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\begin{document}
\title{On a Theorem of Halphen and its Application \newline to Integrable
Systems}
\thanks{Based upon work supported by the
US National Science Foundation under Grant No.~DMS-9970299. \\
\it{J. Math. Anal. Appl. {\bf 251}, 504--526 (2000).}}
\author{F.~Gesztesy}
\address{Department of Mathematics,
University of Missouri, Columbia, MO 65211, USA}
\email{fritz@math.missouri.edu}
\urladdr{http://www.math.missouri.edu/people/fgesztesy.html}
\author{K.~Unterkofler}
\address{Department of Computer Science,
Applied Mathematics Group,
FH-Vorarlberg,
A--6850 Dornbirn,
Austria}
\email{karl.unterkofler@fh-vorarlberg.ac.at}
\author{R.~Weikard}
\address{Department of Mathematics,
University of Alabama at Birmingham, \\
Birmingham, AL 35294--1170, USA}
\email{rudi@math.uab.edu}
\urladdr{http://www.math.uab.edu/rudi}
\subjclass{Primary 33E05, 34C25; Secondary 58F07 }
\begin{abstract}
We extend Halphen's theorem which characterizes the solutions
of certain $n$th-order differential equations with rational
coefficients and meromorphic fundamental systems to a
first-order $n \times n$ system of differential equations.
As an application of this circle of ideas we consider stationary
rational algebro-geometric solutions of the $\kdv$ hierarchy and
illustrate some of the connections with completely integrable
models of the Calogero-Moser-type. In particular, our treatment recovers
the complete characterization of the isospectral class of such rational
KdV solutions in terms of a precise description of the
Airault-McKean-Moser locus of their poles.
\end{abstract}
\keywords{Halphen's theorem, KdV hierarchy}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction} \lb{s1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The purpose of this paper is twofold. First we prove an extension of
Halphen's theorem, which characterizes the fundamental system of
solutions of certain $n$th-order ordinary differential equations with
rational coefficients to first-order $n\times n$ systems. In the second
part of this paper we show how to apply Halphen's theorem to completely
integrable systems of the Calogero-Moser-type, recovering a
complete characterization of the isospectral class of all
algebro-geometric rational solutions of the KdV hierarchy.
We start by describing Halphen's original result. Consider the following
$n$th-order differential equation
\begin{align}
q_n(z)y^{(n)}(z)+q_{n-1}(z) y^{(n-1)}(z)+\dots+
q_0(z) y(z) =0, \label{1.18}
\end{align}
where $q_j(z)$ are polynomials, and the order of $q_n(z)$ is at
least the order of $q_j(z)$ for all $0\leq j\leq (n-1)$, that is,
\begin{subequations} \lb{1.20}
\begin{align}
& q_m(z) \text{ are polynomials, $0\leq m\leq n$,} \lb{1.20a} \\
& q_m(z)/q_n(z) \text{ are bounded near $\infty$ for all
$0\leq m \leq n-1$}.
\lb{1.20b}
\end{align}
\end{subequations}
Then the zeros of $q_n(z)$ are the possible singularities of solutions
of \eqref{1.18}.
Assuming the fundamental system of solutions of \eqref{1.18} to be
meromorphic, the following theorem due to Halphen holds.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem} {\rm (}Halphen \cite{Ha85}, Ince
\cite[p.~372--375]{In56}{\rm)} \lb{t1.6}
Assume \eqref{1.20} and
suppose \eqref{1.18} has a meromorphic fundamental system of
solutions. Then the general solution of \eqref{1.18} is of the form
\begin{align}
y (z ) = \sum_{m=1}^{n} c_m r_m(z) e^{\lambda_m z}, \lb{1.21}
\end{align}
where $r_m(z)$ are rational functions of $z$,
$\lambda_m\in\bbC$, $1\leq m\leq n$, and $c_m$, $1\leq m\leq n$
are arbitrary complex constants.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Moreover, the converse of Halphen's theorem holds as well.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem} {\rm (}Ince \cite[p.~374--375]{In56}{\rm)} \lb{t1.6a}
Suppose $r_m(z)$ are rational functions of $z$ and
$\lambda_m, c_m\in\bbC$, $1\leq m\leq n$. If $r_1(z)e^{\lambda_1 z},
\dots,r_n(z)e^{\lambda_n z}$ are linearly independent, then
\begin{align}
y (z ) = \sum_{m=1}^{n} c_m r_m(z) e^{\lambda_m z} \lb{1.22}
\end{align}
is the general solution of an $n$th-order equation of the type
\eqref{1.18}, whose coefficients satisfy \eqref{1.20}.
\end{theorem}
\begin{rem} \lb{r1.7}
{\em We note that Halphen's main idea of proof in
\cite{Ha85} consists of replacing the rational coefficients in
\eqref{1.18} by appropriate elliptic coefficients {\rm(}as discussed in
\cite{Ha84}{\rm)} followed by an application of Picard's theorem
{\rm(}cf., e.g., \cite[p.~375--378]{In56}{\rm)}. A closer examination of
his argument seems to reveal a lack of proof of the crucial fact that the
associated differential equation with elliptic coefficients necessarily
has a meromorphic fundamental system of solutions. A proof of
Theorem~\ref{t1.6} {\rm(}and Theorem~\ref{t1.6a}{\rm)}, using a
completely different strategy, is provided in Ince's monograph
\cite[p.~372--375]{In56}. }
\end{rem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
One of the principal aims of this note is to prove a first-order
$n\times n$ system generalization of Halphen's Theorem~\ref{t1.6}
and its converse, Theorem~\ref{t1.6a}, in Section~\ref{s2}.
Analogous results hold for $n$th-order equations and first-order systems
with periodic and elliptic coefficients. For a glimpse at the vast
literature in these cases and their applications to completely integrable
systems we refer the interested reader to
\cite{GS98}--\cite{GW98a}, \cite{We99}, \cite{We00} and the literature
therein.
In Section~\ref{s3} we then apply Halphen's theorem to the problem of
characterizing the isospectral class of all stationary rational KdV
solutions. All such (nonconstant) solutions $q$ are well-known to be
necessarily of the form
\begin{equation}
q(z)=q_\infty-\sum_{\ell=1}^M s_\ell(s_\ell+1)(z-\zeta_\ell)^{-2}
\lb{1.23}
\end{equation}
for some $q_\infty\in\bbC$, $\{\zeta_\ell\}_{1\leq\ell\leq
M}\subset\bbC$, $\zeta_\ell^\prime\neq\zeta_\ell$ for
$\ell^\prime\neq\ell$, and
\begin{equation}
s_\ell\in\bbN, \,\,1\leq\ell\leq M
\text{ with } \sum_{\ell=1}^M s_\ell(s_\ell+1)=g(g+1) \lb{1.24}
\end{equation}
for some $g\in\bbN$, and the underlying spectral curve is then of the
especially simple rational type
\begin{equation}
y^2=(E-q_\infty)^{2g+1}. \lb{1.25}
\end{equation}
On the other hand, not every
$q$ of the type \eqref{1.23}, \eqref{1.24} is an algebro-geometric
solution of the KdV hierarchy. In general, the points $\zeta_\ell$ must
satisfy a set of intricate constraints. In fact, necessary and
sufficient conditions on $\zeta_\ell$ for $q$ in \eqref{1.23} to be a
rational KdV solution are given by
\begin{equation}
\sum_{\substack{\ell^\prime=1\\ \ell^\prime\neq \ell}}^M
\frac{s_{\ell^\prime}(s_{\ell^\prime}+1)}{(\zeta_{\ell}
-\zeta_{\ell^\prime})^{2k+1}}=0 \quad
\text{for $k=1, ..., s_{\ell^\prime}$ and $\ell=1,\dots,M$.} \lb{1.26}
\end{equation}
This result was first derived by Duistermaat and Gr\"unbaum
\cite{DG86} (cf.~p.~199) in 1986, as a by-product of their investigations
of bispectral pairs of differential operators. We will provide an
elementary derivation of this result on the basis of Halphen's theorem
and an explicit Frobenius-type analysis in Section~\ref{s3}.
For a fixed $g\in\bbN$, \eqref{1.24} and
\eqref{1.26} yield a complete parametrization of all rational KdV
solutions belonging to the spectral curve \eqref{1.25}. In other words,
they provide a complete characterization of the isospectral class of KdV
solutions corresponding to \eqref{1.25}. The constraints \eqref{1.26}
represent the proper generalization of the locus of poles introduced by
Airault, McKean, and Moser \cite{AMM77} in the sense that they explicitly
describe the situation where poles are permitted to collide (i.e., where
some of the $s_\ell >1$).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Halphen's theorem for first-order systems} \lb{s2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
This section is devoted to a generalization of Halphen's theorem (and its
converse) to first-order systems.
We briefly describe some of the notation used in this section.
$I_n$ denotes the identity in
$\bbC^n$. An $m\times m$ diagonal matrix $D=(d_j\delta_{j,k}
)_{1\leq j,k\leq m}$ will occasionally be denoted by
$\diag(d_1,\dots,d_m)$. The operation of transposition is denoted
by the superscript $t$. Moreover, it will be convenient to denote
the set of all
$m\times n$ matrices whose entries are rational functions with respect
to $z\in\bbC$ by $\calR^{m\times n}$, the subset of
$\calR^{m\times n}$ with rational entries bounded at infinity by
$\calR^{m\times n}_\infty$.
We recall that for $T\in\calR^{n\times n}$ invertible and
differentiable with
respect to $z$, the transformation $y(z)=T(z)u(z)$ turns the
first-order system of differential equations $y'(z)=A(z)y(z)$ into the
system $u'(z)=B(z)u(z)$, with $B(z)=T(z)^{-1}(A(z)T(z)-T'(z))$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{defi} \lb{d2.1}
(i) Two matrices $A, B\in\calR^{n\times n}$ are called \textit{of the
same kind} if
there exists an invertible matrix $T\in\calR^{n\times n}$ such that
\begin{align}
B(z)=T(z)^{-1}(A(z)T(z)-T'(z)). \label{}
\end{align}
(ii) $B\in\calR^{n\times n}$ is called {\it reduced of order $k$} if
$B_{j,\ell}=\delta_{j+1,\ell}$ for all $1\leq j\leq k$ and $1\leq
\ell\leq n$.
\end{defi}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Our approach, including the notion of matrices being ``of the same
kind'', was inspired by Loewy \cite{Lo18}. The relation of being of the
same kind is obviously an equivalence relation on $\calR^{n\times n}$.
The relation of being of the same kind is obviously an equivalence
relation on $\calR^{n\times n}$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemm} \label{l2.3}
Suppose that $A\in\calR^{n\times n}_{\infty}$ is reduced of order
$k-1$. Then either
$A_{k,k+1} = \dots\ =A_{k,n}=0$, or else there exists a matrix
$B\in\calR^{n\times n}_{\infty}$ of the same kind as $A$ and also
reduced of order $k-1$ but with the additional property that
$B_{k,k+1}(\infty)\neq 0$. Moreover,
$A(\infty)$ and $B(\infty)$ have the same
eigenvalues counting algebraic multiplicities.
\end{lemm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof} We assume not all of the entries $A_{k,k+1}$,\dots,
$A_{k,n}$ are equal to zero. Consider the $(n-k+1)\times(n-k)$ matrix
in the lower right corner of $A$ and denote it by $R$. Suppose that
$r$ is the largest nonnegative integer such that $z^r R_{1,j}(z)$ remains
bounded near infinity for every $j\in\{1,\dots,n-k\}$. Then there exists
an
$\ell\in\{1,\dots,n-k\}$ such that $z^r R_{1,\ell}(z)$ does not vanish
at infinity. Denote the constant $(n-k)\times(n-k)$ matrix, which
achieves the exchange of columns $1$ and $\ell$ of $R(z)$, by $C$. Then
the first row of $z^r R(z)C$ is bounded at infinity and the first entry
in that row does not vanish at infinity. Next, define
\begin{align}
T(z)=\begin{pmatrix}I_k&0\\0&z^rC\end{pmatrix},
\label{}
\end{align}
where $I_k$ is the $k\times k$ identity matrix. Let
\begin{align}
A(z)=\begin{pmatrix}
\tilde A_{1,1}(z)&\tilde A_{1,2}(z)\\ \tilde A_{2,1}(z)&\tilde
A_{2,2}(z) \end{pmatrix}, \label{}
\end{align}
where $\tilde A_{1,1}(z)$ and $\tilde A_{2,2}(z)$ are square matrices
with $k$ and $n-k$ rows, respectively. Then
\begin{align}
T(z)^{-1}A(z)T(z)=\begin{pmatrix}
\tilde A_{1,1}(z)&z^r \tilde A_{1,2}(z)C\\ z^{-r} C^{-1} \tilde
A_{2,1}(z) &C^{-1} \tilde A_{2,2}(z) C \end{pmatrix}. \label{}
\end{align}
Since only the last row of $\tilde A_{1,2}(z)$ is different from zero,
and since that row equals the first row of $R(z)$, the matrix
$T(z)^{-1}A(z)T(z)$ remains bounded at infinity and its first $k-1$
rows are the same as those of $A(z)$. The matrix $C$ was chosen so that
the first entry in the last row of $z^r\tilde A_{1,2}(z)C$ does not
vanish at infinity. Since
\begin{equation}
\lim_{z\to\infty}T(z)^{-1}T'(z)=0, \lb{2.5}
\end{equation}
we conclude that $B=T^{-1}(AT-T')\in\calR^{n\times n}_{\infty}$ is
reduced of order $k-1$ and that $B_{k,k+1}(\infty)\neq0$.
Finally we prove that $A(\infty)$ and $B(\infty)$ have the same
eigenvalues counting algebraic multiplicities. Since $T(\infty)$ might not
exist, we first compute
\begin{align}
&\det(\lim_{z\to\infty}((T^{-1}AT)(z)-\lambda I_n))
=\lim_{z\to\infty}\det((T^{-1}AT)(z)-\lambda I_n) \nonumber\\
&=\lim_{z\to\infty}\det(A(z)-\lambda I_n)
=\det(\lim_{z\to\infty}(A(z)-\lambda I_n)). \lb{2.6}
\end{align}
By \eqref{2.5}, the left-hand side of \eqref{2.6} is the characteristic
polynomial of $B(\infty)$, while the right-hand side is the
characteristic polynomial of $A(\infty)$. This completes the proof.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemm} \label{l2.4}
Assume that $A\in\calR^{n\times n}_{\infty}$ is reduced of order $k-1$ and
suppose that
$A_{k,k+1}(\infty)\neq 0$. Then
there exists a matrix $B\in\calR^{n\times n}_{\infty}$ of the same
kind as $A$, which is reduced of order $k$. Moreover, $A(\infty)$ and
$B(\infty)$ are similar and hence isospectral {\rm(}i.e., their
eigenvalues, including algebraic and geometric multiplicities,
coincide{\rm)}.
\end{lemm}
\begin{proof}
Let $T\in\calR^{n\times n}$ denote the $n\times n$ matrix obtained from
the identity matrix $I_n$ by replacing its $(k+1)$st row by
\begin{align}
(-A_{k,1},\dots,-A_{k,k},1,
-A_{k,k+2},\dots,-A_{k,n})/A_{k,k+1}. \label{}
\end{align}
$T^{-1}$ is then the matrix obtained from the identity
matrix $I_n$ by replacing the $(k+1)$st row by
$(A_{k,1},\dots,A_{k,n})$.
Note that the entries of $T$ and $T^{-1}$ are rational and
bounded at infinity. Hence the matrix
$B=T^{-1}(AT-T')$ has rational entries
bounded at infinity. A straightforward calculation then shows that the
first $k$ rows of $B$ have the desired form. Since $T$ and
$T'$ are bounded at infinity, $\lim_{z\to\infty}T(z)^{-1}T'(z)=0$
and hence $B(\infty) =T(\infty)^{-1}A(\infty)T(\infty)$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem} \label{t2.5}
Let $Q\in\calR_\infty^{n\times n}$ and suppose that the
first-order system
$y'(z)=Q(z)y(z)$ has a meromorphic fundamental system of solutions.
Then $y'=Qy$ has a fundamental matrix of the type
\begin{align}
Y(z)=R(z)\exp(\diag(\lambda_1 z,\dots,\lambda_n z)), \label{}
\end{align}
where $\lambda_1,\dots,\lambda_n$ are the eigenvalues of
$Q(\infty)$ and $R\in\calR^{n\times n}$.
\end{theorem}
\begin{proof}
The theorem will be proved by induction on $n$. Let $n=1$. Any
pole of $Q(z)$ must be of first-order with an integer
residue, that is,
\begin{align}
Q(z)=\lambda_1+\sum_{\ell=1}^N \frac{m_\ell}{z-a_\ell}, \label{}
\end{align}
with $m_1,\dots,m_N\in\bb Z$. Then
$Y(z)=\prod_{\ell=1}^N (z-a_\ell)^{m_\ell} \exp(\lambda_1 z)$
proves the claim for $n=1$.
Next, let $n$ be any natural number and assume that Theorem~\ref{t2.5}
has been proven for any natural number strictly less than $n$.\\
By hypothesis, $Q\in\calR^{n\times n}_{\infty}$ and $Q(z)$ can be
regarded to be reduced at least of order zero. We denote the
eigenvalues of $Q(\infty)$ by $\lambda_1,\dots,\lambda_n$.
Repeated, perhaps alternating, applications of Lemmas~\ref{l2.3} and
\ref{l2.4} then yield the existence of an integer
$k\in\{1,\dots,n\}$, a $k\times k$ matrix
$B_1(z)$, an $(n-k)\times k$ matrix $B_3(z)$, and an $(n-k)\times(n-k)$
matrix $B_4(z)$, such that
\begin{align}
B(z)=\begin{pmatrix}B_1(z)&0\\ B_3(z)&B_4(z)\end{pmatrix}
\label{}
\end{align}
has the following properties:
\begin{enumerate}
\item $B\in\calR_\infty^{n\times n}$.
\item $B$ is of the same kind as $Q$, that is, there exists an
invertible matrix $T\in\calR^{n\times n}$ such that
$B(z)=T(z)^{-1}(Q(z)T(z)-T'(z))$.
\item $B_1(z)$ is reduced of order $k-1$.
\item After a suitable relabeling of the eigenvalues of $Q(\infty)$ the
eigenvalues of $B_1(\infty)$ are $\lambda_1,\dots,\lambda_k$ and the
eigenvalues of $B_4(\infty)$ are $\lambda_{k+1},\dots,\lambda_n$.
\item The first-order system $u'(z)=B(z) u(z)$ has a meromorphic
fundamental system of solutions with respect to $z\in\bbC$.
\end{enumerate}
We now have to distinguish whether $k=n$ or $k0$, rather than the case $q=0$
only. But Theorem~\ref{t2.5} can not hold in general for $q>0$ as
shown by the following elementary counterexample.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{exa} \lb{e1.12}
The first-order $2\times 2$ system
$$
y'(z) = \left(\begin{matrix}0 & 1\\z^m & 0
\end{matrix} \right) y(z), \quad m\in {\mathbb{N}}
$$
has no solution in terms of elementary functions, although it clearly
has a meromorphic fundamental system. The particular case $m=1$
represents the well-known Airy equation.
\end{exa}
\begin{rem} \lb{r2.6a}
{\em In the case where all eigenvalues $1\leq\lambda_j\leq n$
of $Q(\infty)$ are distinct, we now sketch an alternative proof of
Theorem~\ref{t2.5}, based
on Theorem~12.3 in Wasow's monograph \cite{Wa87}. Since Theorem~12.3 in
\cite{Wa87} only applies to appropriate sectors of the complex plane with
vertex at the origin, we argue as follows. First one can
find a sufficiently small sector $S_3$, which does not contain
any separation
rays. (We recall that a ray (i.e., a half line), where $\Re(\lambda_j z
-\lambda_k z)=0$ for some pair of distinct integers $j,k$, is called a
separation ray.) Then one chooses two other sectors $S_1,S_2$
with opening
angles $\phi_j < \pi$, $j=1,2$, such that $ S_1 \cup S_2 \cup S_3 =
\cz\backslash\{0\}$. It is then possible to show that the
transition matrix
from sector $S_1 $ to sector $S_2 $ equals the identity matrix.
Hence, the
solution of the form $ Y (z) = R(z) \exp({\diag(\lambda_1,
\ldots, \lambda_n)
z})$ in sector $S_1 $ is valid in sector $S_2$ too and thus can
be continued
into $S_3$ since by hypothesis, the sector $S_3$ contains no
separation rays. }
\end{rem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Finally, we turn to a converse of Theorem~\ref{t2.5}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem} \lb{t2.6}
Suppose $R\in\calR^{n\times n}$, $\det(R)\neq 0$, and
$\lambda_1,\dots,\lambda_n\in\bbC$. Then
\begin{align}
Y(z)=R(z)\exp(\diag(\lambda_1 z,\dots,\lambda_n z)) \label{}
\end{align}
is a fundamental matrix of a first-order linear system of differential
equations $y'(z)=Q(z)y(z)$, where $Q\in\calR^{n\times n}$ and
$Q(z)$ is of the
same kind as a matrix in $\calR^{n\times n}_\infty$. In fact, $Q(z)$ is
of the same kind as the constant diagonal matrix
$\diag(\lambda_1,\dots,\lambda_n)$.
\end{theorem}
\begin{proof} Since
\begin{equation}
Q(z)=R(z)\diag(\lambda_1,\dots,\lambda_n)R(z)^{-1}+R'(z)R(z)^{-1},
\lb{2.27}
\end{equation}
we choose $T(z)=R(z)^{-1}$ and hence obtain $T'=-R^{-1}R'R^{-1}$ and
thus,
\begin{equation}
Q=T^{-1}(\diag(\lambda_1 ,...,\lambda_n)T-T'). \lb{2.28}
\end{equation}
Hence, $Q(z)$ is of the same kind as the constant matrix
$\diag(\lambda_1,\dots,\lambda_n)$.
\end{proof}
\section{Some applications to rational solutions of the stationary
$\kdv$ hierarchy} \lb{s3}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this section we describe the connections between the preceding
results and infinite-dimensional completely integrable Hamiltonian
systems. For reasons of brevity we will only consider the simplest case
of the $\kdv$ hierarchy, and in accordance with Sections~\ref{s1},
\ref{s2}, only study its stationary rational
solutions bounded at infinity (cf.~\cite{AS78}, \cite{AM78}--\cite{CC77},
\cite{Gr82}, \cite{Ka95}, \cite{Kr78}--\cite{Kr74}, \cite{Mo77},
\cite{Oh88}, \cite{Pe94}, \cite{Sh94}, \cite{So78}, \cite{Wi98} and the
literature cited therein). The principal results on the stationary
$\kdv$ hierarchy as needed in this section are summarized in the
appendix, and we freely use these results and the notation
established there in what follows.
The rational $\kdv$ solutions bounded at infinity are usually discussed in
a time-dependent setting and the dynamics of their poles is in an
intimate relationship with completely integrable systems of the
Calogero-Moser-type. In our discussion below, the time-dependence will
generally be suppressed and only occasionally be mentioned in connection
with particular isospectral deformations of rational solutions of the
$\kdv$ hierarchy. Our principal focus will be on stationary
(isospectral) aspects of these rational $\kdv$ solutions and the
implications of Halphen's theorem in this context.
We start by quoting a number of known results on stationary rational
$\kdv$ solutions bounded at infinity.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem} \lb{t3.1}
Let $N\in\bbN$ and $\{z_j\}_{1\leq j\leq N}\subset\bbC$. \\
\noindent {\rm (}i{\rm )} {\rm (}Airault, McKean, and Moser
\cite{AMM77}{\rm )}
%Suppose that
%\begin{equation}
%z_j\neq z_k \text{ for } j\neq k, \,\, 1\leq j,k\leq N. \lb{3.1}
%\end{equation}
Any rational solution $q$ of
{\rm(}some, and hence infinitely many equations of\,{\rm)} the $\kdv$
hierarchy, or equivalently, any rational algebro-geometric potential
$q$, is necessarily of the form
\begin{equation}
q(z)=q_\infty -2\sum_{j=1}^N (z-z_j)^{-2}, \lb{3.2}
\end{equation}
for some $q_\infty\in\bbC$ and with $N\in\bbN$ of the special type
$N=g(g+1)/2$ for some
$g\in\bbN$. \\
\noindent {\rm (}ii{\rm )} {\rm (}Airault, McKean, and Moser
\cite{AMM77} {\rm (}see also \cite{We99}{\rm ))} If one allows for
``collisions'' between the
$z_j$, that is, if the set $\{z_j\}_{1\leq j\leq N}$ clusters into
groups of points, then the corresponding rational algebro-geometric
potential $q$ is necessarily of the form
\begin{equation}
q(z)=q_\infty -\sum_{\ell=1}^M s_\ell(s_\ell +1)(z-\zeta_\ell)^{-2},
\lb{3.5}
\end{equation}
where for some $g\in\bbN$,
\begin{subequations} \lb{3.4}
\begin{align}
&\{z_j\}_{1\leq j\leq
N}=\{\zeta_\ell\}_{1\leq\ell\leq M}\subset\bbC, \text{ with $\zeta_\ell$
pairwise distinct,} \lb{3.4a} \\
& s_\ell\in\bbN,\,\,\, 1\leq\ell \leq M, \no \\
&\sum_{\ell=1}^M s_\ell(s_\ell +1)=2N \text{ for some $N\in\bbN$ of
the type $N=g(g+1)/2$.} \lb{3.4b}
\end{align}
\end{subequations}
\noindent {\rm (}iii{\rm )} The extreme case of all
$z_j$ colliding into one point, say $\zeta_1$, that is,
$\{z_j\}_{1\leq j\leq N}=\{\zeta_1\}\subset\bbC$
yields an algebro-geometric $\kdv$ potential of the elementary form
\begin{equation}
q(z)=q_\infty -g(g+1)(z-\zeta_1)^{-2}, \quad g\in\bbN \lb{3.6}
\end{equation}
and no additional constraints on $\zeta_1\in\bbC$. \\
\noindent {\rm (}iv{\rm )} In all cases {\rm (}i{\rm )}--{\rm (}iii{\rm
)}, if
$q$ is a rational $\kdv$ potential {\rm (}i.e., if $g\in\bbN$ and the
points $z_j$ {\rm (}resp.
$\zeta_\ell${\rm )} satisfy appropriate restrictions,
cf.~Theorem~\ref{t3.6}{\rm)}, the underlying rational hyperelliptic curve
$\calK_g$ is of the especially simple form
\begin{equation}
\calK_g \colon y^2=(E-q_\infty)^{2g+1}. \lb{3.8}
\end{equation}
In particular, the potentials \eqref{3.2}, \eqref{3.5}, and \eqref{3.6}
are all isospectral {\rm (}assuming \eqref{3.2} and \eqref{3.5} are
algebro-geometric $\kdv$ potentials, of course{\rm )}. \\
\noindent {\rm (}v{\rm )} {\rm (}Weikard \cite{We99}{\rm )} $q$ is a
rational $\kdv$ potential if and only if
$\psi''+(q-E)\psi=0$ has a meromorphic fundamental solutions
{\rm (}w.r.t.~$z${\rm )} for all values of the spectral parameter
$E\in\bbC$.\\
\noindent {\rm (}vi{\rm )} If $q$ is a rational KdV potential of the form
\eqref{3.5}, then $y''+qy=Ey$ has linearly independent solutions of
the Baker-Akhiezer-type
\begin{align}
&\psi_\pm(E,z)=\big(\pm E^{1/2}\big)^{-g}\Bigg(\prod_{j=1}^g
\big(\pm E^{1/2}-\nu_j(z)\big)\Bigg) e^{\pm E^{1/2} z}, \lb{3.8a} \\
&\hspace*{5.53cm} E\in\bbC\backslash\{q_\infty\}, \,\, z\in\bbC, \no
\end{align}
with $\mu_j(z)=\nu_j(z)^2$, $1\leq j\leq g$ the zeros of $F_g(z,x)$
as defined in \eqref{A.10}.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent (To avoid annoying case distinctions we will in almost all
circumstances exclude the trivial case $N=g=0$ in this section.)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{rem} \lb{r3.2}
{\em {\rm(}i{\rm)} It must be emphasized that for $N>1$, not any potential
$q$ of the type \eqref{3.2} is
an algebro-geometric $\kdv$ potential. In fact, for $N>1$, there exist
nontrivial constraints on the set $\{z_j\}_{1\leq j\leq N}$ for
\eqref{3.2} to represent an algebro-geometric $\kdv$ potential. For
instance, if the $z_j$ in \eqref{3.2} are pairwise distinct, then
Airault, MacKean, and Moser \cite{AMM77} proved that
\begin{equation}
\sum_{\substack{j^\prime=1\\ j^\prime\neq j}}^N
\frac{1}{(z_{j}
-z_{j^\prime})^{3}}=0 \quad
\text{for $j=1,\dots,N$} \lb{3.8aa}
\end{equation}
are necessary conditions for $q$ in \eqref{3.2} to be a stationary
KdV potential. In the
case of collisions {\rm (}i.e., if $s_{\ell_0} >1$ for some
$1\leq\ell_0\leq M${\rm )} the necessary constraints on
$\{\zeta_\ell\}_{1\leq\ell\leq M}$ are more involved than in the
nondegenerate case above and a complete description
of all constraints were originally obtained by Duistermaat and Gr\"unbaum
\cite{DG86} in 1986. An alternative proof of their result will be given
in Theorem~\ref{t3.6} below. \\
{\rm(}ii{\rm)} In connection with
Theorem~\ref{t3.1}\,(ii) one might naively expect that any decomposition
of $g(g+1)=\sum_{\ell=1}^M s_\ell(s_\ell+1)$ can actually be realized for
some choice of $\{\zeta_\ell\}_{1\leq \ell\leq M}$ with
$\zeta_\ell\neq\zeta_{\ell^\prime}$ for $\ell\neq\ell^\prime$. However,
the simple counterexample
$q(z)=-6(z-\zeta_1)^{-2}-6(z-\zeta_2)^{-2}$, which satisfies
$\kdv_3(q)=-5670(\zeta_1-\zeta_2)^2(\zeta_1+\zeta_2-2z)(z-\zeta_1)^{-6}
(z-\zeta_2)^{-6}$, quickly destroys such hopes. \\
{\rm(}iii{\rm)} Strictly speaking, the version of Theorem~3.1\,(v)
proven in \cite{We99} assumes in addition to $q$ being rational, that
$q$ is bounded at infinity. However, assuming that
$$
q(z)\underset{z\to\infty}{=}\alpha z^k + O(z^{k-1}) \, \text{ for some
$\alpha\neq 0$ and $k\in\bbN$,}
$$
a simple inductive argument using \eqref{A.1} proves
$$
\hat f_j^\prime(z)=\frac{k\alpha^j}{2}\bigg(\prod_{\ell=1}^{j-1}
\frac{2\ell+1}{2\ell} \bigg)z^{jk-1} +O(z^{jk-2}), \quad j\geq 1,
$$
using the usual convention (for $j=1$) that products over empty sets
are put equal to one. Thus, since
$\hat f_j^\prime$ cannot vanish in this case, a rational
$q$ unbounded at infinity cannot satisfy any of the stationary KdV
equations (cf.~\eqref{A.6}). }
\end{rem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Before we discuss
additional facts, we briefly pause and mention some of the ingredients
entering the proof of items (i)--(v) in Theorem~\ref{t3.1}. We start with
a fairly complete treatment of item (iii) and for simplicity of notation
put $q_\infty=\zeta_1=0$ and
\begin{equation}
q_g(z)=-g(g+1)z^{-2}, \quad g\in\bbN, \,\, z\in\bbC\backslash\{0\}.
\lb{3.8b}
\end{equation}
{}From \cite[Ch.~10]{AS72} one infers that
($E\in\bbC\backslash\{0\}$, $z\in\bbC$)
\begin{align}
\psi_\pm (E,z)=\Bigg(\sum_{k=0}^g \frac{(g+k)!}
{k!(g-k)!}(\pm 2E^{1/2}z)^{-k}\Bigg)e^{\mp E^{1/2}z}, \lb{3.9}
\end{align}
are linearly independent solutions of
$\psi''+(q_g-E)\psi=0$, $E\in\bbC\backslash\{0\}$.
Thus, one concludes that
\begin{equation}
\psi_+(E,z)\psi_-(E,z)=\prod_{j=1}^g\Big(1-\frac{\kappa_j}{Ez^2}\Big)
\text{ for some } \kappa_j\in\bbC, \,\, 1\leq j\leq g. \lb{3.10a}
\end{equation}
Hence a comparison with \eqref{A.10}--\eqref{A.13},
\eqref{A.16b}--\eqref{A.16g} yields
\begin{equation}
\hat F_g(E,z)=\prod_{j=1}^g \big(E-\mu_j(z)\big), \quad
\mu_j(z)=\kappa_jz^{-2}, \,\, 1\leq j\leq g, \lb{3.11}
\end{equation}
where $\hat F_g(E,z)$ denotes the polynomial of degree $g$
with respect to $E$ associated with $q_g(z)$ in \eqref{3.8b}, as
introduced in the appendix. Thus, $q_g(z)$ is a $\kdv$ potential
satisfying $\widehat \kdv_g (q_g)=0$
for a particular set of constants $\{c_\ell\}_{1\leq\ell\leq g}$ in
\eqref{A.8}. However, taking into account the simple form of
$q_g(z)$ in \eqref{3.8b}, homogeneity considerations in connection
with the corresponding $\hat f_j$ and \eqref{A.17} then
yield in the special case $q(z)=q_g(z)$,
\begin{align}
& c_\ell=0, \quad 1\leq \ell\leq g, \lb{3.13} \\
& \hat F_g(E,z)=F_g(E,z), \quad \hat f_j(z)=f_j(z), \quad
1\leq j\leq g, \lb{3.14} \\
& f_j(z)=d_j z^{-2j} \text{ for some } d_j\in\bbC\backslash\{0\},
\,\, 1\leq j\leq g, \lb{3.15} \\
& f_{k+1}(z)=0, \quad \sKdV_k(q_g)=0, \quad k\geq g,
\lb{3.16} \\
& y^2=E^{2g+1}, \text{ that is, $\hat E_m=0$, \,\, $0\leq m\leq 2g$}
\lb{3.17}
\end{align}
(and of course $c_0=\hat f_0(z)=f_0(z)=1$). This yields item (iii)
and part of item (iv). Since $q$ in \eqref{3.2} and \eqref{3.5}
in the special case $q_\infty=0$ satisfies
$q(z)\underset{|z|\to\infty}{=} 2Nz^{-2}\big(1
+ O\big(|z|^{-1}\big)\big)$,
one infers that $f_{k+1}=0$ for some $k\in\bbN$ can only happen if
$N=k(k+1)/2$ for some $k\in\bbN$.
This illustrates $N=g(g+1)/2$ and \eqref{3.4b}. Item (v) in \cite{We99}
follows from a careful combination of Frobenius theory for second-order
linear ordinary differential equations in the complex domain, Halphen's
theorem, Theorem~\ref{t1.6} (for $n=2$), and some of the algebro-geometric
formalism briefly sketched in the appendix. As a by-product of a
proof of item (v) one shows that
$\psi''(z)-cz^{-2}\psi(z)=E\psi(z)$, $z\in\bbC\backslash\{0\}$
has a meromorphic fundamental system of solutions for all $E\in\bbC$ if
and only if $c\in\bbC$ is of the special form $c=s(s+1)$ for some
$s\in\bbN_0$.
This illustrates why collisions necessarily must happen as described in
\eqref{3.4a}. This fact was already known to Kruskal \cite{Kr74} in 1974.
That $q$ in \eqref{3.2}, \eqref{3.5}, and
\eqref{3.6} are all isospectral $\kdv$ potentials, that is, they all
belong to the same algebraic curve \eqref{3.8} (assuming \eqref{3.2} and
\eqref{3.5} satisfy the additional restrictions to make them
algebro-geometric $\kdv$ potentials, of course) can be shown by several
methods. Either by invoking time-dependent $\kdv$ flows as in
\cite{AMM77}, or by commutation techniques (i.e., Darboux-type
transformations) as in \cite{AM78}, \cite{EK82}, \cite{Mo77}, \cite{Oh88}
(cf.~also \cite{GH99b}). This fact also follows from the results in
\cite{We99}. Finally, identifying $\psi_\pm(E,z)/\psi_\pm(E,z_0)$ with
the two branches of the Baker-Akhiezer function $\psi(P,z,z_0)$,
$P=(E,y)$ in
\eqref{A.16c}, a combination of \eqref{A.10},
\eqref{A.16f}, and the normalizations
\begin{equation}
\lim_{|z|\to\infty} \psi_\pm(E,z)\exp(\mp E^{1/2}z)=1, \quad
\lim_{|E|\to\infty} \psi_\pm(E,z)\exp(\mp E^{1/2}z)=1, \no
\end{equation}
then proves
$\psi_+(E,z)\psi_-(E,z)=E^{-g}F_g(E,z)=\prod_{j=1}^g \bigg(1-
\frac{\mu_j(z)}{E}\bigg)$, and hence \eqref{3.8a}.
Finally, we study the precise restrictions on the set of
poles $\{z_j\}_{1\leq j\leq N}=\{\zeta_\ell\}_{1\leq\ell\leq M}$ for $q$
in \eqref{3.5} to be a $\kdv$ potential.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemm} \label{l3.4}
Suppose the function $q$ has a Laurent expansion about the point
$z_0\in\bbC$ of the type
\begin{equation}
q(z)=\sum_{j=0}^\infty q_j (z-z_0)^{j-2}, \lb{3.30}
\end{equation}
where $q_0=-s(s+1)$ and, without loss of generality, $\Re(2s+1)\geq0$.
Define for $\sigma\in\bbC$,
\begin{align}
f_0(\sigma)&=-\sigma(\sigma-1)-q_0=(s+\sigma)(s+1-\sigma), \lb{3.31}\\
c_0(\sigma)&=\prod_{j=1}^{2s+1} f_0(\sigma+j), \,\,
c_j(\sigma)=\frac{\sum_{m=0}^{j-1}
q_{j-m} c_m(\sigma)}{f_0(\sigma+j)}, \;\, j\in\bbN, \label{crec} \\
w(\sigma,z)&=\sum_{j=0}^\infty c_j(\sigma) (z-z_0)^{\sigma+j},
\lb{3.33} \\
v(\sigma,z)&=\frac{\partial w}{\partial\sigma}(\sigma,z)
=\sum_{j=0}^\infty \left(\frac{\partial c_j}{\partial\sigma}
+c_j\log(z-z_0)\right) (z-z_0)^{\sigma+j}. \lb{3.34}
\end{align}
If $2s+1$ is not an integer, then $y''+qy=0$
has the linearly independent solutions $y_1=w(s+1,\cdot)$ and
$y_2=w(-s,\cdot)$. If $2s+1$ is an integer, then $y''+qy=0$ has the
linearly independent solutions $y_1=w(s+1,\cdot)$ and $y_2=v(-s,\cdot)$.
Moreover, $y''+qy=0$ has a meromorphic fundamental system of solutions
near $z_0$ if and only if $s\in\bb N_0$ and $c_{2s+1}(-s)=0$.
\end{lemm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
This is a classical result in ordinary differential equations (cf., e.g.,
\cite{In56}, Chs.~XV, XVI). A recent proof can be found in
Section 3 of
\cite{We99}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{defi} \lb{d3.5}
Let $q$ be a rational function. Then $q$ is called a \textit{Halphen
potential} if it is bounded near infinity and if $y''+qy=Ey$ has a
meromorphic fundamental system of solutions {\rm(}w.r.t.~$z${\rm)} for
each value of the complex spectral parameter $E\in\bbC$.
\end{defi}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Of course every constant is a Halphen potential. Moreover, by
Theorem~\ref{t3.1}\,(v), $q$ is a Halphen potential if and only if it
is a rational KdV potential (i.e., if and only if it satisfies one and
hence infinitely many of the equations of the stationary KdV hierarchy).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem} \lb{t3.6}
Let $q$ be a nonconstant rational function. Then $q$ is a Halphen
potential if and only if there are $M\in\bbN$, $s_\ell\in\bbN$,
$1\leq\ell\leq M$, $q_\infty\in\bbC$, and pairwise distinct
$\zeta_\ell\in\bbC$, $\ell=1,\dots,M$, such that
\begin{equation}
q(z)=q_\infty-\sum_{\ell=1}^M s_\ell(s_\ell+1)(z-\zeta_\ell)^{-2}
\lb{3.35}
\end{equation}
and
\begin{equation}
\sum_{\substack{\ell^\prime=1\\ \ell^\prime\neq \ell}}^M
\frac{s_{\ell^\prime}(s_{\ell^\prime}+1)}{(\zeta_{\ell}
-\zeta_{\ell^\prime})^{2k+1}}=0 \quad
\text{for $k=1, ..., s_{\ell}$ and $\ell=1,\dots,M$.} \lb{3.36}
\end{equation}
Moreover, $q$ is a rational KdV potential if and only if $q$ is of
the type \eqref{3.35} and the constraints \eqref{3.36} hold. In
particular, for fixed $g$, the constraints \eqref{3.36} characterize the
isospectral class of all rational KdV potentials associated with the
curve $y^2=(E-q_\infty)^{2g+1}$, where $g(g+1)=\sum_{\ell=1}^M
s_\ell(s_\ell+1)$.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
By Theorem~\ref{t3.1}\,(v), it suffices to prove the characterization of
Halphen potentials. Suppose that
$q$ is a nonconstant Halphen potential. Then a pole
$z_0$ of $q$ is a regular singular point of $y''+qy=Ey$ and hence
$$
q(z)-E=\sum_{j=0}^\infty Q_j (z-z_0)^{j-2}
$$
in a sufficiently small neighborhood of $z_0$,
where $Q_2$ is a first order polynomial in $E$, while $Q_j$ for $j\neq2$
are independent of $E$. The indices associated with $z_0$, defined
as the roots of $\sigma(\sigma-1)+Q_0=0$ (hence they are
$E$-independent), must be distinct integers whose sum
equals one. We denote them by $-s$ and $s+1$ where $s>0$ and note that
$Q_0=-s(s+1)$. We intend to prove that $Q_{2j+1}=0$ whenever
$j\in\{0,...,s\}$ by applying Lemma \ref{l3.4}. Proceeding by way of
contradiction, we thus assume that for some nonnegative integer
$k\in\{0,...,s\}$,
$Q_{2k+1}\neq 0$ and $k$ is the smallest such integer.
We note that $f_0(\cdot+j)$ are positive in $(-s-1,-s+1)$ for
$j=1,...,2s$, whereas $f_0(\cdot+2s+1)$ has a simple zero at
$-s$ and its derivative is negative at $-s$. Next one defines
\begin{equation}
\gamma_0(\sigma)=\prod_{j=1}^{2s+1} f_0(\sigma+j) \quad\text{and}\quad
\gamma_1(\sigma)=\prod_{j=2}^{2s+1} f_0(\sigma+j). \lb{3.38}
\end{equation}
$\gamma_0$ and $\gamma_1$ have simple zeros at $-s$ and
and $\gamma_0'(-s)$ and $\gamma_1'(-s)$ are negative.
The functions $c_0=\gamma_0$ and $c_1=Q_1\gamma_1$ are polynomials
with respect to $E$. Actually, $c_0$ has degree zero in $E$ and $c_1$ is
constant but might equal zero. Hence the relations \eqref{ce},
\eqref{co}, and
\eqref{gamma} below are satisfied for $j=1$. Next we assume that for some
integer $\ell\in\{1, ..., s\}$, the functions
$c_0$, ..., $c_{2\ell-1}$ are polynomials in $E$ and that the relations
\begin{align}
c_{2j-2}(\sigma)&=\gamma_{2j-2}(\sigma) Q_2^{j-1}+ O(E^{j-2}),
\label{ce}\\
c_{2j-1}(\sigma)&=\begin{cases}
\gamma_{2j-1}(\sigma) Q_{2k+1} Q_2^{j-k-1}+ O(E^{j-k-2}),
&j-1\geq k,\\
0,&j-11${\rm)}. Moreover, this appears to be the first systematic
derivation of this locus {\rm(}with or without collisions{\rm)} within a
purely stationary approach {\rm(}i.e., without involving special
time-dependent
$\kdv$ flows, etc.{\rm)}.\\
{\rm(}iii{\rm)} For $k=1$, conditions \eqref{3.36} coincide with the
necessary conditions at collision points found by Airault, McKean, and
Moser \cite{AMM77} in their Remark~1 on p.~113. However, since there
are additional necessary conditions in \eqref{3.36}
corresponding to
$k\geq 2$, this disproves the conjecture made at the end of the proof of
their Remark~1. \\
{\rm(}iv{\rm)} The genus
$g=2$ ($N=3$) example,
$\tilde q_2(z,t)=-6z(z^3+6t)(z^3-3t)^{-2}$, $t\in\bbC$, with
$z_j=(3t)^{1/3}\omega_j$,
$\omega_j=\exp(2\pi ij/3)$, $1\leq j\leq 3$, explicitly illustrates the
locus in \eqref{3.36}. One verifies that $\tilde q_2(t)$
satisfies the $k$th stationary $\kdv$ equation, $\sKdV_k(\tilde q_2(t))=0$
for all $k\geq 2$ and all
$t\in\bbR$, as well as the 1st time-dependent $\kdv$ equation
$\tilde q_{2,t}=4^{-1}\tilde q_{2,xxx}+2^{-1}3\tilde q_2\tilde q_{2,x}$
{\rm(}see, e.g., \cite{Ai78}, \cite{DG86}{\rm)}. }
\end{rem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Extensions of the stationary formalism described in this section to
elliptic $\kdv$ potentials are in preparation.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%% appendices %%%%%%%%%%%%%%%%%%%%%%%%%%%
\appendix{The stationary KdV hierarchy} \lb{sA}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\renewcommand{\theequation}{A.\arabic{equation}}
\setcounter{theorem}{0}
\setcounter{equation}{0}
In this section we review basic facts on the stationary KdV
hierarchy. Since this material is well-known, we confine
ourselves to a brief account. Assuming $q$ to be meromorphic
in $\bbC$, consider the recursion relation
\begin{equation}\lb{A.1}
\hat{f}_0(z)=1, \quad \hat{f}'_{j+1}(z)=4^{-1}
\hat{f}'''_j(z) +
q(z)\hat{f}'_j(z)+2^{-1}q'(z)\hat{f}_j(z)
%\quad j\in\bbN_0
\end{equation}
for $j\in\bbN_0$ (with $\prime$ denoting differentiation with respect to
$z$ and
$\bbN_0=\bbN\cup\{0\}$) and the
associated differential expressions (Lax pair)
\begin{align}
L_2 &=\dfrac{d^2}{dz^2} + q(z), \lb{A.2} \\
\hat{P}_{2g+1}&=\sum^{g}_{j=0} \left[-\dfrac{1}{2}
\hat{f}'_j(z)+
\hat{f}_j(z) \dfrac{d}{dz}\right] L_2^{g-j},\quad g\in\bb N_0. \lb{A.3}
\end{align}
One can show that
\begin{equation}\lb{A.4}
\left[ \hat{P}_{2g+1}, L_2 \right]=2\hat{f}'_{g+1}
\end{equation}
([$\cdot,\cdot$] the commutator symbol) and explicitly computes
from \eqref{A.1},
\begin{equation}\lb{A.5}
\hat{f}_0=1, \; \hat{f}_1=2^{-1} q+c_1,\;
\hat{f}_2=8^{-1}
q'' + 8^{-1} 3q^2 +c_12^{-1} q+c_2, \text{ etc.},
\end{equation}
where $c_j\in\bbC$ are integration constants. Using the convention
that the corresponding homogeneous quantities obtained by setting
$c_{\ell}=0$ for $\ell=1,2,\ldots$ are denoted by $f_j$, that is,
\begin{equation}
f_j=\hat{f}_j\big|_{c_{\ell}=0, \, 1\leq\ell\leq j}\, , \quad j\in\bbN,
\lb{A.5a}
\end{equation}
one obtains
\begin{equation}
\hat f_j=\sum_{\ell=0}^j c_\ell f_{j-\ell}, \quad 0\leq j\leq g.
\lb{A.5b}
\end{equation}
The (homogeneous) stationary KdV
hierarchy is then defined as the sequence of equations
\begin{equation}\lb{A.6}
\sKdV_g(q)=2f'_{g+1} =0, \quad g\in\bb N_0.
\end{equation}
Explicitly, this yields
\begin{equation}\lb{A.7}
\sKdV_0(q)=q^\prime =0, \quad \sKdV_1(q) =4^{-1}q'''+
2^{-1} 3qq'=0, \text{ etc. }
\end{equation}
The corresponding nonhomogeneous version of $\sKdV_g(q)=0$ is
then defined by
\begin{equation}\lb{A.8}
\widehat \sKdV_g (q)=2\hat{f}'_{g+1}=2\sum^{g}_{j=0}c_{g-j}f'_{j+1}=0,
\end{equation}
where $c_0=1$ and $c_1,...,c_g$ are arbitrary complex constants.
If one assigns to $q^{(\ell)}=d^{\ell}q/dz^{\ell}$ the degree
$\deg(q^{(\ell)}) = \ell+2, \; \ell\in\bb N_0$, then the
homogeneous
differential polynomial $f_j$ with respect to $q$ turns
out to have
degree $2j$, that is,
\begin{equation}\lb{A.9}
\deg (f_j)=2j, \quad j\in\bb N_0.
\end{equation}
Next, introduce the polynomial $\hat{F}_g(E,z)$ in $E\in\bb C$,
\begin{equation}\lb{A.10}
\hat{F}_g(E,z) = \sum^{g}_{j=0} \hat{f}_{g-j}(z) E^j
=\prod_{j=1}^\g (E-\mu_j(z)).
\end{equation}
Since $\hat f_0(z)=1$,
\begin{align}
&-2^{-1} \hat F_g''(E,z)\hat F_g(E,z)+4^{-1}\hat
F_g'(E,z)^2+(E-q(z))\hat F_g(E,z)^2 \no \\
&=\hat{R}_{2g+1}(E,z) \lb{A.12}
\end{align}
is a monic polynomial in $E$ of degree $2g+1$.
However, equations \eqref{A.1} and \eqref{A.8} imply that
\begin{equation}\lb{A.11}
2^{-1} \hat{F}'''_g-2(E-q)\hat{F}'_g+q'\hat{F}_g=0
\end{equation}
and this shows that $\hat{R}_{2g+1}(E,z)$ is in fact independent of $z$.
Hence it can be written as
\begin{equation}\lb{A.13}
\hat{R}_{2g+1}(E)=\prod^{2g}_{m=0} (E-\hat{E}_m), \quad
\{\hat{E}_m\}_{0\leq m\leq 2g} \subset \bb C.
\end{equation}
By \eqref{A.4} the nonhomogeneous KdV equation \eqref{A.8} is
equivalent to the commutativity of $L_2$ and $\hat{P}_{2g+1}$.
This shows
that
\begin{equation}\lb{A.14}
[\hat{P}_{2g+1}, L_2 ]=0,
\end{equation}
and therefore, if $L_2 \psi=E\psi$, this implies that
$\hat P^2_{2g+1}\psi
=\hat R_{2g+1}(E)\psi$. Thus $[\hat{P}_{2g+1}, L_2 ]=0$
implies $\hat P^2_{2g+1} = \hat R_{2g+1}(L_2)$
by the Burchnall and Chaundy theorem. This illustrates the intimate
connection between
the stationary KdV equation $\hat{f}'_{g+1}=0$ in
\eqref{A.8} and the
compact (possibly singular) hyperelliptic curve $\hat\calK_g$ of
(arithmetic)
genus $g$ obtained upon one-point compactification of
the curve
\begin{equation}\lb{A.16}
\hat\calK_g\colon y^2=\hat{R}_{2g+1}(E)=\prod^{2g}_{m=0}(E-\hat{E}_m)
\end{equation}
by joining the point at infinity, denoted by $P_\infty$. Points
$P\in\hat\calK_\g\backslash\{P_\infty\}$ will be denoted by $P=(E,y)$,
moreover, the involution (hyperelliptic sheet exchange map) $*$ on
$\hat\calK_\g$ is defined by
\begin{equation}
*\colon\hat\calK_{g}\to\hat\calK_{g}, \quad P=(E,y)\mapsto P^{*}=(E,-y),
\, P_{\pm\infty}^{*}=P_{\mp\infty}.\lb{A.16a}
\end{equation}
Introducing the meromorphic function $\phi(\cdot,z)$ on $\hat\calK_g$,
\begin{equation}
\phi(P,z)=\big[y(P)+(1/2)\hat F^\prime_g(E,z)\big]/\hat F_g(E,z), \quad
P=(E,y)\in\hat\calK_g \lb{A.16b}
\end{equation}
and the stationary Baker-Akhiezer function $\psi(\cdot,z,z_0)$ by
\begin{equation}
\psi(P,z,z_0)=\exp\bigg(\int_{z_0}^z dz^\prime\,\phi(P,z^\prime)
\bigg), \quad P\in\hat\calK_\g\backslash\{P_\infty\}, \lb{A.16c}
\end{equation}
one infers (for $P=(E,y)\in\hat\calK_\g\backslash\{P_\infty\}$,
$(z,z_0)\in\bbC^2$)
\begin{align}
L_2\psi(P,\cdot,z_0)&=E\psi(P,\cdot,z_0), \lb{A.16d} \\
P_{2g+1}\psi(P,\cdot,z_0)&=y\psi(P,\cdot,z_0), \lb{A.16e} \\
\psi(P,z,z_0)\psi(P^*,z,z_0)&=\hat F_g(E,z)/\hat F_g(E,z_0),
\lb{A.16f}
\\ W(\psi(P,\cdot,z_0),\psi(P^*,\cdot,z_0))&=-2y(P)/\hat F_g(E,z_0),
\lb{A.16g}
\end{align}
where $W(f,g)(z)=f(z)g^\prime (z)-f^\prime (z)g(z)$ denotes the Wronskian
of $f$ and $g$. Thus, $\psi(P,z,z_0)$ and $\psi(P^*,z,z_0)$ are linearly
independent solutions of $L_2\psi=E\psi$ as long as $E\in\bbC
\backslash\{\hat E_m\}_{0\leq m\leq 2g}$. The two branches of
$\psi(P,z,z_0)$ will be denoted by $\psi_\pm(E,z,z_0)$, respectively.
The above formalism leads to the following standard definition.
\begin{defi} \label{dA.1}
Any solution $q$ of one of the stationary KdV
equations \eqref{A.8} is called an {\bf algebro-geometric KdV
potential}.
\end{defi}
For brevity of notation we will occasionally call such $q$ simply
$\kdv$ potentials.
Finally, denoting $\hat {\ul E}=(\hat E_0,\dots,\hat E_{2g})$,
consider
\begin{align*}
&\bigg(\prod_{m=0}^{2g} \bigg(1-\frac{\hat E_m}{z}\bigg)
\bigg)^{1/2}=\sum_{k=0}^{\infty}c_k(\hat {\ul E})z^{-k}, \\
&\text{where }\, c_0(\hat {\ul E})=1,\quad
c_1(\hat {\ul E})=-\frac12\sum_{m=0}^N \hat E_m, \,\, \text{ etc.}
\end{align*}
Assuming that $q$ satisfies the $g$th stationary
(nonhomogeneous) KdV equation \eqref{A.8}, the integration constants
$c_\ell$ in \eqref{A.5b} become a functional of the $\hat E_m$ in the
underlying curve \eqref{A.16} and one verifies
\begin{equation}
c_\ell=c_\ell(\hat {\ul E}), \quad \ell=0,\dots,g. \lb{A.17}
\end{equation}
{\bf Acknowledgment.}
We are indebted to Wolfgang Bulla for discussions on this subject.
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\end{document}
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