''
Inverse \ Problems \ {\bf 17} (2001) 1053 and references therein.

\bibitem{OlSo}
P.~J.~Oliver and V.~V.~Sokolov,
%``Integrable evolution equation on associative algebras,''
Commun.\ Math.\ Phys.\ {\bf 193} (1998) 245.

\bibitem{Tsuchida}
T.~Tsuchida,
%``Integrable discretizations of derivative nonlinear Schroedinger equations,''
J.\ Phys.\ A {\bf 35} (2002) 7827
[nlin.si/0105053].
%%CITATION = NLIN-SI 0105053;%%

\end{thebibliography}

\end{document}





