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\begin{document}
\begin{titlepage}
\begin{center}
{\LARGE Multi--soliton Solutions of Two--dimensional Matrix Davey--Stewartson Equation\\

}
\vspace{1cm}
{\large A.~N.~Leznov$^a$~~and~~E.~A.~Yuzbashyan$^b$}\\
\vspace{1cm}
{\small\it$^a$Institute for High Energy Physics,\\
142284 Protvino, Moscow Region, Russia.\\
e-mail: leznov@mx.ihep.su\\
\vspace{0.5cm}
$^b$Bogoliubov Laboratory of Theoretical Physics,\\
Joint Institute for Nuclear Research,\\
141980 Dubna, Moscow Region, Russia.\\
e-mail: emil@thsun1.jinr.dubna.su\\
Tel.: +7 09621 65988  Fax: +7 09621 65084\\

}
\end{center}
\end{titlepage}

\begin{abstract}
 The explicit formulae for
m--soli\-ton solutions of (1+2)--dime\-nsi\-onal
matrix Davey--Stewartson equation
are represented. They are found by means of
known general solution of the matrix Toda chain with the fixed ends \cite{1}.
These solutions are
expressed trough $m+m$ independent solutions of a pair of linear
Shr\"odinger equations with Hermitian potentials.
\end{abstract}

\begin{quote}
{\it MSC:} 35Q51; 58G35\\
{\it PACS:} 02.30.Jr \\
{\it Keywords:} solitons; integrable systems, discrete symmetries.
\end{quote}

\section{Davey--Stewartson Equation}

 Let $u,v$ are two non--singular $s\times s$ matrix functions of
$x,y$, i.e. each matrix element is a function of
  $x,y$ coordinates of
two--dimensional space. Partial derivatives of these functions up to some
sufficient large order are assumed to exist.

 We define the matrix Davey--Stewartson equation~(DSE) as the following
partial differential equation.
\beg
iu_t+au_{xx}+bu_{yy}-2au\int dy(u^*u)_x-2b\int dx (uu^*)_y \cdot u=0\lb{dse}
\en
where $a,b$ are arbitrary real numbers and
$z^*$ is a Hermitian conjugate of a matrix $z$.
It will be convenient to deal not with the equation~(\ref{dse}),
but with the following expanded system, which we call the matrix
Davey--Stewartson system~(DSS).
\beg
\begar{r}
iu_t+a u_{xx}+b u_{yy}-2au\integ dy (vu)_x-2b\integ dx (uv)_y\cdot u=0\\\\
-iv_t+a v_{xx}+b v_{yy}-2a\integ dy (vu)_x\cdot v-2b v\integ
dx (uv)_y =0
\enar\lb{dss}
\en
Bellow for definity we had chosen  $a=b=1$. It is easy to see that the
DSE is the
system~(\ref{dss}) under additional condition $v=u^*$. This condition we
call the condition of reality.


 In the case $s=1$~(scalar case), when $u,v$ are scalar functions and the
order of the multipliers is not essential~(\ref{dse})
is the usual, well--known Davey--Stewartson equation \cite{2}. In the
scalar case soliton solutions of the DSS were obtained in \cite{2a}.


\section{Discrete substitution}
 The method we use to solve the problem is based on
the dis\-crete trans\-for\-ma\-tion investigation. Here we consider concrete
discrete transformation, which is important for our problem.

 By direct calculations can be checked that~(\ref{dss}) is
invariant with respect to the following change of unknown matrices~$u,v$.

\beg \tilde u=v^{-1},\quad \tilde v=\left[vu-(v_x v^{-1})_y\right]
v\equiv v \left[uv-(v^{-1}v_y)_x\right]\lb{ds} \en
Here $\tilde u,\tilde v$ denote  the ``new'', transformed operators.
Invariance means that matrices $\tilde u,\tilde v$ satisfy exactly the same
system~(\ref{dss}), which matrices $u,v$ satisfy. Mapping~(\ref{ds}) is an
invertible one and the ``old'' matrices $u,v$ can be expressed trough the
``new'' ones.
\beg
v=\tilde u^{-1},
\quad u=\left[\tilde u\tilde v-(\tilde u_y \tilde u^{-1})_x\right] \tilde u
\equiv \tilde u \left[\tilde v\tilde u-(\tilde u^{-1}\tilde u_x)_y\right]
\lb{is}
\en
Transformation~(\ref{ds}) can be rewritten in the form of an infinite
chain of equations in two equivalent ways as
\beg
\left((v_n)_x v_n^{-1}\right)_y=v_n v_{n-1}^{-1}-v_{n+1} v_n^{-1},\quad
u_{n+1}= v_n^{-1}\lb{ch1}
\en
or as
\beg \left(v_n^{-1} (v_n)_y\right)_x=v_{n-1}^{-1}v_n-v_n^{-1}v_{n+1}
\lb{ch2}
\en
where $(v_n,u_n)$ is a
result of n--times substitution~(\ref{ds}) applied to some initial matrices
$v_0,u_0$.  Sequences~(\ref{ch1},\ref{ch2}) with $v_{-1}^{-1}=v_N=0$ boundary
conditions we call the matrix Toda chain with  the fixed ends.

In the scalar case $s=1$  general solution of the Toda chain with
the fixed ends was found in \cite{6} for all  series of semi--simple algebras,
except $E_7,E_8$. In \cite{7} this result was reproduced in terms
of invariant root technique applicalible to all semi--simple series.

The explicit general solution of the  matrix Toda chain with the fixed ends
was found in \cite{1}. It was expressed trough $N+N$ arbitrary independent
matrix functions of a single argument~$X_r(x),Y_r(y)$ as
\beg
v_0=\sum_{r=1}^{N} X_r Y_r \lb{dT}
\en
To~(\ref{dT}) correspond the following formula for $u_N$.
\beg
u_{N}=\sum_{r=1}^{N} \tilde Y_r(x)\tilde X_r(y) \lb{iT}
\en
Matrices  $\tilde X , \tilde Y$ here are not arbitrary ones, but in some way
depend from $ X, Y$. Both these results (\ref{dT}) and (\ref{iT})
will be used in further consideration.


\section{General Strategic}

We are going to solve the DSS~(\ref{dss}) under condition of reality $u=v^*$.
Here we describe how the discrete transformation is used for that.
General idea is the following. At first we take some obvious solution of the
DSS~(\ref{dss}). It may be not a solution of the problem (reality condition
may be not satisfied). Then, by means of the discrete transformation~(\ref{ds}),
we get from that initial, obvious solution a solution which is satisfy to
the condition of reality.

For $u_0=0$ the first equation of the system (\ref{dss}) is satisfied
identialy, the second one gives
\beg
-i{v_0}_t+ {v_0}_{xx}+{v_0}_{yy}+V_1(t,x)
v_0+v_0 V_2(t,y)=0\lb{reduced}
\en
where $V_1,V_2$ are arbitrary
$s\times s$ matrix functions of their arguments (these terms arise from
the undefined inte\-grals $\int dx (uv)_y$, $\int dy (uv)_x$ in the
sys\-tem (\ref{dss})). Obviously, condition of reality is not satisfied for
this solution. But
after enough times discrete transformations (\ref{ds})
it is possible to come to the solution for which it is satisfied.
To clarify this, let us consider some solution $u,v$ for which
condition of reality is satisfied, $u=v^*$. Denoting
$u_1,v_1$ and $u_{-1},v_{-1}$ the results of direct~(\ref{ds}) and
inverse~(\ref{is}) substitutions respectively, it can be easily  proved that
$u_{-1}={v_1}^*$ and $v_{-1}=u_1^*$.  On the $m$-th step, we have:
$u_{-m}={v_m}^*$ and $v_{-m}=u_m^*$, where index $m\quad (-m)$
correspond to the result
of the m-times direct (inverse) transformation. And vice versa one can prove
that if we begin from solution $u_0=0, v_0$ and after
2m--times discrete transformation receive $u_{2m}=v^*_0, v_{2m}=0$, the
solution in the middle of the chain automatically satisfy the reality
condition, $u_{m+1}=v_{m+1}^*$.

 The system arising from equations $u_0=v_{2m}=0$ is already resolved by
the formula (\ref{dT}). So it remains to solve the equation: $u_{2m}=v^*_0$.
It leads to the following relations between $X_r, \tilde X_{\sigma[r]}$ and
$Y_r, \tilde Y_{\sigma[r]}$
\beg
X_r^*=\tilde X_{\sigma[r]}\qquad Y_r^*=Y_{\sigma[r]}\lb{cr}
\en
where $\sigma$ denotes one of the $(2m)!$ possible permutations of the
$2m$ low
indexes. To solve~(\ref{cr}) at first it is necessary to find the dependence
of $\tilde X$ and $\tilde Y$ from $X$ and $Y$ respectively.
Finally equation~(\ref{reduced}) in terms of $X_r,Y_r$ can be rewritten as
\beg
-i{X_r}_t+ {X_r}_{xx}+V_1(t,x) X_r=0\quad -i{Y_r}_t+ {Y_r}_{xx}+Y_r V_2(t,y)=0
\lb{reducedtoX}
\en
Thus to find the $m$--soliton solutions of the DSE~(\ref{dse}) it is necessary
to undertake the following steps
\begin{itemize}
\item find the dependence $\tilde X_i\left(X_1,\dots,X_{2m}\right)$ and
$\tilde Y_i\left(Y_1,\dots Y_{2m}\right)$
\item solve the system~(\ref{cr})
\item find such a dependence of matrix functions $X_r,Y_r$ from
the time argument, which will satisfy to the system~(\ref{reducedtoX}).
\end{itemize}
After this substituting $X_r,Y_r$ in~(\ref{dT}) we find $v_0$, for which
$u_{m+1}=v_{m+1}^*$ is some partial (m--soliton) solution of the
Davey--Stewartson equation~(\ref{dse}).

\section{Scalar Case}

To gain some experience firstly we consider the
scalar case $s=1$, for which much of the necessary calculation steps are
well--known and much simpler then in the general matrix case.

In this case for the mentioned above boundary conditions the following
formulae for arbitrary $k$ takes place \cite{4}.
\beg
u_k={Det_{k-1}\over Det_k}\quad v_k={Det_{k+1}\over Det_k},
\quad Det_{-1}\equiv0,\quad Det_0\equiv1\lb{Detk}
\en
where $Det_k$ is  the principle minor of dimension $k$
 of the matrix ($v^0\equiv v_0$)
$$
\pmatrix{    v^0 & v^0_x    & v^0_{xx}  &....\cr
           v^0_y & v^0_{xy} & v^0_{xxy} &.....\cr
        v^0_{yy} & v^0_{xyy} & v^0_{xxyy} &.....\cr
              ...& ..........&  ..........&.....\cr
              ...& ..........&  ..........&.....\cr}
$$
and $v^0$ is determined by~(\ref{dT}), where $X_r,Y_r$ are
arbitrary scalar
functions of their arguments. Substituting~(\ref{dT}) in the formula for
$u_{2m}$ from~(\ref{Detk}) and comparing with~(\ref{iT}) we find
\beg
\tilde X_r(x)={W_{2m-1}( X_1,X_2,...,X_{r-1},X_{r+1},...X_{2m})\over
W_{2m}(X_1,X_2,....X_{2m})}\lb{scX}
\en
Here and bellow  $W_k$ denotes a Wrosnkian of dimension~$k$ constructed from
the functions in the brackets.
\beg
W_k(g_1,\dots,g_k)\equiv
\begar{|llll|}
g_1&g_2&\ldots&g_k\\g_1'&g_2'&\ldots&g_k'\\\vdots&\vdots&\ddots&\vdots\\
g_1^{(k-1)}&g_2^{(k-1)}&\ldots&g_k^{(k-1)}\end{array}\qquad W_0\equiv1\lb{wr}
\en
Expressions for $\tilde Y_r$ can be received from~(\ref{scX}) by the simple
exchange $X\to Y$.

In the condition of reality~(\ref{cr}) we use the permutation
$\sigma[r]=2m-r+1$
To resolve~(\ref{cr}) and~(\ref{reducedtoX}) it is suitable to represent
the functions $X_r,Y_r$ in the Frobeniouse--like form
\beg
\begar{l}
X_1=\phi_1,\quad X_r=\phi_1 \integ dx \phi_2\dots\integ dx \phi_r\\\\
Y_1=\psi_1,\quad Y_r=\psi_1 \integ dx \psi_2\dots\integ dx \psi_r \\
\enar
\lb{frob}
\en
From~(\ref{scX}) we find
\beg
\tilde X_{2m}=\Bigl(\prod^{2m}_{k=1}\phi_k\Bigr)^{-1}\quad\tilde X_r=
\Bigl(\prod^{2m}_{k=1}\phi_k\Bigr)^{-1}\int
dx\phi_{2m} \dots\int dx\phi_{2m-r}\lb{tilde}
\en
After this the reality condition~(\ref{cr}) takes the following form
\beg
\phi^*_r=\phi_{2m-r+2}\quad (r=2,3,...2m), \quad
\phi_{m+1}=\phi^*_{m+1}=\Bigl(\prod_{k=1}^m \phi_k
\phi^*_k\Bigr)^{-1}\lb{crphi}
\en
And from~(\ref{reducedtoX}) we have
\beg
{\phi_r}_t=\biggl(\phi_r \Bigl(\ln\phi_r \prod_{k=1}^{r-1} \phi_k
^2\Bigr)'\biggr)'\lb{reducedtophi}
\en
The imaginary unity~$i$ here is included into the time
variable, which therefore should be treated as a pure imaginary
from this moment.
Independently can be checked that the systems
(\ref{crphi}) and (\ref{reducedtophi})
are compatible and if (\ref{reducedtophi}) is obeyed for some $\phi_r,\quad
r\le m$, for $\phi_{2m-r+2}$ it also holds. Therefore it is enough to
consider only equations with~$r\le m$ in the system~(\ref{reducedtophi}).
Now
we introduce the new unknown functions $f_r^{-1}=\phi_1\cdots\phi_r,
\quad r\le m+1$.  From~(\ref{reducedtophi}) we find
\beg
({f_r^{-1}f_{r-1}})_t=-\left({f_r^{-1}f_{r-1}} (\ln f_r f_{r-1})'\right)'
\lb{reducedtof}
\en
From~(\ref{crphi}) it follows that
$f^*_m=f_{m+1}^{-1}$. Substituting this in the~$(m+1)$-th equation of the
last system, we have
\beg
(f_m f^*_m)_t=\left(f_m f^*_m\left(\ln f_m {f^*_m}_{-1}\right)'\right)'
\lb{m-th}
\en
Equation~(\ref{m-th}) is equivalent to  one--dimensional Shr\"odinger
equation with arbitrary real potential.
\beg
{f_m}_t+f_m''=Uf_m,\quad U=U^*\label{shr}
\en
Now let us consider the $m$--th equation of the system~(\ref{reducedtof}).
\beg
({f_m^{-1}f_{m-1}})_t=-\left({f_m^{-1}f_{m-1}} (\ln f_m f_{m-1})'\right)'
\lb{m-1-th}
\en
Partially resolving it as
\beg
f_m^{-1}f_{m-1}=z',\quad (\ln f_m f_{m-1})'=-{z_t\over z'}
\en
and excluding function $f_{m-1}$, we conclude that function $zf_m$ satisfy
exactly the same equation~(\ref{shr}), which $f_m$ satisfy.
Denoting $u_i,\quad (1\le i\le m)$ $m$ independent solutions of~(\ref{shr})
we find
\beg
f_m=u_1,\quad zf_m=u_2\Longrightarrow  f_{m-1}=z'f_m=
{\begar{|ll|}u_1&u_2\\u_1'&u_2'\enar\over u_1}
\lb{1step}
\en
 In the general case, for arbitrary $i$ the following formula holds.
\beg
f_r={W_{m-r+1}\over W_{m-r}}\quad r\le m\lb{generalstep}
\en
where $W_i=W_i(u_1,\dots,u_i)$.

To prove~(\ref{generalstep}) we use the well--known Jacobi identity for
determinants. Let $T$ is some infinite in both directions matrix;
$D_n(T)$ denotes the determinant of its
$n\times n$ principle minor, $T^s$ is the matrix received from
$T$ by deleting its $s$-th column and $T_p$--by deleting its $p$-th row.
In this
notations Jacobi identity takes the form
\beg
D_n(T)D_n(T^n_n)-D_n(T^n)D_n(T_n)=D_{n+1}(T)D_{n-1}(T)\label{jacobi}
\en
From~(\ref{jacobi}) the following identity can be easily derived.
\beg
W_i\overline{W}_i'-W_i'\overline{W}_i=W_{i-1}W_{i+1}
\label{jacobiderived}
\en
where $\overline{W}_i=W_i(u_i\to u_{i+1})=W_i(u_1,u_2,\dots,u_{i-1},u_{i+1})$.

Now let us partially resolve equation~(\ref{reducedtof}) for arbitrary $r$.
\beg
f_{r-1}=z'f_r \qquad (\ln f_rf_{r-1})'=-{z_t\over z'}\lb{part}
\en
Excluding $f_{r-1}$ from the last system we find that $f_r$ and $zf_r$ are
different solutions of the same equation. And if $f_r$ is given by the
formula~(\ref{generalstep}), $zf_r$ can be determined as
$$
zf_r={\overline{W}_{m-r+1}\over W_{m-r}}
$$
After this from~(\ref{part}) with the help of
the identity~(\ref{jacobiderived}) we easily find
$$
f_{r-1}={W_{m-r+2}\over W_{m-r+1}}
$$
Thus formula~(\ref{generalstep}) is proved by induction.

Finally for functions $\phi_r$ from the definition of $f_r$ and
formulae~(\ref{crphi}) we have
\beg
\begar{ccc}
\phi_{m+1}=v_1v^*_1&\phi_r=\dis{{W_{m-r+2} W_{m-r}\over W_{m-r+1}^2}}&\\&&\\
\phi_1=\dis{{W_{m-1}\over W_m}}&\phi^*_r=\phi_{2m-r+2}&r\le m
\enar\lb{phi}
\en

The analogues expressions take place for functions $\psi_k$
\beg
\begar{ccc}
\psi_{m+1}=v_1v^*_1&\psi_r=\dis{{W_{m-r+2} W_{m-r}\over W_{m-r+1}^2}}&\\&&\\
\psi_1=\dis{{W_{m-1}\over W_m}}&\psi^*_r=\psi_{2m-r+2}&r\le m
\enar\lb{psi}
\en
In~(\ref{psi}) $W_i=W_i(v_1,\dots,v_i)$ and
$v_i\equiv v_i(y)\quad1\le i\le m$
are $m$ independent solutions of $(1+1)$--dimen\-si\-onal
linear Shr\"o\-din\-ger equa\-tion with some arbit\-rary real potential $V$.
\beg
{v_i}_t+v_i''=Vv_i\quad V=V^*\lb{shyy}
\en

\section{Matrix case}

Here we consider a general problem as it was formulated in the sections 1
and 3. We find $m$--soliton solutions of the DSE for an arbitrary
dimension of the
unknown matrix~$u$. We therefore receive matrix generalizations of all
formulae of the
previous section. It turns out that quasi--determinants of matrices with
non--commutative entries play the role of usual determinants. Conception of
a quasi--determinant was introduced recently by
Gelfand and Retarh \cite{quasi}.
We use an independent technique, more appropriate for our particular case,
but  quasi--determinants can be used as well.

With the chain~(\ref{ch1},\ref{ch2}) under mentioned above boundary conditions
we connect the following recurrent relations
\beg
R_n\equiv v_n^{-1} {v_n}_y,\quad S_n^q\equiv \sum_{k=0}^{n-1}
 (S^{q-1}_{k_y}+R_k S^{q-1}_k)\lb{recdef}
\en
with the boundary conditions $S^0_i\equiv 1$ for arbitrary $i$.
From definitions~(\ref{recdef}) and equations~(\ref{ch1},\ref{ch2}) we easily
find
\beg
S^1_n=\sum_{k=0}^n R_n\quad S_0^q=
v_0^{-1}{v_0}_{\underbrace{y\cdots y}_{\scriptstyle q}}\lb{unsderb}
\en
\beg
v_{n+1}=-v_n (S_{n+1}^1)_x=(-1)^{n+1}v_0 (S_1^1)_x(S_2^1)_x\cdots(S_{n+1}^1)_x
\lb{mdetk}
\en
For such introduced~(\ref{recdef}) matrix functions $S^q_n$ the following
relation is true.
\beg
S_n^q=\left[(S_{n-1}^1)_x\right]^{-1} ( S_{n-1}^{q+1})_x\lb{mainrec}
\en

Now let us find the dependence of $\tilde X$ from $X$. For that we use the
fact that
each matrix function $X_i$ is determined only by matrices $X_1,\dots,X_{2m}$
and therefore we can choose matrices $Y_1,\dots,Y_{2m}$ in an arbitrary way.
It is convenient to choose
\beg
Y_i={y^{i-1}\over (i-1)!}E\quad v_0=X_1+{y\over1!}X_2+\cdots+{y^{2m-1}\over
(2m-1)!}X_{2m}\lb{choose}
\en
where $E$ is a unity $s\times s$ matrix. Substituting~(\ref{choose}) in the
expression for $v_{2m-1}$ from~(\ref{mdetk}) we find
\beg
\left.v^{-1}_{2m-1}\right|_{y=0}=
-\left[X_1(T^1_0)_x(T^1_1)_x\cdots(T^1_{2m-2})_x\right]^{-1},\lb{v2m}
\en
where matrices $T^n_q$ are determined by the following relations
\beg
T_n^q=\left[(T_{n-1}^1)_x\right]^{-1}(T_{n-1}^{q+1})_x\lb{recT}
\en
with the boundary conditions
\beg
T^q_0=S_0^q|_{y=0}=X_1^{-1}X_{q+1}
\en
Expression~(\ref{v2m}) correspond to one of the functions $\tilde X_i$. And
because these functions can be enumerated in various ways we can choose
\beg
(\tilde X_1)^{-1}=X_1 (T^1_0)_x(T^1_1)_x\cdots(T^1_{2m-2})_x\equiv
F(x_1,\dots,x_{2m})\lb{X}
\en
Formula for arbitrary $i$ can be received from~(\ref{X}) by the cycler
permutation of the indexes.
\beg
\tilde X_i^{-1}=(-1)^{i-1}F(\sigma_i[x_1,\dots,x_{2m}])\lb{cyrcleX}
\en
Arbitrary multiplier can be added in the formula~(\ref{cyrcleX}). It will be
counted
in the expression for $\tilde Y_i$. We added $(-1)^{i-1}$ to do
further calculations
more convenient. After this using (\ref{cyrcleX}) and (\ref{mdetk}) we find
\beg
\tilde Y_1^{-1}=-(Q^1_{2m-2})_y\cdots(Q^1_0)_y\quad
Y_1\equiv G(Y_1,\ldots,Y_{2m})
\lb{Y}
\en
\beg
\tilde Y_i^{-1}=(-1)^iG(\sigma_i[Y_1,\ldots,Y_{2m}])\lb{cycler}
\en
where
\beg
Q^s_n= (Q^{s+1}_{n-1})_y\left[(Q^1_{n-1})_y\right]^{-1}\quad
Q^s_0=Y_{s+1} Y_1^{-1}
\en
Now as in the previous section we represent the initial functions $X_r, Y_r$
in the Frobenious--like form.
\beg
\begar{lll}
X_1=\phi_1,&X_2=\phi_1\integ dx\phi_2,&X_3=\phi_1 \integ dx \phi_2\integ
dx\phi_3,\quad\ldots\\&&\\
Y_1=\psi_1,&Y_2=\integ dx\psi_2\cdot\psi_1,&
Y_3=\integ dx\left(\integ dx\psi_3\cdot\psi_2\right)\cdot\psi_1,\quad\ldots
\enar
\lb{mfrob}
\en
After permutation of the indexes the formulae for $\tilde X_r$
will coincide with (\ref{tilde}). The only difference is that in the
matrix case
the order of the
multipliers must be taken into account.
\beg
\begar{lll}
\tilde X_1=p,&\tilde X_2=\integ dx\phi_{2m}\cdot p,&
\tilde X_3=\integ dx\left(\integ dx\phi_{2m-1}\cdot\phi_{2m}\right)\cdot p,
\quad\ldots\\&&\\
\tilde Y_1=-s,&\tilde Y_2=-s\integ dx\psi_{2m},&\tilde Y_3=-s\integ dx\psi_{2m}
\integ dx\psi_{2m-1},\quad\ldots
\enar\lb{mtilde}
\en
where $p=(\phi_1\cdots\phi_{2m})^{-1}$ and $s=(\psi_{2m}\cdots\phi_{1})^{-1}$.
The condition of reality taken in the form
$\tilde X_r=X_r^*,\quad\tilde Y_r=Y_r^*$ gives
\beg
\begar{l}
\phi^*_r=\phi_{2m-r+2}\quad 2\le r\le m
,\\\\
\phi^{-1}_{m+1}=(\phi^*_{m+1})^{-1}= (\phi_1 \phi_2\cdots\phi_m)^*
(\phi_1 \phi_2\cdots\phi_m)\\\\
\psi^*_r=\psi_{2m-r+2}\quad 2\le r\le m,\\\\
\psi^{-1}_{m+1}=(\psi^*_{m+1})^{-1}=-(\psi_m \psi_{m-1}\cdots\psi_1)^*
(\psi_m \psi_{m-1}\dots\psi_1)\\
\enar
\lb{crpsi}
\en
The fact that all functions $X_r$ are solutions of the same
equation~(\ref{reducedtoX}) leads to the following system.
\beg
-(\phi_s)_t+(2(\phi_1 \phi_2 ...\phi_{s-1})^{-1} (\phi_1 \phi_2 ...\phi_{s-1})'
\phi_s+\phi_s')'=0 \lb{reducedtomphi}
\en
Introducing new the
functions $f_r^{-1}=\phi_1\phi_2\dots\phi_r$ from~(\ref{reducedtomphi})
we find
\beg
-(f_{r-1} f_r^{-1})_t=\left[f_{r-1}'f_r^{-1}-f_{r-1}(f_r^{-1})'\right]'
\label{reducedtomf}
\en
Then from~(\ref{crpsi}) and (\ref{reducedtomf}) we conclude that $f_m$ is a
solution of $(1+1)$--dimensional linear Shr\"odinger equation with Hermitian
potential.
\beg
{f_m}_t+f_m''=Wf_m,\quad W=W^*\label{mshx}
\en
Solution of the system~(\ref{reducedtomf}) can be found by the same scheme as
in the
previous section. Matrix case does not require to
use the Jacobi identity, because recurrent definitions are used.
\beg
f_1=u_1\quad f_{m-r}=(U^1_r)'\cdots(U^1_0)'u_1\label{f}
\en
Matrix functions $U^q_n$ are determined by the following recurrent relations.
$$
U^q_n= (U^{q+1}_{n-1})'\left[(U^1_{n-1})'\right]^{-1}
$$
with the boundary conditions
$$
U^r_0=u_{r+1} u_1^{-1}
$$
where matrices $u_r$ are different solutions of the equation~(\ref{mshx}).
From this we find the formulae for $\phi_{m-r}$ for arbitrary $r$. Finally we
have
\beg
\begar{lll}
\phi_1=f_1^{-1}&\phi_{m-r}=(U^1_r)'&0\le r\le m-2\\&&\\
\phi_{m+1}=u_1u_1^*&\phi^*_r=\phi_{2m-r+2}&2\le r\le m
\enar
\lb{mphi}
\en
For $\psi_i$ we find
\beg
\begar{l}
\psi_1^{-1}=v_1(V^1_{m-2})'\cdots(V^1_0)'v_1\\\\
\psi_{m-r}=(U^1_r)'\quad0\le r\le m-2\\\\
\psi_{m+1}=-v_1v_1^*\quad\psi^*_r=\psi_{2m-r+2}\quad2\le r\le m
\enar
\lb{mpsi}
\en
where
$$
V^q_n=\left[(V^1_{n-1})'\right]^{-1}(V^{q+1}_{n-1})'\quad V^r_0=v_1^{-1}v_{r+1}
$$
and matrices $v_i(y)$ are  different solutions of
$(1+1)$--dimensional linear
Shr\"odinger equation with  arbitrary Hermitian potential.
$$
{v_i}_t+v_i''=v_iM,\quad M=M^*
$$
Now substituting (\ref{mphi},\ref{mpsi}) directly in (\ref{mfrob}) and
(\ref{dT}) and then in the formula for $v_{m+1}$ from~(\ref{mdetk}) we find
the $m$--soliton solution of the matrix DSE. We do not write down the
corresponding
expression, because it can be easily received, but is too large to
represent it here.

\section{The Simplest Example of One--Soliton\protect\\ Solution}

Substituting $m=1$ in the formulae of the last section we find
$$
v_0=X_1Y_1+X_2Y_2,\quad X_1=\phi_1,\quad X_2=\phi_1\int dx\phi_2
$$
$$
Y_1=\psi_1,\quad Y_2=\int dx\psi_2\cdot\psi_1
$$
After this we find the following expression for the one--soliton solution
of the matrix DSE.
\beg
u_1=\psi_1^{-1}\left(1+\int dx \phi_2\int dy \psi_2\right)^{-1}\phi_1^{-1}
\label{ex}
\en
Matrix functions $\phi(t,x),\psi(t,y)$ are determined by $u,v$ solutions of
the one--dimensional linear Schr\"o\-din\-ger equations.
$$
\begar{l}
\dis
\phi_1=u^{-1},\quad \phi_2=uu^*\\\\
\psi_1=v^{-1},\quad \psi_2=-v^*v\\\\
u_t+u_{xx}+uM_1(t,x)=0\\\\
v_t+v_{yy}+M_2(t,y)v=0,\quad M_{1,2}=M^*_{1,2}\\\\
\enar
$$

\section{Conclusion}
The main result of the paper is the explicit expressions for the m--soliton
solutions of the (1+2)--dimensional matrix
Davey--Stewartson equation. By means of
the corresponding formulae of the sections 4 and 5 these solutions are
expressed trough $m+m$ independent solutions of a pair of linear
(1+1)--dimensional Shr\"odinger equations.

From the group--theoretical point of view it means that we had realized
the finite--dimensional representation of the group of integrable mappings.
This viewpoint remained beyond our concrete calculations.

Note that restriction with the finite--dimensional matrices is absolutely
nonessential. We had never used this restriction and moreover the dimension
(s) was not included in any expression.


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\end{document}



