\documentstyle{lamuphys}
%\documentclass{article}
%\usepackage{a4}
%\usepackage{lucidbrb}
%\usepackage{epsfig}
%\usepackage{amsmath}
%\usepackage{amsfonts}

%\setlength{\hoffset}{-1.5cm}
%\setlength{\voffset}{-2cm}
%\setlength{\mboxwidth}{15cm}
%\setlength{\mboxheight}{23cm}

%\renewcommand{\baselinestretch}{1.8}
%\numberwithin{equation}{section}
%\renewcommand{\theequation}{\thesection.\arabic{equation}}

\begin{document}



\title{Constrained and Rational Reductions of the KP Hierarchy}

\author{Gerard Helminck and Johan van de Leur
\thanks{JvdL is financially
supported by the Netherlands Organization for Scientific Research (NWO).}}

\institute{
Faculty of Applied Mathematics,\\
University of Twente,\\
P.O.Box 217, 7500 AE  Enschede,
The Netherlands
}
\maketitle

%\date{}

\pagestyle{headings}

\newtheorem{axiom}{Axiom}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{definition}{Definition}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{theorem}{Theorem}[section]
\newtheorem{example}{Example}[section]
\newtheorem{remark}{Remark}[section]






%  \begin{tabular}[h]{lcl}
%  1991 MSC &:& 22E65, 22E70, 35Q53, 35Q58, 58B25\\
%  Keywords &:& KP hierarchy, Constrained KP, Grassmann manifold
%  \end{tabular}

\begin{abstract}
We consider Segal and Wilson's description of the KP hierarchy on the
Hilbert--Schmidt Grassmannian. In this setting we show that the
(vector) constrained KP
hierarchy and Krichever and Dickey's rational reductions of the KP
hierarchy are the
same. To prove this, we use some results on elementary
B\"acklund--Darboux transformations.
\end{abstract}

\section{Introduction to the KP hierarchy}
The KP hierarchy consists of a tower of nonlinear differential
equations in infinitely many variables $\{ t_n|n\ge 1\}$. It is named
after the simplest nontrivial equation in this tower,
the Kadomtsev--Petviashvili equation:
\begin{equation}
\label{KP}
\frac{3}{4}\frac{\partial^2v}{\partial t_2^2}=
\frac{\partial}{\partial t_1}\Big (\frac{\partial v}{\partial t_3}
-3v\frac{\partial v}{\partial t_1}-\frac{1}{4}
\frac{\partial^3v}{\partial t_1^3}\Big ).
\end{equation}
The solutions of these equations  belong to a ring of functions $R$
that is stable  under the operators
$\partial_n=\frac{\partial}{\partial t_n}$. The form in which one
presents these equations, is the so-called Lax form. This is an
equality between operators in $\partial=\partial_1$ of a specific nature. It is
instructive to digress for a moment on this form. We start with a
simple example. To $v$ in $R$ we associate the differential operators
\begin{equation}
\label{1.1}
{\cal{L}}=\partial^2+2v\quad\mbox{and}\quad{\cal{P}}=\partial^3
+3v\partial+\frac{3}{2}\frac{\partial v}{\partial t_1}.
\end{equation}
Both ${\cal{L}}$ and ${\cal{P}}$ are regarded as operators acting on $R$ and
as such we can speak of the commutant of these operators. A direct
computation shows
\[
[{\cal{P}},{\cal{L}} ]=\frac{1}{2}\frac{\partial^3 v}{\partial t_1^3}+
6\frac{\partial v}{\partial t_1}v.
\]
In particular, $[{\cal{P}},{\cal{L}} ]$ is a zero-th  order operator in
$\partial$. Thus we see that the condition that $v$ is a solution of
the Korteweg--de Vries (KdV) equation,
\begin{equation}
\label{KdV}
\frac{\partial v}{\partial t_3}=\frac{1}{4}\frac{\partial^3 v}{\partial
t_1^3}
+3v\frac{\partial v}{\partial t_1},
\end{equation}
is equivalent to the following condition for the Schr\"odinger
operator ${\cal{L}}$:
\begin{equation}
\label{1.2}
\frac{\partial {\cal{L}}}{\partial t_3}=0\cdot\partial
+2\frac{\partial v}{\partial t_3}=[{\cal{P}},{\cal{L}} ].
\end{equation}
The relation (\ref{1.2}) is called the Lax form of the KdV equation.
Equation (\ref{1.2}) gives rise e.g. to the following questions:
\begin{enumerate}
\item[(Q1)]Is there a systematic way to come up with ${\cal{P}}$ once one
has ${\cal{L}}$?
\item[(Q2)]Are there more differential operators in $\partial$ such
that the commutator with ${\cal{L}}$ has order $0$?
\item[(Q3)]What can be done for other operators besides the
Schr\"odinger operator?
\end{enumerate}
The first two questions lead in a natural way to an extension of
$R[\partial]=\{\sum_{i=0}^n a_i\partial^i|\ a_i\in R\}$.
By adding, namely, suitable integral operators to the ring, it becomes
possible to take roots of differential operators like the
Schr\"o\-dinger operator and this framework provides the answer to
these questions.
Consider the
ring $R[\partial , \partial^{-1})$ of pseudodifferential operators
with coefficients in $R$. It consists of all expressions
\[
\sum^N_{i=-\infty} a_i \partial^i\quad , \quad a_i \in R \quad
\mbox{{}for all} \;\; i ,
\]
that are added in an obvious way and multiplied according to
\[
\partial^j\circ a \partial^i
=\sum_{k=0}^\infty {j \choose k}\partial^k(a)\partial^{i+j-k}.
\]
Each operator $P=\sum
p_j \partial^j$ decomposes as $P=P_+ + P_-$ with $P_+ =
\sum\limits_{j\geq 0} p_j \partial^j$ its differential operator part
and $P_- = \sum\limits_{j<0} p_j \partial^j$ its integral operator
part. We denote by $Res_\partial P=p_{-1}$ the {\it residue } of $P$.
For each ${\cal{L}}\in R[\partial]$ of the form
\begin{equation}
\label{1.3}
{\cal{L}}=\partial^k+\sum_{i=0}^{k-2} a_i\partial^i,
\end{equation}
one shows recursively that there is a unique operator $L\in
R[\partial,\partial^{-1})$
of the form
\begin{equation}
  \label{1.1.1}
  L=\partial + \sum_{j<0} \ell_j \partial^j\quad , \ell_j \in R\quad
  \mbox{{}for all}\;\; j<0.
\end{equation}
such that ${\cal{L}}=L^k$ and consequently one denotes $L$ also  as
${\cal{L}}^{1/k}$. Any element in $R[\partial,\partial^{-1})$ of the
form (\ref{1.1.1}) is called a {\it Lax operator}. In particular, the operator
${\cal{P}}$ equals
$({\cal{L}}^{3/2})_+$ and this answers question Q1. Since ${\cal{L}}$ and
${\cal{L}}^{1/2}$ commute, we
have for all $i\ge 1$
\begin{equation}
\label{1.5}
[({\cal{L}}^{i/2})_+,{\cal{L}}]=-[({\cal{L}}^{i/2})_-,{\cal{L}}].
\end{equation}
As the left-hand side of (\ref{1.5}) belongs to $R[\partial ]$ and
the right-hand side is of order $\le 0$, we see that all
$\{({\cal{L}}^{i/2})_+|\ i\ge 1\}$ are examples of elements in $R[\partial
]$
whose commutator with ${\cal{L}}$ is of order zero. This answers
question Q2. Hence it makes sense
to consider the system of equations
\[
\partial_n ({\cal{L}})=[({\cal{L}}^{n/2})_+,{\cal{L}}],\quad n\ge 1,
\]
which is called the KdV hierarchy. Of course, for all even $n$ this gives
$\partial_n(v)=0$. Likewise one can consider for each differential
operator ${\cal{L}}$ of the form (\ref{1.3}), the equations
\begin{equation}
\label{1.6} \partial_n ({\cal{L}})=[({\cal{L}}^{n/k})_+,{\cal{L}}],\quad n\ge
1.
\end{equation}
This system of equations is called the $k$-th Gelfand--Dickey
hierarchy and shows that the Schr\"odinger operator is not an isolated
case (this answers question Q3). It is not difficult to show
that the equations (\ref{1.6}) for ${\cal{L}}$ are equivalent to the
following ones for $L={\cal{L}}^{1/k}$
\begin{equation}
  \label{1.1.2}
  \partial_n (L) = \sum_{j<0} \partial_n (\ell_j) \partial^j =
  [(L^n)_+ , L],\quad n\geq 1.
\end{equation}
Thus we have come to a system of equations, called the
{\it Lax equations} for a Lax operator $L$ that
includes all Gelfand--Dickey hierarchies. Because of the reason
mentioned before, it is called the KP hierarchy.

Equation (\ref{1.1.2}) has at least the trivial solution
$L=\partial$ and can be seen as the compatibility equation of the
linear system
\begin{equation}
  \label{1.1.3}
  L\psi = z\psi \quad \mbox{and}\quad \partial_n (\psi) = (L^n)_+ (\psi)
\end{equation}
One needs a context in which the actions of (\ref{1.1.3}) make
sense and that allows you to derive (\ref{1.1.2}) from
(\ref{1.1.3}). {}For the trivial solution (\ref{1.1.3}) becomes
\[
\partial \psi = z\psi \quad \mbox{and}\quad \partial_n \psi = z^n \psi
\quad \mbox{{}for all}\;\; n\geq 1.
\]
Hence,
the function $\gamma(t) = \exp (\sum\limits_{i\geq 1} t_i z^i)$ is a
solution. The space $M$ of so-called {\it oscillating functions} is a
space {}for
which we can make sense of (\ref{1.1.3}) and that can
be seen as a collection of perturbations of this solution. It is
defined as
\[
M=\{ (\sum_{j\leq N} a_j z^j) e^{\sum t_i z^i} \mid a_i \in R,
\;\;\mbox{{}for all}\;\; i\}.
\]
Expressing $e^{\sum t_i z^i}$ in terms of the elementary Schur
functions, i.e.,
\[
e^{\sum t_i z^i}=\sum_{k=0}^\infty p_k(t)z^k
\]
one notices that the product
\[
(\sum_{j\leq N} a_j z^j)e^{\sum t_i z^i}=\sum_{\ell\in\mathbbz}
(\sum_{k=0}^\infty a_{\ell-k}p_k)z^\ell
\]
is still formal. To make sense of this expression as an infinite
series in $z$ and $z^{-1}$, the coefficients
$\sum_{k=0}^\infty a_{\ell-k}p_k$ for all $\ell\in\mathbbz$ have to be
well defined functions of $t=(t_n)$. In that light it is natural to
have a context such that for each element $t$
\[
\sum_{j\leq N} |a_j(t)|^2<\infty
\quad\mbox{and}\quad
\sum_{k=0}^\infty |p_k(t)|^2<\infty.
\]
Such a context has been given in \cite{SW} and will be considered in
section 2.



The space $M$ becomes a $R[\partial, \partial^{-1})$-module by the
natural extension of the actions
\[
\begin{array}[h]{lcl}
b\{(\sum_j a_j z^j) e^{\sum t_i z^i}\} &=& (\sum_j ba_j z^j) e^{\sum t_i
  z^i}\\
\partial\{ (\sum_j a_j z^j) e^{\sum t_i z^i}\} &=& (\sum_j \partial(a_j)
z^j + \sum_j a_j z^{j+1} ) e^{\sum t_i z^i} .
\end{array}
\]
It is even a free $R[\partial , \partial^{-1})$-module, since we have
\[
(\sum p_j \partial^j) e^{\sum t_i z^i} = (\sum p_j z^j )e^{\sum t_i
  z^i} .
\]

An element $\psi$ in $M$ is called  an {\it oscillating function of type}
$z^{\ell}$,
if it has the
form
\[
\psi (z) = \psi(t,z)=\{ z^{\ell} + \sum_{j<\ell} \alpha_j z^j\} e^{\sum t_i
z^i} .
\]
The fact that $M$ is a free $R[\partial, \partial^{-1})$-module,
permits you to show that each oscillating function of type $z^{\ell}$ that
satisfies (\ref{1.1.3}) gives you a solution of (\ref{1.1.2}). This
function is then called a {\it wavefunction} of the KP-hierarchy.
Notice that we can write
$\psi(z)=P(\partial)e^{\sum t_i z^i}$. It
is not difficult to see that the equations (\ref{1.1.2}) for
$L=P\partial P^{-1}$ are
equivalent to the Sato--Wilson equations for $P$:
\begin{equation}
  \label{1.3.1}
   \partial_n( P) P^{-1} = - (P
   \partial^n P^{-1})_-.
\end{equation}

On $R[\partial , \partial^{-1})$ we have an anti-algebra
morphism called {\it taking the adjoint}. The adjoint of $P=\sum p_i
\partial^i$ is given by
\[
P^* = \sum_i (-\partial)^i p_i .
\]
%
Using this, we associate to an oscillating function
(resp. wavefunction) $\psi(z)$ its {\it adjoint oscillating function}
(resp.{\it wavefunction})
$\psi^*(z)=P(\partial)^{*-1}e^{-\sum t_i z^i}$. The adjoint wavefunction
satisfies
\begin{equation}
\label{1}
L^*\psi^* = z\psi^* \quad \mbox{and}\quad \partial_n (\psi^*) = -(L^n)_+^*
(\psi^*).
\end{equation}
Moreover, an oscillating function is a wavefunction if and only if  $\psi(t,z)$
and its adjoint oscillating function $\psi^*(t,z)$ satisfy the famous
bilinear identity (see \cite{DJKM}):
\begin{equation}
\label{bil}
Res_{z=0} \psi(t,z)\psi^*(s,z)=0.
\end{equation}

\section{The Segal--Wilson Grassmannian}
An analytic approach to construct
wavefunctions of the KP-hierarchy is considered in \cite{SW}. Segal
and Wilson consider the Hilbert space
\[
H=\{ \sum_{n\in \mathbbz} a_n z^n \mid a_n \in \mathbbc ,
\sum_{n\in\mathbbz} \mid a_n\mid^2 < \infty\},
\]
with decomposition $H=H_+ \oplus H_-$, where
\[
H_+ = \{ \sum_{n\geq 0} a_n z^n \in H\} \quad \mbox{and}\quad H_- = \{
\sum_{n<0} a_n z^n \in H\}
\]
and inner product
$<\cdot \mid \cdot >$ given by
\[
< \sum_{n\in \mathbbz} a_n z^n \mid \sum_{m\in \mathbbz} b_m z^m >
  = \sum_{n\in \mathbbz} a_n \overline{b_n} .
\]
To this decomposition is associated the Grassmannian $Gr(H)$ consisting
of all closed subspaces $W$ of $H$ such that the orthogonal projection
$p_+ : W\rightarrow H_+$ is Fredholm and the orthogonal projection
$p_- : W \rightarrow H_-$ is Hilbert-Schmidt. The connected components
of $Gr(H)$ are given by
\[
Gr^{(\ell)} (H) = \left\{
 W\in Gr(H) |\;
p_+:z^{-\ell}W\rightarrow  H_+ \;\;\mbox{has index zero}
\right\}.
\]
On each of these components we have a natural action by multiplication
of the group of commuting flows
\[
\Gamma_+ = \{ \exp (\sum_{i\geq 1} t_i z^i )\mid t_i \in \mathbbc ,\
\sum \mid t_i \mid (1+\epsilon)^i < \infty \quad \mbox{{}for some}\;\;
\epsilon > 0\}.
\]
Now we take {}for $R$ the ring of meromorphic functions on $\Gamma_+$
and {}for $\partial_n$ the partial derivative w.r.t. the parameter
$t_n$ of $\Gamma_+$.
 For each $W\in Gr^{(\ell)} (H)$, let $\Gamma_+^W$ be given by
\[
\Gamma_+^W=\{\gamma(t)=\exp (\sum_{i=1}^\infty t_iz^i)\in \Gamma_+|
\ p_+:\gamma^{-1}z^{-\ell}W\to  H_+\
\mbox{is a bijection}\}.
\]
In a similar way as in \cite{SW}, one shows that $\Gamma_+^W$ is
nonempty. For $\gamma \in \Gamma_+^W$, let $P_W(t,z)$ be $z^\ell$
times the inverse
image of $1$ under the projection $p_+:\gamma^{-1}z^{-\ell}W\to
H_+$. Then  we associate to $W$ an oscillating function
$\psi_W(t,z)=P_W(t,z)e^{\sum_{i=1}^\infty t_iz^i}$
of type $z^\ell$, which has the properties that it is
defined on a dense open subset of $\Gamma_+$
and that it takes values in $W$.
Moreover, it is known that the range of
$\psi_W$ spans a dense subspace of $W$.
A crucial property of the elements $\gamma(t)$ in this dense open subset of
$\Gamma_+$ is that
\[
W\cap (z^{\ell} H_+)^{\perp} \gamma(t) = \{ 0 \} .
\]
By exploiting this property, one shows that $\psi_W$ is a wavefunction
for the KP hierarchy.
Hence, if we write $\psi_W(t,z)=
P_W(t,\partial) \cdot e^{\sum t_i z^i}$ with $P_W \in R[\partial ,
\partial^{-1})$, then $L_W = P_W \partial P_W^{-1}$ is a solution of
the KP-hierarchy. Each component of $Gr(H)$ generates in this way the
same set of solutions of the KP-hierarchy, so it would suffice, as is done
in \cite{SW}, to consider only $Gr^{(0)} (H)$. However, we need to
consider here all components.

Let $W\in Gr^{(\ell)} (H)$, then $W^\perp$ is a closed subspace of
$H$, moreover
$p_-:W^\perp\to H_-$ is a Fredholm operator of index $-\ell$ and
$p_+:W^\perp\to H_+$ is a Hilbert--Schmidt operator. Interchanging the
role of $H_+$ and $H_-$, we see that $W^\perp$ is a plane in the
adjoint Grassmannian $Gr^{*} (H)$ consisting of planes $U$ for which
$p_-$ is a Fredholm operator and $p_+$ is a Hilbert--Schmidt operator.
We will now show that the closure of the linear span  of all
$\overline{z\psi^*_W(t,z)}$ is in fact $W^\perp$. Since the
wavefunction $\psi_W$ and its adjoint wavefunction $\psi_W^*$ satisfy
the bilinear identity (\ref{bil}), one has that
\[
<\psi_W(t,z)|\overline{z\psi_W^*(s,z)}>=0.
\]
Which means that $\overline{z\psi_W^*(s,z)}\in W^\perp$ for all
relevant $s$.
Let $U$ be the closure of the linear span  of all
$\overline{z\psi^*_W(s,z)}$, then $U\in Gr^{*} (H)$ and moreover
$p_-:U\to H_-$ has index $-\ell$. Hence $U=W^\perp$ (see
\cite{HP} for more
details). The following proposition is now obvious.
\begin{proposition}
\label{p1}
let $V,\ W$ be planes in $Gr(H)$. Then $V\subset W$ (and hence
$W^\perp\subset V^\perp$) if and only if
\[
<\psi_V(t,z)|\overline{z\psi_W^*(s,z)}>=0.
\]
\end{proposition}


\section{Elementary B\"acklund--Darboux Transformations}

Let $\psi$ be a wavefunction of type $z^\ell$, not necessarily in the
Segal--Wilson setting. Like in \cite{Oe},
we define elementary B\"acklund--Darboux transformations
that produce new wavefunctions of type $z^{\ell\pm 1}$. For $q,r \in
R$, one
considers the first order differential
operators
$q\partial q^{-1}$ and $r^{-1}\partial r$. An obvious question now is:
When are  $q\partial q^{-1}\psi$ and $r^{-1}\partial^{-1} r \psi$
again wavefunctions? The answer is given by the following theorem:

\begin{theorem}
  \label{t1}
Let $\psi$ be a wavefunction of type $z^\ell$ and let $\psi^*$ be its
adjoint wave function. Then the following holds:

\noindent
(a)\
Let  $q$ and $r\in R$ satisfy
\begin{eqnarray}
\label{2}
\partial_n (q)&=&(L^n)_+(q),\\
\label{3}
\partial_n (r)&=&-(L^n)_+^*(r)\quad\mbox{for all }n=1,2,\ldots
\end{eqnarray}
Then $q\partial  q^{-1}\psi$ (resp. $r^{-1}\partial^{-1}  r\psi$) is a
wavefunction of type $z^{\ell+1}$ (resp. $z^{\ell-1}$)
and
$-q^{-1}\partial^{-1}q\psi^*$ (resp. $-r\partial r^{-1}\psi^*$)
is its adjoint wavefunction.

\noindent
(b)\
If $q\partial  q^{-1}\psi$ (resp. $r^{-1}\partial^{-1}  r\psi$)
is again a wavefunction
and  hence $-q^{-1}\partial^{-1}q\psi^*$ (resp. $-r\partial
r^{-1}\psi^*$) its adjoint wavefunction,
then one can find an up to a scalar multiple unique $\hat q$ (resp. $\hat r)\in
R$,
such that \[
q\partial  q^{-1}=\hat q\partial  \hat q^{-1}\quad (\mbox{resp. }
r^{-1}\partial r=\hat r^{-1}\partial \hat r),
\]
which satisfies the equations (\ref{2}) (resp. (\ref{3})).
\end{theorem}
The proof of this theorem will be given in a forthcoming publication,
it uses e.g. the
following identity (which can be proved using
(\ref{1.3.1}) and some elementary identities of residues of
pseudodifferential operators, see \cite{HvdL})
\begin{equation}
\label{4}
\begin{array}[lcl]{lcl}
\partial_n (q\partial q^{-1}P)(q\partial q^{-1}P)^{-1}
\ &=&-(q\partial q^{-1}P\partial^n(q\partial q^{-1}P)^{-1})_-
\\
\ &\ & +q^{-1}\partial_n (q)
-q \partial q^{-2} \partial_n (q) q
\partial^{-1} q^{-1}\\ \ &\ &
-q^{-1} (L^n)_+(q)
+
q \partial q^{-1}(L^n)_+(q)
\partial^{-1} q^{-1}.
\end{array}
\end{equation}
For $(b)$ one needs the Zakharov--Shabat equations for the Lax
operator $L$, which follow from the equations (\ref{1.1.2}).



In the Segal--Wilson setting one has the following consequence of Theorem
\ref{t1}.
\begin{corollary}
\label{c1}
Let W be a plane in $Gr(H)$, $L_W$ the corresponding Lax operator and
let $\psi_W$ and $\psi_W^*$ be
the corresponding wavefunction and its adjoint.
Then there exists a function
$q$ (resp. $r$) $\in R$, defined on  an open dense subset of
$\Gamma_+$ which satisfies (\ref{2}) (resp. (\ref{3})) for $L=L_W$,
if and only if there exists a plane $W_q$ (resp. $W_r$) in $Gr(H)$
satisfying the codimension  $1$
inclusion:
\begin{equation}
\label{6}
W_q\subset W\quad(\mbox{resp.}\quad W\subset W_r)
\end{equation}
such that the corresponding  (adjoint) wavefunctions are connected by
$\psi_{W_q}=q\partial  q^{-1}\psi_W$
and $\psi_{W_q}^*=-q^{-1}\partial^{-1}q\psi_W^*$ (resp.
$\psi_{W_r}=r^{-1}\partial^{-1}  r\psi_W$
 and $\psi_{W_r}^*=-r\partial r^{-1}\psi_W^*$).
\end{corollary}
Clearly the B\"acklund--Darboux operators of Theorem \ref{t1} are
invertible. Moreover, considering elementary properties of the residue
of pseudodifferential operators, one cane prove the following useful
lemma.
\begin{lemma}
\label{l1}
Let $q,\ r\in R$, then $q$ (resp. $r$) satisfies (\ref{2}) (resp.
(\ref{3}))
if and only if $q^{-1}$ (resp. $r^{-1}$)
satisfies
\[
\begin{array}{rcl}
\partial_n (  q^{-1})
& = &-(q\partial  q^{-1}L^nq\partial^{-1}q^{-1})_+^* (q^{-1}),\\
\partial_n (  r^{-1})
& = &\phantom{-}(r^{-1}\partial^{-1}  rL^nr^{-1}\partial r)_+(r^{-1}),\quad
\mbox{respectively}.
\end{array}
\]
\end{lemma}
We will use the above results in the proof of the main theorem of this paper.

\section{Reductions of the KP Hierarchy}

There exist  many reductions of the KP hierarchy. The most famous ones
are the ones described in section 1, viz., the ones related to
the Gelfand-Dickey hierarchies. Recall, that this is a
subsystem of the KP-hierarchy consisting of all solutions $L$ that are
the $k$-th root of a differential operator. This gives solutions
of the KP-hierarchy that do not depend on the $\{ t_{kn},\;
\mbox{with}\;\; n\geq 1\}$. Those operators satisfy the condition $L^k
= (L^k)_+$. These
$k$-th Gelfand--Dickey reductions contain the
Korteweg--de Vries (KdV) equation (\ref{KdV})
for $k=2$ and the Boussinesq equation for
$k=3$:
\[
3\partial_2^2(u)+\partial_1^4(u)+6u(\partial_1(u))^2=0.
\]

Another reduction, the so-called (vector) constrained KP hierarchies,
were introduced as natural generalizations of the Gelfand--Dickey
hierarchies. Instead of assuming that $L^k$ has no integral operator
part, one allows an $(L^k)_-$ of a very simple form, viz.,
\begin{equation}
  \label{1.2.4}
  L^k = (L^k)_+ + \sum^m_{j=1} q_j \partial^{-1} r_j,
\end{equation}
such that the following conditions on the functions $q_j$ and $r_j$
hold:
\begin{eqnarray}
  \label{1.2.2}
\partial_n (q_j)&=& (L^n)_+ (q_j),\\
  \label{1.2.3}
  \partial_n (r_j) &=&-(L^n)^*_+ (r_j) \quad \mbox{{}for all} \;\;n\geq 1.
\end{eqnarray}
Many interesting integrable hierarchies  e.g.
AKNS, Yajima--Oikawa and Melnikov
hierarchies appear amongst these
constrained  families. For $k=1$ one finds the following equations:
\begin{equation}
\label{7}
\partial_2(q_j)=\partial_1^2(q_j)+2q_j\sum_{i=1}^m q_ir_i,\quad
\partial_2(r_j)=-\partial_1^2(r_j)-2r_j\sum_{i=1}^m q_ir_i.
\end{equation}
If one moreover assumes that $q_j=r_j^*$ then (\ref{7}) becomes the
vector non-linear Schr\"odinger equation.
For $k=2$, $m=1$ one finds;
\begin{equation}
\label{8}
\partial_2(u)=\partial_1(qr),\quad
\partial_2(q)=\partial_1^2(q)+2uq,\quad
\partial_2(r)=-\partial_1^2(r)-2uq,
\end{equation}
which is the Yajima--Oikawa system. For more information on the
(vector constrained hierarchy we refer the reader to the
references at the end of this paper.

Finally we discuss the
rational reductions of the KP hierarchy introduced in \cite{K} and
\cite{D1}. In this case one assumes that the Lax operator
$L$ satisfies
\begin{equation}
\label{9}
L^k=L_2^{-1}L_1,
\end{equation}
where $L_1$ and $L_2$
are coprime differential operators  of order $k+m$ and $m$,
respectively. This generalizes the Gelfand--Dickey reduction in a
different way. It can be shown (see \cite{K}) that
in this case the Lax equations (\ref{1.1.2})
are equivalent to
\begin{eqnarray}
\label{5.4}
\partial_iL_1&=&((L_1L_2^{-1})^{i/k})_+L_1-L_1((L_2^{-1}L_1)^{i/k})_+,\\
\label{5.5}
\partial_iL_2&=&((L_1L_2^{-1})^{i/k})_+L_2-L_2((L_2^{-1}L_1)^{i/k})_+.
\end{eqnarray}
For $k=m=1$ this reduction gives the AKNS hierarchy. As the first
equations one finds (\ref{7}) with $m=1$.

It is well known
that, among the solutions coming from the Segal-Wilson Grassmannian,
the ones that satisfy the $k$-th Gelfand-Dickey hierarchy are exactly
characterized by
\begin{equation}
\label{10}
z^k W \subset W.
\end{equation}
In the next section we consider a
generalization of this condition  and we show that that condition
describes both generalizations of the Gelfand--Dickey reduction.







\section{An Extension of the Condition $z^k W\subset W$}

In this section we consider, {}for each $k$ and $m$  in
$\mathbbn=\{ 0,1,2,\ldots\}$, $k\ne 0$
subspaces $W$ in $Gr(H)$ that satisfy the {\bf  $m$-Vector
$k$-Constrained} ($mVkC$){\bf -condition}:
\begin{equation}
\label{1.2.1}
There\ is\ a\ subspace\ W'\ of\ W\ of\ codimension\ m\ such\ that\ z^k(W')
\subset W.
\end{equation}
This is a natural generalization of the condition that describes
inside $Gr(H)$ the solutions of the $k$-th Gelfand-Dickey
hierarchy. We will  associate
to each $W$, satisfying the $mVkC$-condition, $2m$ functions $\{ q_j
\mid 1\leq j\leq m\}$ and $\{ r_j \mid 1 \leq j \leq m\}$ which satisfy
the equations (\ref{1.2.4}), (\ref {1.2.2}), (\ref{1.2.3}) for $L=L_W$.
For details we refer to \cite{HvdL}.

Take any $W$ in $Gr^{(\ell)}$ that satisfies the $mVkC$-condition. It
is no restriction to assume that the $m$ occurring in (\ref{1.2.1}) is
optimal, i.e. there is a basis $\{ u_1, \ldots , u_m\}$
of the orthocomplement of $W'$ in $W$ such that
\[
(\mbox{Span}\{ z^k u_1 , \ldots , z^k u_m\})\cap W = \{0\}.
\]
It is no restriction to assume that the vectors $\{ z^k (u_1) ,
\ldots , z^k (u_m)\}$ are an orthonormal basis of the orthocomplement
of $W$ in $z^k W + W$. To the space $W$ we associate the subspaces
\[
W_j = W \oplus \mathbbc z^k u_j , 1\leq j \leq m.
\]
Clearly the $W_j$ all belong to $Gr^{(\ell -1)}(H)$ and hence, they have
wavefunctions $\psi_{W_j}$ of type $z^{\ell -1}$.
Recall that $\psi_{W_j} (t,z)$ is well-defined {}for all $t$ belonging
to the open dense subset
\[
\Gamma^{W_j}_+ = \{ \gamma (t) = \exp (\sum t_i z^i)\in
\Gamma_+ |\ p_+:\gamma^{-1}z^{-\ell+1}W_j\to  H_+\
\mbox{is a bijection} \}.
\]
On $\Gamma^{W_j}_+$ we consider the function
\begin{equation}
  \label{1.2.6}
  s_j (t) = < \psi_{W_j} (t,z) \mid z^k u_j >.
\end{equation}
Since the vectors $\{ \psi_{W_j} (t,z) \mid t\in \Gamma^{W_j}_{+}\}$
are lying dense in $W_j$ and $m$ was assumed to be optimal, the
functions $\{s_j\}$ do not vanish.
Denote by $r_j=s_j^{-1}$, then on a dense open subset of $\Gamma_+$, there is
defined the
function
\begin{equation}
  \label{1.2.7}
  \varphi_j = \frac{1}{s_j} \psi_{W_j}  = r_j \psi_{W_j} .
\end{equation}
It takes values in $W_j$ and has moreover the following useful
property
\begin{equation}
  \label{1.2.8}
  \varphi_j (t) - z^k u_j \in W ,
\end{equation}
{}for all $t$ in a dense open subset of $\Gamma_+$. This property is a
consequence of the facts that $\varphi_j (t) - z^k u_j$ is by
construction orthogonal to $z^k u_j$ and that $W$ is the
orthocomplement of $\mathbbc z^k u_j$ inside $W_j$. In \cite{Z},
similar functions $ \varphi_j$ are introduced, only not using
the geometry, but as solutions of a certain system of differential
equations. In particular, we can dispose of the condition (a) in the
Proposition  of \cite{Z}. Thus we have obtained  $m$  functions $r_j$.

We define the $ q_j$ as follows

\[
q_j (t)= <\psi_W (t,z) \mid u_j>.
\]
Because $m$ is optimal, the functions $q_j$ are non-zero on an
open dense subset of $\Gamma_+$. Since $u_j$ does not depend on $t$,
(\ref{1.1.3}) implies the equations (\ref {1.2.2}) for all $q_j$.

First we derive an expression {}for $(L^k_W)_- (\psi_W)$. Thereto we
consider
\begin{equation}
  \label{1.2.11}
  \Phi (t) = z^k \psi_W - (L^k_W)_+ (\psi_W) - \sum_{j=1}^m q_j \varphi_j .
\end{equation}
Since $\varphi_j$ takes values in $W_j$, the function $(L^k_W)_+
(\psi_W)$ does so in the
space $W$ and $z^k \psi_W$ in $z^k W$. Hence we have that $\Phi (t)$ belongs
to $W + z^k W$ {}for all relevant $t$. By construction we have that {}for
all $j, 1\leq j \leq m, \Phi (t)$ is orthogonal to $z^k u_j$, hence
$\Phi (t)$ even belongs to $W$. From the {}form of the $\varphi_j$, we
see that on an open dense set of $\Gamma_+$ one has
\[
\Phi (t) = \{ \sum_{s\geq 0} c_s z^{\ell - 1 - s}\} e^{\sum t_i z^i} .
\]
Since
$W\cap (z^{\ell} H_+)^{\perp} \gamma(t) = \{ 0 \} ,$
we arrive at
\begin{equation}
  \label{1.2.12}
  z^k \psi_W - (L^k_W)_+ (\psi_W) = \sum^m_{j=1} q_j \varphi_j .
\end{equation}
This equation is part of the system of differential equations {}for the
$\varphi_j$ as used in $[Z]$. Recall that $\varphi_j$ has the {}form
\[
\varphi_j = \{ r_j z^{\ell - 1} + \;\;\mbox{lower order terms in}\;\;
z\} e^{\sum t_i z^i}.
\]
Hence,
\[
\frac{\partial \varphi_j}{\partial x} = \frac{\partial
  \varphi_j}{\partial t_1} = \{ r_j z^{\ell} + \;\;\mbox{lower order
  terms}\; \} e^{\sum t_i z^i}.
\]
On the other hand we know that $\varphi_j (t) - z^k u_j$ belongs to
$W$ {}for all $t$. Thus also $\frac{\partial \varphi_j}{\partial x} (t)$
belongs to $W$. In $W$ we have that
\[
\frac{\partial \varphi_j}{\partial x} - r_j \psi_W = \{ \sum_{s\geq 0}
\alpha_s z^{\ell - 1 - s} \} e^{\sum t_i z^i} \in (z^{\ell}
  H_+)^{\perp} \gamma
\]
and this has to be zero. By definition we have $\varphi_j = r_j
\psi_{W_j}$ and differentiation w.r.t. $x$ gives
\begin{equation}
  \label{1.2.13}
  \psi_W = \frac{1}{r_j} \partial (r_j \psi_{W_j}) = (r^{-1}_j
  \partial r_j)(\psi_{W_j}).
\end{equation}
Consequently, we have that $\phi_j
=r_j \psi_{W_j} =
\partial^{-1} r_j \psi_{W}$.
We substitute this in equation (\ref{1.2.12}) and obtain
\begin{equation}
  \label{1.2.14}
  (L^k_W)_- (\psi_W) = \{ \sum^m_{j=1} q_j \partial^{-1} r_j \} \psi_W.
\end{equation}
Since the pseudodifferential operators act freely on wavefunctions, we
see that $L_W$ and the functions $q_j$ and $ r_j$ are exactly
connected by equation (\ref{1.2.4})
and we can conclude that $L_W$, the $q_j$ and the $r_j$ {}form
a solution of the $m$-vector $k$-constrained KP-hierarchy.

What remains to be shown are the differential equations (\ref{1.2.3})
for the $r_j$. Recall that $r_j=s_j^{-1}$, hence from (\ref{1.2.6})
and (\ref{1.2.13}) one deduces that $s_j$ satisfies
\[
\partial_n(s_j)=(L_{W_j}^n)_+s_j
=(r_j^{-1}\partial^{-1} r_jL_W^nr_j^{-1}\partial r_j)_+s_j.
\]
Now apply Lemma \ref{l1} to this equation and one obtains
the equations (\ref{1.2.3}).
\begin{remark}
Notice that,
using the terminology of \cite{Oe}, \cite{A0} and \cite{ANP3},
$\phi_j$ is the squared eigenfunction
potential of $\psi_W$ and $r_j$.
\end{remark}



Next we will show that
the $mVkC$-condition is also  connected to the rational reductions
introduced in \cite{K} and \cite{D1}.
For that purpose we assume that $W$ is a plane
in $Gr(H)$ that satisfies the $mVkC$-condition, where we choose $m$ to
be as minimal as is  possible for that plane.
Let $L_W=P_W\partial P_W^{-1}$, with
$P_W$ of the form
\begin{equation}
\label{4.0}
P_W = \partial^{\ell} + \sum_{j<\ell} p_j \partial^j = \{ 1 +
\sum_{s<0} p_{\ell + s} \partial^s \} \partial^{\ell},
\end{equation}  be the corresponding solution of the KP
hierarchy and let $W^1\subset W$ be the subspace of codimension $m$
such that $W_1=z^kW^1\subset W$. Notice first that  $W_1$ is  a
subspace of $W$ and $z^kW$ of codimension $k+m$ and $m$, respectively.
Hence using Corollary \ref{c1} iteratively, there exist
differential operators $L_1$ and $L_2$ of order $k+m$ and $m$,
respectively, such that
\begin{equation}
\label{5.1}
L_1\psi_W=\psi_{W_1},\quad L_2z^k\psi_W=\psi_{W_1}
\end{equation}
and that $\psi_{W_1}$ is again a wavefunction. From (\ref{5.1})
one immediately deduces that
\begin{equation}
\label{5.2}
L_W^k=L_2^{-1}L_1.
\end{equation}
%
Now use the following lemma:
\begin{lemma}
  \label{L5.1}
Let $L=P\partial^kP^{-1}$ be a pseudodifferential operator of order
$k$ and let $L_1$ and $L_2$ be differential operators of order $k+m$
and $m$, respectively, such that $L=L_2^{-1}L_1$. Then one has the
following identities:
\[
L_1(L_2^{-1}L_1)^{i/k}=(L_1L_2^{-1})^{i/k}L_1,\quad
L_2(L_2^{-1}L_1)^{i/k}=(L_1L_2^{-1})^{i/k}L_2.
\]
\end{lemma}
Its proof is straightforward (see e.g. \cite{HvdL}).
Since both $\psi_W$ and $\psi_{W_1}$ are wavefunctions that are
connected by equations (\ref{5.1}),
we find, using (\ref{5.2}) and Lemma \ref{L5.1},  that
\begin{equation}
\label{5.3}
L_W=(L_2^{-1}L_1)^{1/k}\quad\mbox{and}\quad
L_{W_1}=L_1(L_2^{-1}L_1)^{1/k}L_1^{-1}=(L_1L_2^{-1})^{1/k}.
\end{equation}
Hence
\[
\partial_i\psi_{W_1}=((L_1L_2^{-1})^{i/k})_+\psi_{W_1}
=((L_1L_2^{-1})^{i/k})_+L_1\psi_{W}
\]
and on the other hand it is also equal to
\[
\partial_i(L_1\psi_{W})
=\partial_i(L_1)\psi_{W}+L_1((L_2^{-1}L_1)^{i/k})_+\psi_{W}.
\]
{}From these relations one deduces exactly the equations
(\ref{5.4}). One obtains (\ref{5.5}) in a similar fashion (see \cite{HvdL}.
Hence we have obtained
the rational reductions of the KP hierarchy.
It is not difficult to see that our operators must be coprime, since
we have chosen our $m$ to be minimal.

So we have shown in this section that the
$mVkC$-condition leads both to the (vector) constrained KP hierarchy
and to the  rational reductions of KP.

\section{The Main Theorem}

We will now prove the converse of the result from the
{}foregoing section and thus come to the main theorem:


\begin{theorem}
  \label{T1.3.4}
Let $W$ be a plane in $Gr(H)$ and let $L_W$ be the corresponding
solution of the KP-hierarchy. Then {}for $m, k\in\mathbb N$,
$k\ne 0$ and $m$ optimal, the following 3 conditions are
equivalent
\begin{enumerate}
\item[(a)] The space $W$ satisfies the $mVkC$-condition:
\[There\ is\ a\ subspace\ W'\ of\ W\ of\ codimension\ m\ such\ that\ z^k(W')
\subset W.
\]
\item[(b)] There exist functions $\{q_j \mid 1\leq j \leq m\}$ and $\{
  r_j \mid 1\leq j\leq m\}$ defined on an open dense subset of
  $\Gamma_+$ such that the following conditions are fulfilled:
  \begin{enumerate}
  \item[(i)] $\partial_n (q_j) = (L_W^n )_+ (q_j)$
           \quad  {}for all $n\geq 1$,
  \item[(ii)] $\partial_n (r_j) = -(L_W^n)^*_+
              (r_j)$\quad  {}for all $n\geq 1$,
  \item[(iii)] $L_W^k = (L_W^k)_+ + \sum\limits^m_{j=1} q_j
    \partial^{-1} r_j$.
\end{enumerate}
  \item[(c)] There exist coprime differential operators $L_1$ and $L_2$
of order $k+m$ and $m$, respectively,
such that the following conditions are fulfilled:
  \begin{enumerate}
  \item[(i)] $L_W^k=L_2^{-1}L_1$
  \item[(ii)]
$\partial_iL_1=((L_1L_2^{-1})^{i/k})_+L_1-L_1((L_2^{-1}L_1)^{i/k})_+$,
  \item[(iii)]
$\partial_iL_2=((L_1L_2^{-1})^{i/k})_+L_2-L_2((L_2^{-1}L_1)^{i/k})_+$.
\end{enumerate}
\end{enumerate}
\end{theorem}
\noindent {\bf Proof.}
We have already shown that (a) implies (b) and (c).
So assume first that condition
(b) holds. Moreover assume that
$W$ in $Gr^{(\ell)}(H)$.
Recall that there is a unique pseudodifferential operator $P_W$ of
the {}form
(\ref{4.0}),
such that $\psi_W = P_W (e^{\sum t_i z^i})$. This $P_W$ satisfies
the Sato-Wilson equations
(\ref{1.3.1}).

Next we consider {}for each $j, 1\leq j\leq m$, the operators $Q_j$ and
$R_j$ defined by
\begin{equation}
  \label{1.3.2}
  Q_j := q_j \partial q_j^{-1} \quad \mbox{and}\quad R_j =
  r_j^{-1} \partial^{-1} r_j.
\end{equation}
Since all $q_j$ and $r_j$ satisfy (\ref{1.2.2}), (\ref{1.2.3}),
respectively,
$Q_j$ and $R_j$ are elementary B\"acklund--Darboux transformations.
Applying Theorem \ref{t1}, one obtains that the oscillating
functions
  \begin{equation}
    \label{1.3.6}
    \psi_{Q_j} = q_j \partial q^{-1}_j \cdot \psi_W \quad
    \mbox{and}\quad \psi_{R_j} = r_j^{-1} \partial^{-1} r_j \cdot \psi_W.
  \end{equation}
of type $z^{\ell +1}$, respectively $z^{\ell -1}$,
are wavefunctions of
planes $W_{Q_j}$ and $W_{R_j}$. Moreover we have the following
codimension 1 inclusions:
\[
W_{Q_j} \subset W \quad \mbox{and}\quad W\subset W_{R_j}.
\]
The relation (b) (iii) leads to
\[
\begin{array}[h]{lcl}
L_W^k (\psi_W) &=& z^k \psi_W\\
               &=& (L^k_W)_+ (\psi_W) + \sum\limits^m_{j=1} q_j \partial^{-1}
                   r_j \psi_W\\
               &=& (L^k_W)_+ (\psi_W) + \sum\limits_{j\atop r_j \neq 0}
                      q_j r_j \psi_{R_j}.
\end{array}
\]
Thus we see with the usual density argument that
\[
z^k W\subset W + \sum_j W_{R_j} = \sum_j W_{R_j} = \tilde{W} .
\]
Since each $W$ has codimension one in $W_{R_j}$, we see that the
codimension of $W$ in $\tilde{W}$ is $\leq m$. As one assumes in
the conditions of the Theorem that $m$ is optimal, this codimension is
exactly $m$.
Let $W_1$ be the
orthocomplement of $W$ in $\tilde{W}$ and $p_1 : H\rightarrow W_1$ the
orthogonal projection on $W_1$. Inside $W$ we consider
\[
W^1 = \{ w\in W\mid p_1 (z^k w) = 0\}.
\]
Since $\dim (W_1)= m$, we see that $W^1$ is a subspace of $W$ of
codimension $ m$ and by construction $z^k W^1 \subset W$. This
completes the proof (b)$\Rightarrow$(a).

Next assume that (c) holds. Let $\psi_1$ be the oscillating function
$L_1\psi_W$,
then by using Lemma \ref{L5.1}:
\[
(L_1L_2^{-1})^{1/k}\psi_{1}=(L_1L_2^{-1})^{1/k}L_1\psi_{W}
=L_1(L_2^{-1}L_1)^{1/k}\psi_{W}=zL_1\psi_{W}=z\psi_{1}.
\]
Now consider
\[
\begin{array}[h]{lcl}
\partial_i\psi_{1}&=&\partial_i(L_1)\psi_{W}+L_1\partial_i\psi_{W}\\
&=&(((L_1L_2^{-1})^{i/k})_+L_1-L_1((L_2^{-1}L_1)^{i/k})_+
+L_1((L_2^{-1}L_1)^{i/k})_+)\psi_{W}\\
&=&((L_1L_2^{-1})^{i/k})_+L_1\psi_{W}\\
&=&((L_1L_2^{-1})^{i/k})_+\psi_{1}.
\end{array}
\]
Hence $\psi_{1}$ is again a wavefunction of the KP hierarchy. If we
let $W_1$ be the closure of the span of the $\psi_1(t,z)$ then
$\psi_{W_1}=\psi_1$.
Since
$z^k\psi_{W}$ is also a wavefunction,
\[
L_2z^k\psi_{W}=\psi_{W_1}.
\]
Thus we see with the usual density argument that
\begin{equation}
\label{5.6}
\begin{array}[h]{lcl}
W_1&\subset &z^kW\quad\mbox{of codimension }m,\\
W_1&\subset &W\quad\mbox{of codimension }k+m.
\end{array}
\end{equation}
Hence $W^1=z^{-k}W_1$ is a subset of $W$ of codimension $m$ such that
$z^kW^1\subset W$. Since our differential operators are coprime, one
cannot find lower order operators $M_1$ and $M_2$ such that
$L_W=M_2^{-1}M_1$. Therefore there is no smaller subspace $W_1$ and no
smaller $m$ such that (\ref{5.6}) is satisfied.
\hfill{$\Box$}
\
\\

\noindent
As a consequence of this, we obtain that in the  Segal--Wilson setting,
the vector constrained KP hierarchy and rational
reduction of the KP hierarchy define the same reduction of the KP hierarchy.

\
\\

\noindent
{\Large \bf  Acknowledgements.} JvdL would like to thank the
participants of the workshop, especially
Henrik Aratyn, Maarten Bergvelt, John Harnad, Manuel Ma\~nas and Alex
Kasman for useful discussions. We both want to thank Henrik Aratyn for
sending us \cite{A0}. In this article, published in these proceedings,
he presents a different proof that the vector constrained KP and the
rational reduction's of KP describe the same hierarchies. His proof is
based on kernels of differential operators and properties of Wronskians
and is quite different from the proof given in this paper.

\begin{thebibliography}

\bibitem{}{A0}
{Aratyn (1997)}
Aratyn\, H. (1997):
The constrained KP hierarchy as ratio of differential operators.
in the proceedings of this workshop

\bibitem{}{A} {Aratyn (1995)}
Aratyn\, H. (1995):
Integrable Lax hierarchies, their symmetry
reductions and multi-matrix models.
hep-th 9503211

\bibitem{}{AFGZ}{Aratyn et al. (1997a)}
Aratyn\, H., Ferreira\, L., Gomes\, J.F., Zimerman\, A.H. (1997a):
Constrained KP models as integrable matrix hierarchies.
Journ. Math. Phys. {\bf 38},1559--  (hep-th
9509096)

\bibitem{}{AGZ}{Aratyn  et al. (1995a)}
Aratyn\, H., Gomes\, J.F., Zimerman\, A.H. (1995a):
Affine Lie algebraic origin of constrained KP hierarchies.
Journ. Math. Phys {\bf 36}, 3419--
(hep-th 9408104)

\bibitem{}{ANP}{Aratyn et al. (1997b)}
Aratyn\, H.,  Nissimov\, E., Pacheva\, S. (1997b):
Virasoro symmetry of constrained KP hierarchies.
Phys. Letters {\bf 228A}, 164--
(hep-th 9602068)

\bibitem{}{ANP2}{Aratyn et al. (1997c)}
Aratyn\, H.,  Nissimov\, E., Pacheva\, S. (1997c):
Constrained KP hierarchies: Additional symmetries,
Darboux-B\"acklund solutions and relations to multi-matrix models.
Int. J. Mod. Phys. {\bf A12}, 1265--1340.
(hep-th 9607234)

\bibitem{}{ANP3}{Aratyn et al. (1997d)}
Aratyn\, H.,  Nissimov\, E., Pacheva\, S. (1997d):
Methods of squared eigenfunction potentials in integrable hierarchies of KP
type.
solv-int 9701017

\bibitem{}{ANPZ}{Aratyn et al. (1995b)}
Aratyn\, H.,  Nissimov\, E., Pacheva\, S., Zimerman\, A.H. (1995b):
Two-matrix string model as constrained (2+1)-dimensional integrable
system.
Int.J. Mod Phys. {\bf A10}, 2537-- (hep-th 9407017)

\bibitem{}{C}{Cheng (1992)}
Cheng\, Y. (1992):
Constraints of the Kadomtsev-Petviashvili hierarchy.
Journ. Math.Phys. {\bf33}, 3747--3782

\bibitem{}{C1}{Cheng (1995)}
Cheng\, Y. (1995):
Modifying the KP, the $n^{th}$ constrained KP
hierarchies and their Hamiltonian structures.
Commun. Math. Phys. {\bf 171}, 661--682

\bibitem{}{CSZ1}{Cheng et al. (1995a)}
Cheng\, Y., Strampp\, W., Zhang\,Y-J. (1995a):
Bilinear B\"acklund transformations for the KP and $k$-constrained KP
hierarchy.
Acta Appl. Math. {\bf 41}, 341--348

\bibitem{}{CSZ}{Cheng et al. (1995b)}
Cheng\, Y., Strampp\, W., Zhang\,B. (1995ab):
Constraints of theKP hierarchy and multilinear forms.
Commun. Math. Phys.
{\bf 168}, 117--135

\bibitem{}{CZ}{Cheng and Zhang (1994)}
Cheng\, Y., Zhang\,Y-J. (1994):
Bilinear equations for the constrained KP hierarchy.
Inverse Problems  {\bf 10}. L11--L17

\bibitem{}{DJKM}{Date et al. (1983)}
Date\, E., Jimbo\, M., Kashiwara \, M., Miwa\, T. (1983):
Transformation groups for soliton equations.
in: {\it Nonlinear integral systems -- classical theory and quantum
theory} eds. Jimbo\, M. and Miwa\, T., World Scientific, 39--120

\bibitem{}{D}{Dickey (1995a)}
Dickey\, L.A. (1995a):
On the constrained KP.
Letters Math. Phys {\bf 34}, 379--384

\bibitem{}{D1}{Dickey (1995b)}
Dickey\, L.A. (1995b):
On the constrained KP hierarchy II.
Letters Math. Phys {\bf 35}, 229--236

\bibitem{}{DS}{Dickey and Strampp (1996)}
Dickey\, L., Strampp\, W. (1996):
On new identities for KP Baker
functions and their application to constrained hierarchies,
preprint.

\bibitem{}{HvdL}{Helminck and van de Leur (1997)}
Helminck\, G.F., Leur\, J.W.  van de (1997):
An analytic description of the vector constrained KP hierarchy.
preprint solv-int 9706004

\bibitem{}{HP}
{Helminck and Post (1988)}
Helminck\, G.F., Post\, G.F. (1988):
Geometrical interpretation of the bilinear equations for the KP
hierarchy.
Letters Math. Phys. {\bf 16}, 359--364

\bibitem{}{K}{Krichever (1995)}
Krichever\, I. (1995):
General rational reductions
of the KP hierarchy and their symmetries.
Funct. Anal. Appl. {\bf 29}, 75--80

\bibitem{}{L}{van de Leur (1996a)}
Leur\, J. van de (1996a):
A geometrical interpretation of the constrained KP hierarchy.
preprint

\bibitem{}{L1}{van de Leur (1996b)}
Leur\, J. van de (1996b):
The vector
constrained KP hierarchy and Sato's Grassmannian. preprint to appear
in Journ. Geom. Phys. (q-alg 9609001)

\bibitem{}{Li}{Liu (1996)}
Liu\, Q.P. (1996)
Bi-hamiltonian structures of coupled AKNS
hierarchy and coupled Yajima-Oikawa hierarchy.
Journ. Math. Phys.,
2307--2314

\bibitem{}{LW}{Loris and Willox (1996)}
Loris\, I., Willox\, R. (1996):
Bilinear form and solutions of
the $k$--constrained Kadomtsev--Petviashvili hierarchy.
preprint.

\bibitem{}{LW1} {Loris and Willox (1997)}
Loris\, I., Willox\, R. (1997):
On solutions of constrained
KP equations.
Journ. Math. Phys.{\bf 38},
283--291

\bibitem{}{MR}{Mas and Ramos (1995)}
Mas\, J., Ramos\, E. (1995):
The constrained KP hierarchy
and the generalised Miura transformation.
q-alg 9501009

\bibitem{}{Oe}{Oevel and Schief (1993)}
Oevel\, W., Schief\, W. (1993):
Darboux theorems and the KP hierarchy.
in: {\it applications of analytic and geometric Methods in
Differential Equations} (Proceedings of the NATO Advanced Research
Workshop, Exeter, 14--17 July 1992, UK) P.A. Clarkson (ed.), Kluwer
Publ., 193--206

\bibitem{}{OS}{Oevel and Strampp (1993)}
Oevel\, W. Strampp\, W. (1993):
Constrained KP hierarchies
and bi--hamiltonian structures.
Commun. Math. Phys. {\bf 157}, 51--81

\bibitem{}{OS1}{Oevel and Strampp (1995)}
Oevel\, W. Strampp\, W. (1995):
Wronskian solutions of the
constrained Kadomtsev-Petviashvili hierarchy.
Journ. Math. Phys. {\bf 37}, 6213--6219

\bibitem{}{Or}{Orlov (19  )}
Orlov\,A.Yu. (19  ):
Symmetries for unifying different soliton
systems into a single integrable hierarchy.
preprint IINS/Oce04/03

\bibitem{}{Or1}{Orlov (19  )}
Orlov\, A.Yu. (1991):
Volterra operator algebra Zero curvature
representation. Universality of KP.
in: {\it Nonlinear Processes in
physics}, proceeding of the III Potsdam-V Kiev Workshop at Clarkson
Univ., Potsdam , N.Y., USA, eds. A.S. Fokas, D.J. Kaup, A.C. Newell and
V.E. Zakharov, Springer series in Nonlinear Dynamics, Springer Verlag,
Berlin, 126--131

\bibitem{}{S}{Sato (1981)}
Sato\, M. (1981):
Soliton equations as dynamical systems
on infinite dimensional Grassmann manifolds.
Res. Inst. Math.
Sci. Kokyuroku {\bf 439} 30--46

\bibitem{}{SS}{Sidorenko and Strampp (1993)}
Sidorenko\, J., Strampp\, W. (1993):
Multicomponent
integrable reductions in the Kadomtsev--Petviashvilli hierarchy.
Journ. Math Phys.{\bf 34}, 1429--1446

\bibitem{}{SW}{Segal and Wilson (1985)}
Segal\, G., Wilson\, G. (1985):
Loop groups and equations of
KdV type.
Publ. Math IHES {\bf 63}, 1--64

\bibitem{}{Z}{Zhang (1996)}
Zhang\, Y-J. (1996):
On Segal-Wilson's construction for the
$\tau$-functions of the constrained KP hierarchies.
Letters Math. Physics {\bf 36}, 1--15

\bibitem{}{ZC}{Zhang and Cheng (1994)}
Zhang\, Y.-J, Cheng\, Y. (1994):
Solutions for the vector $k$--constrained KP hierarchy.
Journ. Math. Phys. {\bf 35}, 5869--5884
\end{thebibliography}
\end{document}



% Local Variables:
% mode: latex
% TeX-master: t
% End:


