%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% @texfile{ %% filename="halphen.tex", %% version="8-8-00", %% date="May-2000", %% filetype="LaTeX2e", %% copyright="Copyright (C) F. Gesztesy, K. Unterkofler, R. Weikard" %% } \documentclass[reqno]{amsart} %\usepackage{amsmath,amssymb} \newcommand{\field}[1]{{\mathbb{#1}}} \newcommand{\cz}{\field{C}} \newcommand{\diag}{\operatorname{diag}} \newcommand{\e}{\hbox{\rm e}} \newcommand{\bb}[1]{{\mathbb{#1}}} \newcommand{\bbN}{{\mathbb{N}}} \newcommand{\bbR}{{\mathbb{R}}} \newcommand{\bbP}{{\mathbb{P}}} \newcommand{\bbZ}{{\mathbb{Z}}} \newcommand{\bbC}{{\mathbb{C}}} \newcommand{\bbCinf}{{\mathbb{C}_{\infty}}} \newcommand{\calK}{{\mathcal K}} \newcommand{\calR}{{\mathcal R}} \newcommand{\no}{\nonumber} \newcommand{\lb}{\label} \newcommand{\ul}{\underline} \newcommand{\ol}{\overline} \newcommand{\hatt}{\widehat} % use only for capital letters \newcommand{\ti}{\tilde} % use only for capital letters \newcommand{\wti}{\widetilde } \newcommand{\kdv}{\operatorname{KdV}} \newcommand{\genus}{g} \newcommand{\g}{g} \renewcommand{\Re}{\text{\rm Re}} \renewcommand{\Im}{\text{\rm Im}} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\KdV}{KdV} \DeclareMathOperator{\sKdV}{s-KdV} \DeclareMathOperator{\KP}{KP} \DeclareMathOperator{\Bsq}{Bsq} \allowdisplaybreaks \setlength{\parindent}{0pt} \setlength{\parskip}{5pt plus 2pt minus 1pt} \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemm}[theorem]{Lemma} \newtheorem{defi}[theorem]{Definition} \newtheorem{rem}[theorem]{Remark} \newtheorem{exa}[theorem]{Example} \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. \begin{thebibliography}{99} \bibitem{AS78} M.~J.~Ablowitz and J.~Satsuma, \textit{Solitons and rational solutions of nonlinear evolution equations}, J. Math. 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