%% @texfile{
%%     filename="Toda.tex",
%%     version="1.0",
%%     date="Okt-2001",
%%     cdate="20011010",
%%     filetype="LaTeX2e",
%%     stylefiles="dmv.cls",
%%     pics="Toda1,Toda2,Toda3,Toda4,Toda5",
%%     journal="Jahresber. Deutsch. Math.-Verein. 103, no. 4, 149-162 (2001)",
%%     url="https://www.math.uni-bielefeld.de/jb_dmv/#Band103",
%%     copyright="Gerald Teschl".
%%     }

\documentclass{dmv}
\usepackage{latexsym}
\usepackage{amssymb}
\usepackage{hyperref}
\usepackage{graphicx}
 
\title{Almost Everything You Always Wanted to Know About the Toda Equation}
\author{Gerald Teschl, Wien}

%%%%%%%%%%% ABBRS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\nn}{\nonumber}
\newenvironment{neweqnarray}{\arraycolsep=2pt\begin{eqnarray}}{\end{eqnarray}}
\newcommand{\bea}{\begin{neweqnarray}}
\newcommand{\eea}{\end{neweqnarray}}
\newcommand{\mb}{\mbox}
\newcommand{\ba}{\begin{array}}
\newcommand{\ea}{\end{array}}

\newcommand{\D}{\displaystyle}
\newcommand{\ol}{\overline}
\newcommand{\ul}{\underline}
\newcommand{\ti}{\tilde}
\newcommand{\bs}{\backslash}
\newcommand{\hr}{\mathfrak{H}}
\newcommand{\db}{\mathfrak{D}}
\newcommand{\sgn}{\mathrm{sgn}}
\newcommand{\mod}{\,\mathrm{mod}\, }
\newcommand{\ord}{\mathrm{ord}}
\newcommand{\res}{\mathrm{res}}
\newcommand{\re}{\mathrm{Re}}
\newcommand{\im}{\mathrm{Im}}
\newcommand{\I}{\mathrm{i}}
\newcommand{\id}{\mathrm{1\hspace{-0.57ex}l}}
\newcommand{\tr}{\mathrm{tr}}
\newcommand{\Ran}{\mathrm{Ran}}
\newcommand{\Ker}{\mathrm{Ker}}
\newcommand{\hull}{\mathrm{hull}}
\newcommand{\lz}{\ell^2(\Z)}
\newcommand{\mzer}{\backslash \{ 0\}}
\newcommand{\spr}[2]{\langle #1 , #2 \rangle}
\newcommand{\intp}[1]{[\![ #1 ]\!]}
\newcommand{\gele}{\ba{c} \vspace*{-2.5mm} > \\ < \ea}
\newcommand{\zeroone}{\mbox{$\genfrac{}{}{0pt}{}{0}{1}$}}
\newcommand{\zeromone}{\mbox{$\genfrac{}{}{0pt}{}{0}{-1}$}}

\newcommand{\tl}{\mathrm{TL}}
\newcommand{\km}{\mathrm{KM}}
\newcommand{\ultau}{\ul{\tau}}
\newcommand{\ulm}{\ul{m}}
\newcommand{\uln}{\ul{n}}
\newcommand{\ulzeta}{\ul{\zeta}}
\newcommand{\ulz}{\ul{z}}
\newcommand{\hmu}{\hat{\mu}}
\newcommand{\dimu}[1]{\di_{\ul{\hat{\mu}}(#1)}}
\newcommand{\dimuz}{\di_{\ul{\hat{\mu}}}}
\newcommand{\dilam}[1]{\di_{\ul{\hat{\lam}}^\beta(#1)}}
\newcommand{\dilamz}{\di_{\ul{\hat{\lam}}^\beta}}
\newcommand{\vrc}{\ul{\Xi}_{p_0}}
\newcommand{\hvrc}{\ul{\hat{\Xi}}_{p_0}}
\newcommand{\Wp}{W^{p_0}}
\newcommand{\hWp}{\hat{W}^{p_0}}
\newcommand{\di}{\mathcal{D}}
\newcommand{\divm}{{\rm Div}(M)}
\newcommand{\dee}[1]{{\rm deg}(#1)}
\newcommand{\mfu}{\mathcal{M}(M)}
\newcommand{\mfo}{\mathcal{M}^1 (M)}
\newcommand{\hfo}{\mathcal{H}^1 (M)}
\newcommand{\lfu}{\mathcal{L}}
\newcommand{\hF}{\hat{F}}
\newcommand{\Amap}{\ul{A}_{p_0}}
\newcommand{\amap}{\ul{\alpha}_{p_0}}
\newcommand{\hAmap}{\ul{\hat{A}}_{p_0}}
\newcommand{\hamap}{\ul{\hat{\alpha}}_{p_0}}
\newcommand{\zpno}{\zeta_{p_0}}
\newcommand{\upno}{U_{p_0}}
\newcommand{\mus}{{(\mu,\sig)}}
\newcommand{\mmus}{{\{\mu,\sig\}}}
\newcommand{\Rg}[1]{R_{2g+2}^{1/2}(#1)}
\newcommand{\Rr}[1]{R_{2r+2}^{1/2}(#1)}
%\newcommand{\RR}[1]{R\!R_{2r+2,j}(#1)}


%%%%%%%%%%% GREEK %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\vphi}{\varphi}
\newcommand{\eps}{\varepsilon}
\newcommand{\sig}{\sigma}
\newcommand{\lam}{\lambda}
\newcommand{\gam}{\gamma}
\newcommand{\om}{\omega}

%%%%%%%%%%% FONTS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\C}{\mathbb{C}}


\newfont{\symb}{cmr12 scaled 1100}
\newcommand{\ltb}{\mb{\raisebox{-0.5pt}[-0.5pt]{\symb (}}}
\newcommand{\rtb}{\mb{\raisebox{-0.5pt}[-0.5pt]{\symb )}}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\unitlength1cm

\newtheorem{theorem}{Theorem}


\begin{document}

\noindent\parbox{.98\textwidth}{\small
Jahresber. Deutsch. Math.-Verein. {\bf 103}, no. 4, 149-162 (2001)\\[1ex]
Mathematics Subject Classification: 35Q51, 37K10; 37K15, 39A70\\[1ex]
Keywords and Phrases: Toda equation, Kac van Moerbeke equation, Solitons, Lax pair}
%\subjclass{Primary 35Q51, 37K10; Secondary 37K15, 39A70}
%\keywords{Toda equation, Kac van Moerbeke equation, Solitons, Lax pair}
\vspace{1.5cm}


\maketitle

\begin{abstract}
The present article reviews methods from spectral theory and algebraic geometry for finding
explicit solutions of the Toda equation, namely for the $N$-soliton solution
and quasi-periodic solutions. Along they way  basic concepts like Lax pairs, associated
hierarchies, and B\"acklund transformations for the Toda equation are introduced.
\end{abstract}



\section*{Preface}


This article is supposed to give an introduction to some aspects of
completely integrable nonlinear wave equations and soliton mathematics
using one example, the Toda equation. Moreover, the aim is not
to give a complete overview, even for this single equation. Rather
I will focus on only two methods (reflecting my personal bias) and I will
try to give an outline on how explicit solutions can be obtained. More
details and many more references can be found in the monographs
by Gesztesy and Holden \cite{ghbok}, myself \cite{tejocil}, and Toda \cite{ta}.

The contents constitutes an extended version of my talk given at
the joint annual meeting of the \"Osterreichische Mathematische Gesellschaft
and the Deutsche Mathematiker-Vereinigung in
September 2001, Vienna, Austria.




\section{The Toda equation}




In 1955 Enrico Fermi, John Pasta, and Stanislaw Ulam carried out a seemingly innocent
computer experiment at Los Alamos, \cite{fpu}. They considered a simple model for a nonlinear
one-dimensional crystal describing the motion of a chain of particles with nearest neighbor
interaction.\\
\begin{picture}(10,2.5)
\put(4,1.5){\circle*{0.3}}
\put(6.2,1.5){\circle*{0.3}}
\put(8.4,1.5){\circle*{0.3}}
\thicklines
\put(3.7,1.5){\line(1,0){0.6}}
\put(4.3,1.5){\line(1,1){0.4}}
\put(4.7,1.9){\line(1,-1){0.8}}
\put(5.5,1.1){\line(1,1){0.4}}
\put(5.9,1.5){\line(1,0){0.6}}
\put(6.5,1.5){\line(1,1){0.4}}
\put(6.9,1.9){\line(1,-1){0.8}}
\put(7.7,1.1){\line(1,1){0.4}}
\put(8.1,1.5){\line(1,0){0.6}}
\thinlines
\put(6.2,1.5){\line(0,-1){1.2}}
\put(6.2,0.7){\vector(-1,0){0.5}}
\put(6.2,0.7){\vector(1,0){0.5}}
\put(6.4,0.2){$q(n,t)$}
\end{picture}

The Hamiltonian of such a system is given by
\begin{equation}
\mathcal{H}(p,q) = \sum_{n\in\Z} \Big(\frac{p(n,t)^2}{2} + V(q(n+1,t) - q(n,t)) \Big),
\end{equation}
where $q(n,t)$ is the displacement of the $n$-th particle from its equilibrium position,
$p(n,t)$ is its momentum (mass $m=1$), and $V(r)$ is the interaction potential.

Restricting the attention to finitely many particles (e.g., by imposing periodic boundary
conditions) and to the {\bf harmonic interaction} $V(r)=\frac{r^2}{2}$, the equations of motion form a
linear system of differential equations with constant coefficients. The solution
is then given by a superposition of the associated {\em normal modes}. It was general belief at
that time that a generic nonlinear perturbation would yield to {\em thermalization}.
That is, for any initial condition the energy should eventually be equally distributed
over all normal modes. The aim of the experiment was to investigate the rate of
approach to the equipartition of energy. However, much to everybody's surprise,
the experiment indicated, instead of the expected thermalization, a quasi-periodic motion
of the system! Many attempts were made to explain this result but it was not until ten years later
that Martin Kruskal and Norman Zabusky, \cite{zakr}, revealed the connections with solitons.

This had a big impact on soliton mathematics and led to an explosive growth in the last
decades. In particular, it led to the search for a potential $V(r)$ for which the above
system has soliton solutions. By considering addition formulas for elliptic functions,
Morikazu Toda came up with the choice
\begin{equation}
V(r) = {\rm e}^{\D - r} + r -1.
\end{equation}
The corresponding system is now known as the {\bf Toda equation}, \cite{ta1}.

\begin{figure}[th]
\centering
\fbox{\includegraphics[width=4cm]{Toda1}}
\caption{Toda potential $V(r)$}
\end{figure}

{\scriptsize This model is of course only valid as long as the relative displacement is
not too large (i.e., at least smaller than the distance of the particles in the
equilibrium position). For small displacements it is approximately equal to a harmonic
interaction.}\\

The equation of motion in this case reads explicitly
\bea \nn
\frac{d}{dt} p(n,t) &=& -\frac{\partial\mathcal{H}(p,q)}{\partial q(n,t)}\\ \nn
&=& {\rm e}^{\D -(q(n,t) - q(n-1,t))} - {\rm e}^{\D -(q(n+1,t) - q(n,t))},\\
\frac{d}{dt} q(n,t) &=& \frac{\partial\mathcal{H}(p,q)}{\partial p(n,t)} = p(n,t).
\eea

The important property of the Toda equation is the existence of so called {\bf soliton} solutions,
that is, pulslike waves  which spread in time without changing their size and shape.
This is a surprising phenomenon, since for a generic linear equation one would expect spreading
of waves (dispersion) and for a generic nonlinear force one
would expect that solutions only exist for a finite time (breaking of waves). Obviously our
particular force is such that both phenomena cancel each other giving rise to a stable
wave existing for all time!

In fact, in the simplest case of one soliton you can easily verify that this solution is given by
\begin{equation}
q_1(n,t) = q_0 - \ln\frac{1 + \gam \exp(-2\kappa n \pm 2\sinh(\kappa)t)}{1
+ \gam \exp(-2\kappa (n-1) \pm 2\sinh(\kappa)t)}, \quad \kappa,\gam>0.
\end{equation}
\begin{figure}[th]
\centering
\fbox{\includegraphics[width=8cm]{Toda2}}
\caption{One soliton}
\end{figure}
It describes a single bump traveling through the crystal with speed
$\pm\sinh(\kappa)/ \kappa$ and width proportional to $1/ \kappa$.
In other words, the smaller the soliton the faster it propagates. It results in
a total displacement $2 \kappa$ of the crystal.

Such {\em solitary waves} were first observed by the naval architect John
Scott Russel \cite{rus}, who followed the bow wave of a barge which moved along a channel
maintaining its speed and size (see the review article by Palais \cite{pal} for further
information).

Existence of soliton solutions is usually connected to complete integrability of
the system, and this is also true for the Toda equation.

{\scriptsize The motivation as a simple model in solid state
physics presented here is of course not the only application of the Toda equation. In fact,
the Toda equation and related equations are used to model Langmuir oscillations in plasma
physics, to investigate conducting polymers, in quantum cohomology, etc.. Some
general books dealing with the Toda lattice are the monographs by Toda \cite{ta}, \cite{ta1},
by Eilenberger \cite{eil}, by Faddeev and Takhtajan \cite{fad} and by Teschl
\cite{tejocil}. Another good source on soliton mathematics is the recent review article
by Palais \cite{pal}. Finally, it should also be mentioned that the Toda equation can be
viewed as a discrete version of the Korteweg-de Vries equation (see \cite{ta} or
\cite{pal} for informal treatments).}



\section{Complete integrability and Lax pairs}



To see that the Toda equation is indeed integrable we introduce Flaschka's variables
\begin{equation}
a(n,t) = \frac{1}{2} {\rm e}^{\D -(q(n+1,t) -
q(n,t))/2}, \qquad b(n,t) = -\frac{1}{2} p(n,t)
\end{equation}
and obtain the form most convenient for us
\bea \nn
\dot{a}(t) &=& a(t) \Big(b^+(t)-b(t)\Big), \\ \label{todeqfl}
\dot{b}(t) &=& 2 \Big(a(t)^2-a^-(t)^2\Big).
\eea
Here we have used the abbreviation
\begin{equation}
f^\pm(n)= f(n\pm1).
\end{equation}

To show complete integrability it suffices to find a so-called {\bf Lax pair}, that
is, two operators $H(t)$, $P_2(t)$ in $\lz$ such that the Lax equation
\begin{equation} \label{laxeq}
\frac{d}{dt} H(t) = P_2(t) H(t) - H(t) P_2(t)
\end{equation}
is equivalent to (\ref{todeqfl}). One can easily convince oneself that the right
choice is
\bea \nn
H(t) &=& a(t) S^+ + a^-(t) S^- + b(t),\\
P_2(t) &=& a(t) S^+ - a^-(t) S^-,
\eea
where $(S^\pm f)(n) = f^\pm(n)= f(n\pm1)$ are the shift operators.
Now the Lax equation (\ref{laxeq}) implies that the operators $H(t)$ for
different $t\in\R$ are unitarily equivalent:

\begin{theorem}
Let $P_2(t)$ be a family of bounded skew-adjoint operators, such that $t\mapsto
P_2(t)$ is differentiable. Then there exists a family of unitary propagators $U_2(t,s)$
for $P_2(t)$, that is,
\begin{equation}
\frac{d}{dt} U_2(t,s) = P_2(t) U_2(t,s), \qquad U_2(s,s)=\id.
\end{equation}
Moreover, the Lax equation (\ref{laxeq}) implies
\begin{equation}
H(t)= U_2(t,s) H(s) U_2(t,s)^{-1}.
\end{equation}
\end{theorem} 

If the Lax equation (\ref{laxeq}) holds for $H(t)$ it automatically also holds
for $H(t)^j$. Taking traces shows that
\begin{equation}
\tr\ltb H(t)^j - H_0^j\rtb, \quad j\in\N,
\end{equation}
are conserved quantities, where $H_0$ is the operator corresponding to
the constant solution $a_0(n,t) = \frac{1}{2}$, $b_0(n,t)=0$ (it is
needed to make the trace converge). For example,
\bea \nn
\tr\ltb H(t) - H_0\rtb &=& \sum_{n\in\Z} b(n,t) = -\frac{1}{2} \sum_{n\in\Z}
p(n,t) \mbox{ and}\\
\tr\ltb H(t)^2 - H_0^2\rtb &=& \sum_{n\in\Z} b(n,t)^2 + 2(a(n,t)^2 - \frac{1}{4}) =
\frac{1}{2} \mathcal{H}(p,q)
\eea
correspond to conservation of the total momentum and the total energy,
respectively.

{\scriptsize The Lax pair approach was first advocated by Lax \cite{lax} in connection with
the Korteweg-de Vries equation. The results presented here are due to Flaschka
\cite{fl1}, \cite{fl2}. More informations on the trace formulas and conserved
quantities can be found in Gesztesy and Holden \cite{conl} and Teschl
\cite{tist}.}



\section{Types of solutions}




The reformulation of the Toda equation as a Lax pair is the key to methods
for solving the Toda equation based on spectral and inverse spectral theory for
the {\bf Jacobi operator} $H$ (tridiagonal infinite matrix). But before we go
into further details let me first show what kind of solutions one can obtain
by these methods.


The first type of solution is the general $N$-soliton solution
\begin{equation}
q_N(n,t) = q_0 - \ln\frac{\det(\id + C_N(n,t))}{\det(\id + C_N(n-1,t))},
\end{equation}
where
\begin{equation}
C_N(n,t) = \left(\frac{\sqrt{\gam_i \gam_j}}{1-
\mathrm{e}^{-(\kappa_i+\kappa_j)}}
\mathrm{e}^{ -(\kappa_i+\kappa_j)n -(\sig_i\sinh(\kappa_i) +
\sig_j\sinh(\kappa_j)) t}
\right)_{1\le i,j\le N}
\end{equation}
with $\kappa_j,\gam_j>0$ and $\sig_j\in\{\pm1\}$. The case $N=1$ coincides with
the one soliton solution from the first section. Two examples with $N=2$ are depicted below.
\begin{figure}[th]
\centering
\fbox{\includegraphics[width=8cm]{Toda3}}
\caption{Two solitons, one overtaking the other}
\end{figure}
\begin{figure}[th]
\centering
\fbox{\includegraphics[width=8cm]{Toda4}}
\caption{Two solitons traveling in different directions}
\end{figure}
These solutions can be obtained by either factorizing the underlying Jacobi operator
according to $H=A A^*$ and then commuting the factors or, alternatively, by the {\bf inverse
scattering transform}.

The second class of solutions are (quasi-)periodic solutions which can be found
using techniques from {\bf Riemann surfaces} (respectively algebraic curves). Each such
solution is associated with a {\bf hyperelliptic curve} of the type
\begin{equation}\label{hyelcrv}
w^2 = \prod_{j=0}^{2g+1} (z-E_j), \quad E_j\in\R,
\end{equation}
where $E_j$, $0\le j \le 2g+1$, are the band edges of the spectrum of $H$ (which is
independent of $t$ and hence determined by the initial conditions). One
obtains
\begin{equation} \label{thetasol}
q(n,t) = q_0 - 2(t\ti{b} + n\ln(2\ti{a})) - \ln \frac{\theta(\ul{z}_0-
2n\Amap(\infty_+) - 2t\ul{c}(g))}{\theta(\ul{z}_0- 2(n-1) \Amap(\infty_+) -
2t \ul{c}(g))},
\end{equation}
where $\ul{z}_0\in\R^g$, $\theta:\R^g \to \R$ is the Riemann theta function
associated with the hyperelliptic curve (\ref{hyelcrv}), and $\ti{a}, \ti{b} \in\R$,
$\Amap(\infty_+), \ul{c}(g) \in\R^g$ are constants depending only on the curve (i.e.,
on $E_j$, $0\le j \le 2g+1$). If $q(n,0)$, $p(n,0)$ are (quasi-) periodic with average
$0$, then $\ti{a}=\frac{1}{2}$, $\ti{b}=0$.
\begin{figure}[th]
\centering
\fbox{\includegraphics[width=8cm]{Toda5}}
\caption{A periodic solution associated with $w^2 = (z^2-2) (z^2-1)$}
\end{figure}

How these solutions can be obtained will be outlined in the following sections.
These methods can also be used to combine both types of solutions and {\em put $N$
solitons on top} of a given periodic solution.




\section{The Toda hierarchy}




The Lax approach allows for a straightforward generalization of the
Toda equation by replacing $P_2$ with more general operators $P_{2r+2}$ of order
$2r+2$. This yields the {\bf Toda hierarchy}
\begin{equation} \label{laxeqr}
\frac{d}{dt} H(t) = P_{2r+2}(t) H(t) - H(t) P_{2r+2}(t)
\qquad\Leftrightarrow\qquad
\tl_r(a,b)=0.
\end{equation}
To determine the admissible operators $P_{2r+2}$ (i.e., those for which the
commutator with $H$ is of order $2$) one restricts them to the algebraic
kernel of $H-z$
\begin{equation} \label{defrz}
( P_{2r+2}|_{\Ker(H-z)} ) = 2 a G_r(z) S^+ - H_{r+1}(z),
\end{equation}
where
\begin{equation}
G_r(z)= \sum_{j=0}^r g_{r-j} z^j, \qquad H_{r+1}(z)= z^{r+1} +
\sum_{j=0}^r h_{r-j} z^j - g_{r+1}.
\end{equation}
Inserting this into (\ref{laxeqr}) shows after a long and tricky calculation
that the coefficients are given by the diagonal and off-diagonal matrix
elements of $H^j$,
\begin{equation}\label{defgh}
g_j(n)= \spr{\delta_n}{H^j \delta_n}, \qquad
h_j(n)= 2 a(n) \spr{\delta_{n+1}}{H^j \delta_n}.
\end{equation}
Here $\spr{.}{..}$ denotes the scalar product in $\lz$ and $\delta_n(m)=1$ for
$m=n$ respectively $\delta_n(m)=0$ for $m\ne n$ is the canonical basis.
The $r$-th Toda equation is then explicitly given by
\bea \nn
\dot{a}(t) &=& a(t) ( g^+_{r+1}(t) - g_{r+1}(t) ),\\
\dot{b}(t) &=& h_{r+1}(t) - h^-_{r+1}(t).
\eea
The coefficients $g_j(n)$ and $h_j(n)$ can be computed recursively.

{\scriptsize The Toda hierarchy was first considered by Ueno and Takasaki
\cite{ut}, \cite{ut1}. The recursive approach for the standard Lax formalism,
\cite{lax} was first advocated by Al'ber \cite{al}. Here I followed Bulla, Gesztesy,
Holden, and Teschl \cite{bght}.}





\section{The Kac-van Moerbeke hierarchy}




Consider the {\bf super-symmetric Dirac operator}
\begin{equation} 
D(t) = \left( \ba{cc} 0 & A(t)^* \\ A(t) & 0 \ea \right),
\end{equation}
and choose
\begin{equation}
A(t) =\rho_o(t) S^+ + \rho_e(t), \qquad A(t)^* =\rho_o^-(t) S^- +\rho_e(t),
\end{equation}
where
\begin{equation}
\rho_e (n,t) =\rho(2n,t), \qquad \rho_o (n,t) =\rho (2n+1,t)
\end{equation}
are the ``even'' and ``odd'' parts of some bounded sequence $\rho(t)$.
Then $D(t)$ is associated with two Jacobi operators
\begin{equation}
H_1(t) =A(t)^* A(t), \qquad H_2 (t) =A(t) A(t)^*,
\end{equation}
whose coefficients read
\bea \nn 
a_1(t) = \rho_e(t) \rho_o(t), &\qquad& b_1(t) = \rho_e(t)^2 +\rho_o^-(t)^2,
\\ \label{defakt}
a_2(t) = \rho_e^+(t) \rho_o(t), &\qquad& b_2(t) = \rho_e(t)^2 +\rho_o(t)^2.
\eea
The corresponding Lax equation
\begin{equation}\label{laxeqd}
\frac{d}{dt} D(t) = Q_{2r+2}(t) D(t) - D(t) Q_{2r+2}(t),
\end{equation}
where
\begin{equation}
Q_{2r+2}(t) = \left( \ba{cc} P_{1,2r+2}(t) & 0 \\ 0 & P_{2,2r+2}(t) \ea\right),
\end{equation}
gives rise to evolution equations for $\rho(t)$ which are
known as the {\bf Kac-van Moerbeke hierarchy}, $\km_r(\rho)=0$. The first one
(the Kac-van Moerbeke equation) explicitly reads
\begin{equation}
\km_0(\rho)= \dot{\rho}(t) - \rho(t)(\rho^+(t)^2 - \rho^-(t)^2) =0.
\end{equation}
Moreover, from the way we introduced the Kac-van Moerbeke hierarchy, it is not
surprising that there is a close connection with the Toda hierarchy. To
reveal this connection all one has to do is to insert
\begin{equation}
D(t)^2 = \left( \ba{cc} H_1(t) & 0 \\ 0 & H_2(t) \ea\right)
\end{equation}
into the Lax equation
\begin{equation}
\frac{d}{dt} D(t)^2 = Q_{2r+2}(t) D(t)^2 - D(t)^2 Q_{2r+2}(t),
\end{equation}
which shows that the Lax equation (\ref{laxeqd}) for $D(t)$ implies the Lax equation
(\ref{laxeqr}) for both $H_1$ and $H_2$. This observation gives a
{\bf B\"acklund transformation} between the Kac-van Moerbeke and the Toda hierarchies:

\begin{theorem}
For any given solution $\rho(t)$ of $\km_r(\rho)=0$
we obtain, via (\ref{defakt}), two solutions $(a_j(t),b_j(t))_{j=1,2}$ of $\tl_r(a,b)=0$.
\end{theorem}

This is the analog of the Miura transformation between the modified and the
original Korteweg-de Vries hierarchies. 

{\scriptsize The Kac-van Moerbeke equation has been first introduced by Kac and van
Moerbeke in \cite{km}. The B\"acklund transformation connecting the Toda and the Kac-van
Moerbeke equations has first been considered by Toda and Wadati in \cite{tawa}.}



\section{Commutation methods}



Clearly, it is natural to ask whether this transformation can be inverted. In
other words, can we factor a given Jacobi operator $H$ as $A^* A$ and then compute
the corresponding solution of the Kac-van Moerbeke hierarchy plus the second solution
of the Toda hierarchy?

This can in fact be done. All one needs is a positive solution of the system
\begin{equation} \label{requnt}
H(t) u(n,t) = 0, \qquad \frac{d}{dt} u(n,t) = P_{2r+2}(t) u(n,t)
\end{equation}
and then one has
\bea \nn
\rho_{o}(t) &=& -\sqrt{\frac{-a(t) u(t)}{u^+(t)}}, \\
\rho_{e}(t) &=& \sqrt{\frac{-a(t) u^+(t)}{u(t)}}.
\eea
In particular, starting with the trivial solution $a_0(n,t)=-\frac{1}{2}$, $b_0(n,t)=0$
and proceeding inductively one ends up with the $N$-soliton solutions.

The method of factorizing $H$ and then commuting the factors is known as {\bf Darboux
transformation} and is of independent interest since it has the property of
inserting a single eigenvalue into the spectrum of $H$.


{\scriptsize Commutation methods for Jacobi operators in connection with the
Toda and Kac-van Moerbeke equation were first considered by
Gesztesy, Holden, Simon, and Zhao \cite{TKvM}. For further generalizations,
see Gesztesy and Teschl \cite{gtjc} and Teschl \cite{ttkm}. A second way to obtain the
$N$-soliton solution is via the inverse scattering transform, which was first
worked out by Flaschka in \cite{fl2}.}




\section{Stationary solutions}



In the remaining sections I would like to show how two at first sight unrelated
fields of mathematics, spectral theory and algebraic geometry, can be combined
to find (quasi-)periodic solutions of the Toda equations.

To reveal this connection, we first look at stationary solutions of the Toda
hierarchy or, equivalently, at commuting operators
\begin{equation}
P_{2r+2} H - H P_{2r+2}  =0.
\end{equation}
In this case a short calculation gives
\begin{equation}
(P_{2r+2} |_{\Ker(H-z)})^2 = H_{r+1}(z)^2 - 4 a^2 G_r(z) G_r^+(z) =:
R_{2r+2}(z),
\end{equation}
where $R_{2r+2}(z)$ can be shown to be independent of $n$. That is, it is
of the form
\begin{equation}
R_{2r+2}(z)= \prod_{j=0}^{2r+1} (z- E_j)
\end{equation}
for some constant numbers $E_j\in\R$. In particular, this implies
\begin{equation}
(P_{2r+2})^2 = \prod_{j=0}^{2r+1} (H- E_j)
\end{equation}
and the polynomial $w^2= \prod_{j=0}^{2r+1} (z- E_j)$ is known as the
{\bf Burchnall-Chaundy polynomial} of $P_{2r+2}$ and $H$. In particular,
the connection between the stationary Toda hierarchy and the
{\bf hyperelliptic curve}
\begin{equation}
\mathcal{K} = \{ (z,w)\in \C^2 | w^2 = \prod_{j=0}^{2r+1} (z- E_j)\}
\end{equation}
is apparent. But how can it be used to solve the Toda equation? This will be shown
next. We will for simplicity assume that our curve is nonsingular, that is,
that $E_j < E_{j+1}$ for all $j$.


{\scriptsize The fact that two commuting differential or difference operators satisfy
a polynomial relation, was first shown by Burchnall and Chaundy \cite{bc1},
\cite{bc2}. The approach to stationary solutions presented here follows again
Bulla, Gesztesy, Holden, and Teschl \cite{bght}.}




\section{Jacobi operators associated with stationary solutions}



Next some spectral properties of the Jacobi operators associated with stationary
solutions are needed. First of all, one can show that
\bea\nn
g(z,n) &=& \frac{G_r(z,n)}{R_{2r+2}^{1/2}(z)} = \spr{\delta_n}{(H-z)^{-1} \delta_n},\\ \label{defgjhj}
h(z,n) &=& \frac{H_{r+1}(z,n)}{R_{2r+2}^{1/2}(z)} = \spr{\delta_{n+1}}{(H-z)^{-1} \delta_n}.
\eea
This is not too surprising, since $g_j$ and $h_j$ are by (\ref{defgh}) just the expansion
coefficients in the Neumann series of the resolvent.

But once we know the diagonal of the resolvent we can easily read off the
spectrum of $H$. The open branch cuts of $R_{2r+2}^{1/2}(z)$ form an essential
support of the absolutely continuous spectrum and the branch points support
the singular spectrum. Since at each branch point we have a square root
singularity, there can be no eigenvalues and since the singular continuous
spectrum cannot be supported on finitely many points, the spectrum is
purely absolutely continuous and consists of a finite number of bands.\\
\begin{picture}(10,2)
\put(2.5,1){\vector(1,0){8.5}}
\put(4,0.9){\rule{9mm}{2mm}}
\put(4,0.3){$\scriptstyle E_0$}
\put(4.8,0.3){$\scriptstyle E_1$}
\qbezier(4.9,1)(5.5,2)(5.98,1)
\qbezier(4.9,1)(5.5,0)(5.98,1)
\put(5.6,1.35){$\bullet$}
\put(5.5,1.6){$\scriptstyle \mu_1(n)$}
\put(6,0.9){\rule{11mm}{2mm}}
\put(6,0.3){$\scriptstyle E_2$}
\put(7,0.3){$\scriptstyle E_3$}
\qbezier(7.1,1)(7.9,2)(8.68,1)
\qbezier(7.1,1)(7.9,0)(8.68,1)
\put(7.4,0.56){$\bullet$}
\put(7.45,0.78){$\scriptstyle \mu_2(n)$}
\put(8.7,0.9){\rule{7mm}{2mm}}
\put(8.7,0.3){$\scriptstyle E_4$}
\put(9.3,0.3){$\scriptstyle E_5$}
\end{picture}\\
The points $\mu_j(n)$ are the zeros of $G_r(z,n)$,
\begin{equation} \label{grmuj}
G_r(z,n) = \prod_{j=1}^r (z- \mu_j(n)),
\end{equation}
and can be interpreted as the
eigenvalues of the operator $H_n$ obtained from $H$ by imposing an additional
{\bf Dirichlet boundary condition} $u(n)=0$ at $n$. Since $H_n$ decomposes into a
direct sum $H_{-,n}\oplus H_{+,n}$ we can also associate a sign $\sig_j(n)$
with $\mu_j(n)$, indicating whether it is an eigenvalue of $H_{-,n}$ or $H_{+,n}$.

\begin{theorem}
The band edges $\{ E_j \}_{0 \le j \le 2r +1}$ together with the Dirichlet data
$\{ (\mu_j(n)$, $\sig_j(n)) \}_{1 \le j \le r}$ for one $n$ uniquely determine
$H$. Moreover, it is even possible to write down explicit formulas for
$a(n+k)$ and $b(n+k)$ for all $k\in\Z$ as functions of these data. Explicitly one has
\bea \nn
b(n) &=& b^{(1)}(n)\\ \nn
a(n - {\textstyle {0 \atop 1}})^2 &=& \frac{b^{(2)}(n) - b(n)^2}{4} \pm \sum_{j=1}^r
\frac{\sig_j(n) R_{2r+2}^{1/2}(\mu_j(n))}{2 \prod_{k\ne j} (\mu_j(n)-\mu_k(n))}\\ \nn
b(n\pm1) &=& \frac{1}{a(n-{0 \atop 1})^2} \Big( \frac{2
b^{(3)}(n) - 3 b(n) b^{(2)}(n) + b(n)^3}{12}\\ \nn
&&{} \pm \sum_{j=1}^r \frac{\sig_j(n) R_{2r+2}^{1/2}(\mu_j(n))
\mu_j(n)}{2 \prod_{k\ne j} (\mu_j(n)-\mu_k(n))} \Big)\\ \label{traceform}
&\vdots&
\eea
where
\begin{equation}
b^{(\ell)}(n) = \frac{1}{2} \sum_{j=0}^{2r+1} E_j^\ell - \sum_{j=1}^r \mu_j(n)^\ell.
\end{equation}
\end{theorem}

These formulas already indicate that $\hat{\mu}_j(n)=(\mu_j(n),\sig_j(n))$ should be
considered as a point on the Riemann surface $\mathcal{K}$ of $R_{2r+2}^{1/2}(z)$, where
$\sig_j(n)$ indicates on which sheet $\mu_j(n)$ lies.


{\scriptsize The result for periodic operators is due to van Moerbeke
\cite{vm}, the general case was given by Gesztesy, Krishna, and Teschl \cite{gkt}. Trace
formulas for Sturm-Liouville and also for Jacobi operators have a long history.
The formulas for $b^{(\ell)}$, $\ell=1,2$, were already given in
\cite{vm} for the periodic case. The formulas presented here and in particular
the fact that the coefficients $a$ and $b$ can be explicitly written down in terms of
minimal spectral data are due to Teschl \cite{ttr}. Most proofs use results on
orthogonal polynomials and the moment problem. One of the classical references is
\cite{ak}, for a recent review article see Simon \cite{simp}.}



\section{Algebro-geometric solutions of the Toda equations}



The idea now is to choose a stationary solution of $\tl_r(a,b)=0$ as the
initial condition for $\tl_s(a,b)$ and to consider the time evolution in our
new {\em coordinates} $\{ E_j \}_{0 \le j \le 2r +1}$ and
$\{ (\mu_j(n), \sig_j(n)) \}_{1 \le j \le r}$. From unitary equivalence of the
family of operators $H(t)$ we know that the band edges $E_j$ do not depend on
$t$. Moreover, the time evolution of the Dirichlet data follows from the
Lax equation
\begin{equation}
\frac{d}{dt} (H(t)-z)^{-1} = [P_{2s+2}(t),(H(t)-z)^{-1}].
\end{equation}
Inserting (\ref{defgjhj}) and (\ref{grmuj}) yields
\begin{equation}
\frac{d}{dt} \mu_j(n,t) = -2 G_s(\mu_j(n,t),n,t) \frac{\sig_j(n,t) R_{2r+2}^{1/2}(\mu_j(n,t))}{
\prod_{k\ne j} (\mu_k(n,t)-\mu_j(n,t))},
\end{equation}
where $G_s(z)$ has to be expressed in terms of $\mu_j$ using (\ref{traceform}).
Again, this equation should be viewed as a differential equation on $\mathcal{K}$
rather than $\R$. A closer investigation shows that each Dirichlet eigenvalue
$\mu_j(n,t)$ rotates in its spectral gap.

At first sight it looks like we have not gained too much since this flow is
still highly nonlinear, but it can be straightened out using Abel's map from
algebraic geometry. So let us review some basic facts first.

Our hyperelliptic curve $\mathcal{K}$ is in particular a compact Riemann surface
of genus $r$ and hence it has a basis of $r$ holomorphic differentials which
are explicitly given by
\begin{equation}
\zeta_j = \sum_{k=1}^r c_j(k) \frac{z^{k-1}dz}{R_{2r+2}^{1/2}(z)}.
\end{equation}
(At first sight these differentials seem to have poles at each band edge, but
near such a band edge we need to use a chart $z-E_j=w^2$ and $dz=2w dw$ shows
that each zero in the denominator cancels with a zero in the numerator). Given a
homology basis $a_j$, $b_j$ for $\mathcal{K}$ they are usually normalized such that
\begin{equation}
\int_{a_j} \zeta_k = \delta_{j,k} \qquad\mbox{and one sets}\qquad \int_{b_j} \zeta_k =: \tau_{jk}.
\end{equation}
Now the {\bf Jacobi variety} associated with $\mathcal{K}$ is the $r$-dimensional torus
$\C^r\!\!\mod L$, where $L= \Z^r + \tau\Z^r$ and the {\bf Abel map} is given by
\begin{equation}
\Amap(p) = \int_{p_0}^p \ul{\zeta} \mod L, \qquad p,p_0\in\mathcal{K}.
\end{equation}

\begin{theorem}
The Abel map straightens out the dynamical system $\hat{\mu}_j(0,0) \to \hat{\mu}_j(n,t)$
both with respect to $n$ and $t$
\begin{equation}
\sum_{j=1}^r \Amap(\hat{\mu}_j(n,t)) = \sum_{j=1}^r \Amap(\hat{\mu}_j(0,0)) - 2n
\Amap(\infty_+) - t \ul{U}_s,
\end{equation}
where $\ul{U}_s$ can be computed explicitly in terms of the band edges $E_j$.
\end{theorem}

{\em Sketch of proof}. Consider the function (compare (\ref{defrz}))
\begin{equation}
\phi(p,n,t)= \frac{H_{r+1}(p,n,t) + R_{2r+2}^{1/2}(p)}{2 a(n,t) G_r(p,n,t)}
= \frac{2 a(n,t) G_r(p,n+1,t)}{H_{r+1}(p,n,t) - R_{2r+2}^{1/2}(p)}, \quad
p\in\mathcal{K},
\end{equation}
whose zeros are $\hat{\mu}_j(n+1,t)$, $\infty_-$ and whose poles are
$\hat{\mu}_j(n,t)$, $\infty_+$. Abel's theorem implies
\begin{equation}
\Amap(\infty_+) + \sum_{j=1}^r \Amap(\hat{\mu}_j(n,t)) =
\Amap(\infty_-) + \sum_{j=1}^r \Amap(\hat{\mu}_j(n+1,t)),
\end{equation}
which settles the first claim. To show the second claim we compute
\bea \nn
\frac{d}{dt} \sum_{j=1}^r \Amap(\hat{\mu}_j) &=&
\sum_{j=1}^r \dot{\mu}_j \sum_{k=1}^r \ul{c}(k)
\frac{\mu_j^{k-1}}{\sig_j R_{2r+2}^{1/2}(\mu_j)} \\
&=& -2 \sum_{j,k}^r \ul{c}(k) \frac{G_s(\mu_j)}{\prod_{\ell\ne j}
(\mu_j-\mu_\ell)} \mu_j^{k-1}.
\eea
The key idea is now to rewrite this as an integral
\begin{equation}
\int_\Gamma \frac{G_s(z)}{G_r(z)} z^{k-1} dz,
\end{equation}
where $\Gamma$ is a closed path encircling all points $\mu_j$. By
(\ref{grmuj}) this is equal to the above expression by the
residue theorem. Moreover, since the integrand is rational we can also compute
this integral by evaluating the residue at $\infty$, which is given by
\bea \nn
\frac{G_s(z)}{G_r(z)} &=& \frac{G_s(z)}{g(z)} \frac{1}{R_{2r+2}^{1/2}(z)} =
\frac{z^{s+1}(1 + O(z^{-s}))}{R_{2r+2}^{1/2}(z)}\\
&=& -2 \sum_{\ell = \max\{1,r-s\}} \ul{c}(\ell) d_{s-r+\ell}(\ul{E}) =: \ul{U}_s,
\eea
since the coefficients of $G_s$ coincide with the first $s$ coefficients in the
Neumann series of $g(z)$ by (\ref{defgh}). Here $d_j(\ul{E})$ are just the
coefficients in the asymptotic expansion of $1/R_{2r+2}^{1/2}(z)$.
\hfill$\Box$\\

Since the poles and zeros of the function $\phi(z)$, which appeared in
the proof of the last theorem, as well as their image under the Abel map are known,
a function having the same zeros and poles can be written down using
{\bf Riemann theta functions} ({\bf Jacobi's inversion problem} and {\bf Riemann's
vanishing theorem}). The {\bf Riemann--Roch theorem} implies that both
functions coincide. Finally, the function $\phi(z)$ has also a spectral
interpretation as Weyl $m$-function, and thus explicit formulas for the coefficients
$a$ and $b$ can be obtained from the asymptotic expansion for $|z|\to\infty$.
This produces the formula in equation (\ref{thetasol}).


{\scriptsize The first results on algebro-geometric solutions of the Toda equation were
given by Date and Tanaka \cite{dt}. Further important contributions were made by Krichever,
\cite{kr4} -- \cite{kr1}, van Moerbeke and Mumford \cite{vmm}, \cite{mumtl}.
The presentation here follows Bulla, Gesztesy, Holden, and Teschl \cite{bght} respectively
Teschl \cite{tejocil}. Another possible approach is to directly use the spectral
function of $H$ and to consider its $t$ dependence, see Berezanski\u\i{} and
coworkers \cite{be2}--\cite{besh}. For some recent developments based on Lie theoretic
methods and loop groups I again recommend the review by Palais \cite{pal} as starting point.}



\section*{Acknowledgments}

I thank Fritz Gesztesy for his careful scrutiny of this article leading
to several improvements, as well as Wolfgang Bulla and Karl Unterkofler
for many valuable suggestions.




\begin{thebibliography}{XXXX}

\bibitem{ak} N. Akhiezer, {\em The Classical Moment Problem}, Oliver and
Boyd, London, 1965.
\bibitem{al} S. J. Al'ber, {\em Associated integrable systems},
J. Math. Phys. {\bf 32}, 916--922 (1991).
\bibitem{be2} Yu. M. Berezanski\u\i, {\em Integration of nonlinear
difference equations by the inverse spectral problem method},
 Soviet Math. Dokl., {\bf 31} No. 2, 264--267 (1985).
\bibitem{be3} Yu. M. Berezanski, {\em The integration of semi-infinite
Toda chain by means of inverse spectral problem},
 Rep. Math. Phys., {\bf 24} No. 1, 21--47 (1985).
\bibitem{be4} Yu. M. Berezansky, {\em Integration of nonlinear
nonisospectral difference-differential equations by means of the inverse spectral problem},
in ``{\it Nonlinear Physics. Theory and experiment\/}'', (eds E.~Alfinito, M. Boiti, L.
Martina, F. Pempinelli), World Scientific, 11--20 (1996).
\bibitem{bege} Yu. M. Berezansky and M. I. Gekhtman, {\em Inverse problem of the
spectral analysis and non-Abelian chains of nonlinear equations}, Ukrain. Math. J.,
{\bf 42}, 645--658 (1990).
\bibitem{besh} Yu. Berezansky and M. Shmoish, {\em Nonisospectral flows on semi-infinite
Jacobi matrices}, Nonl. Math. Phys., {\bf 1} No. 2, 116--146 (1994).
\bibitem{bght} W. Bulla, F. Gesztesy, H. Holden, and G. Teschl {\em
Algebro-Geometric Quasi-Periodic Finite-Gap Solutions of the Toda and Kac-van
Moerbeke Hierarchies}, Mem. Amer. Math. Soc. {\bf 135-641}, 1998.
\bibitem{bc1} J.~L.~Burchnall and T.~W.~Chaundy, {\em Commutative
ordinary differential operators}, Proc. London Math. Soc. Ser.~2,
{\bf 21}, 420--440 (1923).
\bibitem{bc2} J.~L.~Burchnall and T.~W.~Chaundy, {\em Commutative
ordinary differential operators}, Proc. Roy. Soc. London {\bf A118},
557--583 (1928).
\bibitem{dt} E. Date and S. Tanaka {\em Analogue of inverse scattering theory
for the discrete Hill's equation and exact solutions for the periodic Toda
lattice}, Prog. Theoret. Phys. {\bf 56}, 457--465 (1976).
\bibitem{dlt} P. Deift, L.C. Li, and C. Tomei, {\em Toda flows with infinitely many
variables}, J. Func. Anal. {\bf 64}, 358--402 (1985).
\bibitem{eil} S. N. Eilenberger, {\em An Introduction to Difference Equations}, Springer,
New York, 1996.
\bibitem{fad} L. Faddeev and L. Takhtajan, {\em Hamiltonian Methods in the
Theory of Solitons}, Springer, Berlin, 1987.
\bibitem{fpu} E. Fermi, J. Pasta, S. Ulam, {\em Studies of Nonlinear Problems},
Collected Works of Enrico Fermi, University of Chicago Press, Vol.II,978--988,1965.
Theory, Methods, and Applications, 2nd ed., Marcel Dekker, New York, 2000.
\bibitem{fl1} H. Flaschka, {\em The Toda lattice. I. Existence of integrals}, Phys. Rev. B {\bf 9},
1924--1925 (1974).
\bibitem{fl2} H. Flaschka, {\em On the Toda lattice. II. Inverse-scattering solution}, Progr. Theoret. Phys.
{\bf 51}, 703--716 (1974).
\bibitem{conl} F. Gesztesy and H. Holden, {\em Trace formulas and conservation
laws for nonlinear evolution equations}, Rev. Math. Phys. {\bf 6}, 51--95 (1994).
\bibitem{ghbok} F. Gesztesy and H. Holden, {\em Soliton Equations and their
Algebro-Geometric Solutions I -- III}, Cambridge Series in Advanced Mathematics, in preparation.
\bibitem{gtjc} F. Gesztesy and G. Teschl, {\em  Commutation
methods for Jacobi operators}, J. Diff. Eqs. {\bf 128}, 252--299 (1996).
\bibitem{gkt} F. Gesztesy, M. Krishna, and G. Teschl, {\em On isospectral sets
of Jacobi operators}, Com. Math. Phys. {\bf 181}, 631--645 (1996).
\bibitem{TKvM} F. Gesztesy, H. Holden, B. Simon, and Z. Zhao, {\em On the Toda
and Kac-van Moerbeke systems}, Trans. Amer. Math. Soc. {\bf 339}, 849--868 (1993).
\bibitem{km} M. Kac and P. van Moerbeke, {\em On an explicitly soluble system of
nonlinear differential equations, related to certain Toda lattices}, Adv. Math.
{\bf 16}, 160--169 (1975).
\bibitem{kr4} I. M. Krichever, {\em Algebraic curves and nonlinear difference
equations}, Russian Math. Surveys. {\bf 334}, 255--256 (1978).
\bibitem{kr2} I. M. Krichever, {\em Nonlinear equations and elliptic curves},
Rev. of Science and Technology {\bf 23}, 51--90 (1983).
\bibitem{kr3} I. M. Krichever, {\em Algebro-geometric spectral theory of the
Schr\"odinger difference operator and the Peierls model}, Soviet Math. Dokl.
{\bf 26}, 194--198 (1982).
\bibitem{kr5} I. M. Krichever, {\em The Peierls model}, Funct. Anal. Appl. {\bf 16},
248--263 (1982).
\bibitem{kr1} I. Krichever, {\em Algebraic-geometrical methods in the theory of
integrable equations and their perturbations}, Acta Appl. Math. {\bf 39},
93--125 (1995).
\bibitem{lax} P. D. Lax {\em Integrals of nonlinear equations of evolution and
solitary waves}, Comm. Pure and Appl. Math. {\bf 21}, 467--490 (1968).
\bibitem{vm} P. van Moerbeke, {\em The spectrum of Jacobi Matrices}, Inv. Math.
{\bf 37}, 45--81 (1976).
\bibitem{vmm} P. van Moerbeke and D. Mumford {\em The spectrum of difference
operators and algebraic curves}, Acta Math. {\bf 143}, 97--154 (1979).
\bibitem{mumtl} D. Mumford, {\em An algebro-geometric construction
of commuting operators and of solutions to the Toda lattice equation,
Korteweg de Vries equation and related non-linear equations},
Intl. Symp. Algebraic Geometry, 115--153, Kyoto, 1977.
\bibitem{pal} R. S. Palais, {\em The symmetries of solitons},  Bull. Amer. Math. Soc.,
{\bf 34}, 339--403 (1997).
\bibitem{rus} J. S. Russel, {\em Report on waves}, 14th Mtg. of the British Assoc. for
the Advance of Science, John Murray, London, pp. 311--390 + 57 plates, 1844.
\bibitem{simp} B. Simon, {\em The classical moment problem as a self-adjoint finite
difference operator}, Advances in Math. {\bf 137}, 82--203 (1998).
\bibitem{ttr} G. Teschl, {\em Trace Formulas and Inverse Spectral Theory for
Jacobi Operators}, Comm. Math. Phys. {\bf 196}, 175--202 (1998).
\bibitem{tist} G. Teschl, {\em Inverse scattering transform for the Toda hierarchy},
Math. Nach. {\bf 202}, 163--171 (1999).
\bibitem{ttkm} G. Teschl, {\em On the Toda and Kac-van Moerbeke hierarchies}, Math. Z.
{\bf 231}, 325--344 (1999).
\bibitem{tejocil} G. Teschl, {\em Jacobi Operators and Completely Integrable Nonlinear Lattices}, Math. Surv. and Monographs {\bf 72}, Amer. Math. Soc., Rhode Island, 2000
\bibitem{ta} M. Toda, {\em Theory of Nonlinear Lattices}, 2nd enl. ed.,
Springer, Berlin, 1989.
\bibitem{ta1} M. Toda, {\em Theory of Nonlinear Waves and Solitons}, Kluwer,
Dordrecht, 1989.
\bibitem{tas} M. Toda, {\em Selected Papers of Morikazu Toda}, ed. by M. Wadati,
World Scientific, Singapore, 1993.
\bibitem{tawa} M. Toda and M. Wadati, {\em A canonical transformation for the
exponential lattice}, J. Phys. Soc. Jpn. {\bf 39}, 1204--1211 (1975).
\bibitem{ut} K. Ueno and K. Takasaki, {\em Toda lattice
hierarchy}, in ``{\it Advanced Studies in Pure Mathematics 4\/}'',
(ed. K. Okamoto), North-Holland, Amsterdam, 1--95 (1984).
\bibitem{ut1} K. Ueno and K. Takasaki, {\em Toda lattice hierarchy. I}',
Proc. Japan Acad., Ser. A {\bf 59}, 167--170 (1983). 
\bibitem{ut2} K. Ueno and K. Takasaki, {\em Toda lattice hierarchy. II}',
Proc. Japan Acad., Ser. A {\bf 59}, 215--218 (1983).
\bibitem{zakr} N. J. Zabusky and M. D. Kruskal, {\em Interaction of solitons in a
collisionless plasma and the recurrence of initial states}, Phys. Rev. Lett. {\bf 15},
240--243 (1965).
\end{thebibliography}
\vspace{2ex}

\parbox{11.02cm}{\small
Gerald Teschl\\
Institut f\"ur Mathematik\\
Strudlhofgasse 4\\ A-1090 Wien\\
Email: \href{mailto:Gerald.Teschl@univie.ac.at}{Gerald.Teschl@univie.ac.at}\\
URL: \href{http://www.mat.univie.ac.at/~gerald/}{http://www.mat.univie.ac.at/\string~gerald/}
\hfill\it
Eingegangen 5.11.2001}


\end{document}
