%% @texfile{
%%     filename="TodaKM.tex",
%%     version="1.1",
%%     date="May-1997",
%%     cdate="19970510",
%%     filetype="LaTeX2e",
%%     journal="Math. Z. 231, 325-344 (1999)",
%%     doi="10.1007/PL00004732",
%%     copyright="Springer".
%%     }


\documentclass{amsart}
%\newcommand{\href}[2]{ #2 }
\usepackage{hyperref}



%%%%%%%%%THEOREMS%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newtheorem{thm}{Theorem}[section]
\newtheorem{lemma}[thm]{Lemma}
\newtheorem{coro}[thm]{Corollary}
\newtheorem{hypo}[thm]{Hypothesis {\bf H.}\hspace*{-0.6ex}}
\newtheorem{rem}[thm]{Remark}

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

%%%%%%%%%%%%%%%%%%ABBRS%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\ti}{\tilde}
\newcommand{\bs}{\backslash}
\newcommand{\nn}{\nonumber}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}
\newcommand{\ba}{\begin{array}}
\newcommand{\ea}{\end{array}}
\newcommand{\D}{\displaystyle}
\newcommand{\ol}{\overline}
\newcommand{\ul}{\underline}

\newcommand{\hr}{{\mathfrak H}}
\newcommand{\db}{{\mathfrak D}}
\newcommand{\spr}[2]{\langle #1 , #2 \rangle}
\newcommand{\id}{{\rm 1\hspace{-0.6ex}l}}
\newcommand{\I}{\mathrm{i}}
\newcommand{\re}{\mathrm{Re}}
\newcommand{\Ker}{\mathrm{Ker}}
\newcommand{\im}{\mathrm{Im}}
\newcommand{\lz}{\ell^2(\Z)}
\newcommand{\tl}{\mathrm{TL}}
\newcommand{\km}{\mathrm{KM}}


%%%%%%%%%%%%%%%GREEK%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\eps}{\varepsilon}
\newcommand{\sig}{\sigma}
\newcommand{\lam}{\lambda}
\newcommand{\gam}{\gamma}


%%%%%%%%%%%%%%%%%%%%%%%%NUMBERING%%%%%%%%%%%%%%%%%%

\numberwithin{equation}{section}
\newcounter{me}

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


\begin{document}

\title{On the Toda and Kac-van Moerbeke Hierarchies}

\author{Gerald Teschl}
\address{Institut f\"ur Reine und Angewandte Mathematik\\
RWTH Aachen\\ 52056 Aachen\\ Germany}
\curraddr{Institut f\"ur Mathematik\\
Strudlhofgasse 4\\ 1090 Wien\\ Austria}
\email{Gerald.Teschl@univie.ac.at}
\urladdr{http://www.mat.univie.ac.at/\string~gerald/}

\thanks{{\it Math. Z. {\bf 231}, 325-344 (1999)}}

\keywords{Toda hierarchy, Kac-van Moerbeke hierarchy, commutation method}
\subjclass{Primary 58F07, 58F37; Secondary 39A10, 47B39}


\begin{abstract}
We provide a comprehensive treatment of the single and double commutation method 
as a tool for constructing soliton solutions of the Toda and Kac-van Moerbeke
hierarchy on arbitrary background. In addition, we present a novel construction based on
the single commutation method. As an illustration we compute the 
$N$-soliton solution of the Toda and Kac-van Moerbeke hierarchy.
\end{abstract}

\maketitle


\section{Introduction}

In 1968 Peter Lax \cite{lax} presented his famous approach for completely integrable
nonlinear evolution equations by rewriting such equations as linear evolution
equations for linear operators, viz.
\begin{equation} \label{laxappr}
\frac{d}{dt} H(t) = [P(t), H(t)],
\end{equation}
where $[P,H]=PH-HP$ denotes the usual commutator. Under suitable conditions,
(\ref{laxappr}) will imply existence of a unitary propagator $U(s,t)$ for $H(t)$, that is,
\begin{equation}
H(t) = U(t,s) H(s) U(s,t), \quad U(t,s)^* = U(t,s)^{-1} = U(s,t).
\end{equation}
As a trivial consequence one concludes that the norm $\|H(t)\|$ of $H(t)$ is independent of
$t$. And it is this, at first sight innocent looking, fact which provides a {\em uniform}
bound on the matrix coefficients of $H(t)$ and hence implies uniqueness and (global)
existence of solutions for the associated nonlinear evolution equation (see
Theorem~\ref{thmexistandunique} below). It seems like this last fact has not been 
used in the literature before and that Theorem~\ref{thmexistandunique} is the
first general existence and uniqueness result for bounded solutions of the 
Toda lattice on the whole line (for the case of the half line see \cite{dlt}, Proposition~1).

The purpose of the present paper is to revisit methods for constructing soliton solutions
on arbitrary background and exploit the abovementioned uniqueness and existence result
to obtain short and elegant proofs for these methods. In particular, we want to simplify
and improve the results of \cite{TKvM} and, at the same time, extend all methods to the entire
hierarchy.

To set the stage we review some basic facts on the Toda and
Kac-van Moerbeke hierarchy in our first two sections. Then we provide a
detailed investigation of the system
\begin{equation}
H(t) u = z u, \quad \frac{d}{dt} u = P(t) u, \quad z\in\C
\end{equation}
(in the weak sense) whose solutions are needed in the explicit construction of soliton
solutions. This section, in some sense, constitutes the technical heart of this paper.

Our final section  will then establish the single and double commutation method as a tool
for constructing soliton solutions on arbitrary background for the entire Toda and
Kac-van Moerbeke hierarchy. In addition, we will show how the single commutation
method can be used in a way which has not been noted in the literature before
(Theorem~\ref{thmsolddm}). As an explicit illustration we compute the
$N$-soliton solution of the Toda and Kac-van Moerbeke hierarchy.





\section{The Toda hierarchy}


In this section we introduce the Toda hierarchy using the standard Lax formalism
(\cite{lax}). We first review some basic facts following \cite{bght} and then
we prove existence and uniqueness for the initial value problem.

We will only consider bounded solutions and hence require

\begin{hypo} \label{habt}
Suppose $a(t),b(t)$ satisfy
\begin{equation}
a(t) \in \ell^\infty(\Z,\R), \:\: b(t) \in \ell^\infty(\Z,\R), \:\: a(n,t)\ne 0
\:\: (n,t) \in \Z\times\R,
\end{equation}
and  let $t \mapsto (a(t),b(t))$ be Fr\'{e}chet differentiable in the Banach space
$\ell^\infty(\Z)\oplus\ell^\infty(\Z)$. 
\end{hypo}

Associated with $a(t), b(t)$ is a Jacobi operator
\begin{equation}
\ba{lccl} H(t) :& \lz & \to & \lz \\ & f &\mapsto& \tau(t) f \ea,
\end{equation}
where
\begin{equation}
\tau(t) f(n)= a(n,t) f(n+1) + a(n-1,t) f(n-1) + b(n,t) f(n)
\end{equation}
and $\lz$ denotes the Hilbert space of square summable (complex-valued) sequences
over $\Z$. Moreover, choose constants $c_0=1$, $c_j$, $1\le j \le r$, $c_{r+1}=0$, set
\bea \nn
g_j(n,t) &=& \sum_{\ell=0}^j c_{j-\ell} \spr{\delta_n}{H(t)^\ell \delta_n}, \\
\label{rectodah}
h_j(n,t) &=& 2 a(n,t) \sum_{\ell=0}^j c_{j-\ell}  \spr{\delta_{n+1}}{H(t)^\ell
\delta_n} + c_{j+1}
\eea
and consider the Lax operator
\begin{equation}  \label{btgptdef}
P_{2r+2}(t) = -H(t)^{r+1} + \sum_{j=0}^r ( 2a(t) g_j(t) S^+ -h_j(t)) H(t)^{r-j} +
g_{r+1}(t),
\end{equation}
where $S^\pm f(n) = f(n\pm1)$. Clearly, (H.\ref{habt}) implies
Fr\'{e}chet differentiability of $t \mapsto H(t)$ and $t \mapsto P_{2r+2}(t)$.
Restricting to the two-dimensional nullspace $\Ker(\tau(t) -z)$, $z\in\C$ of
$\tau(t)-z$ (in $\ell(\Z)$), we have the following representation of $P_{2r+2}(t)$
\begin{equation} \label{btqptFG}
P_{2r+2}(t)\Big|_{\Ker(\tau(t)-z)} =2a(t) G_r(z,t) S^+ - H_{r+1}(z,t),
\end{equation}
where $G_r(z,n,t)$ and $H_{r+1}(z,n,t)$ are monic polynomials in $z$ of the
type
\bea \nn
G_r(z,n,t) &=& \sum_{j=0}^r z^j g_{r-j}(n,t),\\ \label{fgdef}
H_{r+1}(z,n,t) &=& z^{r+1} + \sum_{j=0}^r z^j h_{r-j}(n,t) - g_{r+1}(n,t).
\eea
A straightforward computation shows that the Lax equation
\begin{equation} \label{laxp}
\frac{d}{dt} H(t) -[P_{2r+2}(t), H(t)]=0, \quad t\in\R
\end{equation}
is equivalent to
\bea \label{tlrabo}
\tl_r (a(t), b(t))_1 &=& \dot{a}(t) -a(t) \Big(g_{r+1}^+(t) -
g_{r+1}(t) \Big)=0,\\
\tl_r (a(t), b(t))_2 &=& \dot{b}(t) - \Big(h_{r+1}(t) -h_{r+1}^-(t) \Big)=0,
\eea
where the dot denotes a derivative with respect to $t$ and $f^\pm(n)=f(n\pm 1)$.
Varying $r\in \N_0$ yields the Toda hierarchy (TL hierarchy)
\begin{equation} \label{todahi}
\tl_r(a,b) =(\tl_r (a,b)_1, \tl_r (a,b)_2) =0, \quad r\in\N_0.
\end{equation}

Next, we want to make use of the fact that $H(t)$, $\I P_{2r+2}(t)$ are both bounded
and self-adjoint operators. We start with some preliminary definitions.

Let $P(t)$, $t\in\R$, be a family of bounded skew-adjoint operators in some (separable)
Hilbert space $\hr$. A two parameter family of operators
$U(t,s)$, $(t,s)\in\R^2$, is called a unitary propagator for $P(t)$, if
\begin{enumerate}
\item $U(t,s)$, $s,t\in\R^2$ is unitary.
\item $U(t,t) = \id$ for all $t\in\R$.
\item $U(t,s) U(s,r) = U(t,r)$ for all $(r,s,t) \in\R^3$.
\item The map $t\mapsto U(t,s)$ is Fr\'{e}chet differentiable
in the Banach space ${\mathfrak B}(\lz)$ of bounded linear operators and
\begin{equation}
\frac{d}{dt} U(t,s) = P(t)U(t,s), \quad (t,s)\in\R^2.
\end{equation}
\end{enumerate}
With this notation the following well-known theorem from functional analysis
holds (essentially \cite{rs2}, Theorem~X.69).

\begin{thm}
Let $P(t)$, $t\in\R$ be a family of bounded skew-adjoint operators
such that $t\mapsto P(t)$ is Fr\'{e}chet differentiable.
Then $P(t)$ has a unitary propagator $U(t,s)$ in $\hr$.
\end{thm}

Note also $d/dt\, U(s,t) = -U(s,t) P(t)$, $(t,s)\in\R^2$. Applied to our
situation this gives another well-known result. 

\begin{lemma}
Let $a(t),b(t)$ satisfy $\tl_r(a,b)=0$ and (H.\ref{habt}). Then equation
(\ref{laxp}) implies the existence of a unitary propagator $U_r(t,s)$ for $P_{2r+2}(t)$
such that
\begin{equation} \label{htuehs}
H(t) = U_r(t,s) H(s) U_r(t,s)^{-1}, \quad (t,s)\in \R^2. 
\end{equation}
Thus all operators $H(t)$, $t \in\R$ are unitarily equivalent.

In addition, if $\psi(s) \in \lz$ solves $H(s) \psi(s) = z \psi(s)$ then the
function
\begin{equation} \label{unittranst}
\psi(t) = U_r(t,s) \psi(s),
\end{equation}
fulfills
\begin{equation} \label{syshp}
H(t) \psi(t) = z \psi(t), \qquad \frac{d}{dt} \psi(t) = P_{2r+2}(t) \psi(t).
\end{equation}
\end{lemma}

Before we proceed with our investigations of the Toda equations, we want
to ensure existence and uniqueness of global solutions. For the half line
this has been done in \cite{dlt}, Proposition~1. Unfortunately, this proof
cannot be easily adapted to the present setting on the full line. However,
since we are using Fr\'{e}chet rather than weak differentiability here, a
much simpler argument works.

We now regard the Toda equations as a flow on the Banach space
\begin{equation}
M = \ell^\infty(\Z) \oplus \ell^\infty(\Z).
\end{equation}

\begin{thm} \label{thmexistandunique}
Suppose $(a_0,b_0) \in M$. Then there exists a unique integral curve
$t \mapsto (a(t),b(t))$ in $C^\infty(\R,M)$ of the Toda equations, that is,
$\tl_r(a(t),b(t))=0$, such that $(a(0),b(0)) = (a_0,b_0)$.
\end{thm}

\begin{proof}
The Toda equation gives rise to a vector field $X_r$ on $M$, that is,
\begin{equation}
\frac{d}{dt} (a(t),b(t)) = X_r(a(t),b(t)) \quad\Leftrightarrow\quad
\tl_r(a(t),b(t))=0.
\end{equation}
Since this vector field has a simple polynomial dependence in $a$ and $b$ it
is clearly smooth (i.e., of class $C^\infty$ --- however, Lipschitz
continuous would be sufficient for our purpose). Hence by \cite{amr},
Theorem~4.1.5 solutions for the initial value problem exist locally and are
unique. In addition, by equation (\ref{htuehs}) we have $\| a(t) \|_\infty +
\| b(t) \|_\infty \le 2\| H(t) \| = 2\| H(0) \|$ (at least locally). Thus
any integral curve $(a(t),b(t))$ is bounded on finite $t$-intervals and
Proposition~4.1.22 of \cite{amr} implies global existence.
\end{proof}


\section{The Kac-van Moerbeke hierarchy and its relation to the Toda hierarchy}



In this section we review some basic properties of the Kac-van Moerbeke hierarchy and
its connection with the Toda hierarchy. 

Suppose $\rho(t)$ satisfies

\begin{hypo} \label{hrho}
Let
\begin{equation} 
\rho(t) \in\ell^\infty(\Z,\R), \quad \rho(n,t)\neq 0, \;  (n,t)\in\Z\times\R
\end{equation}
and  let $t \mapsto \rho(t)$ be Fr\'{e}chet differentiable in the Banach space
$\ell^\infty(\Z)$. 
\end{hypo}

Define the ``even'' and ``odd'' parts of $\rho(t)$ by
\begin{equation} \label{rhoeo}
\rho_e (n,t) =\rho(2n,t), \; \rho_o (n,t) =\rho (2n+1,t),\quad (n,t)
\in\Z\times \R
\end{equation}
and consider the bounded operators (in $\ell^2 (\Z)$)
\begin{equation}
A(t) =\rho_o(t) S^+ + \rho_e(t), \; A(t)^* =\rho_o^-(t) S^- +\rho_e(t).
\end{equation}
In addition, we set
\begin{equation}
H_1(t) =A(t)^* A(t), \quad H_2 (t) =A(t) A(t)^*,
\end{equation}
with
\begin{equation}
H_k(t) =a_k (t) S^+ +a_k^- (t) S^- +b_k (t), \qquad k =1,2
\end{equation}
and
\bea \label{defaot}
a_1(t) = \rho_e(t) \rho_o(t), &\qquad& b_1(t) = \rho_e(t)^2 +\rho_o^-(t)^2,
\\ \label{defatt}
a_2(t) = \rho_e^+(t) \rho_o(t), &\qquad& b_2(t) = \rho_e(t)^2 +\rho_o(t)^2.
\eea

Now we define operators $D(t)$, $Q_{2r+2}(t)$ (the Lax pair) in
$\ell^2(\Z,\C^2)$ as follows,
\bea
D(t) &=& \left( \ba{cc} 0 & A(t)^* \\ A(t) & 0 \ea \right),\\ \label{defQtrpt}
Q_{2r+2}(t) &=& \left( \ba{cc} P_{1,2r+2}(t) & 0 \\ 0 & P_{2,2r+2}(t)
\ea \right),
\eea
$r\in\N_0$. Here $P_{k,2r+2}(t)$, $k=1,2$ are defined as in (\ref{btgptdef}),
that is,
\bea \nn
&& P_{k,2r+2} (t) = -H_k(t)^{r+1} +\sum_{j=0}^r ( 2 a_k(t)
g_{k,j}(t) S^+ - h_{k,j}(t)) H_k(t)^j +g_{k,r+1},\\
&& P_{k,2r+2}(t) \Big|_{\Ker (\tau_k (t) -z)}
= 2a_k(t) G_{k,r}(z,t) S^+ - H_{k,r+1}(z,t),
\eea
$\{g_{k,j}(n,t)\}_{0\le j \le r}$, $\{h_{k,j}(n,t)\}_{0\le j
\le r+1}$ are defined as in (\ref{rectodah}), and the polynomials $G_{k,r}(z,n,t)$,
$H_{k,r+1}(z,n,t)$ are defined as in (\ref{fgdef}). Moreover, we choose the same
integration constants in $P_{1,2r+2}(t)$ and $P_{2,2r+2}(t)$ (i.e.,
$c_{1,\ell}=c_{2,\ell} \equiv c_\ell, \: 1 \le \ell \le r$).

Analogous to equation (\ref{laxp}) one obtains that
\begin{equation} \label{laxkm}
\frac{d}{dt} D(t) - [Q_{2r+2}(t), D(t)] =0
\end{equation}
is equivalent to
\bea \nn
\ul{\km}_r(\rho) &=& (\km_r (\rho)_e, \; \km_r(\rho)_o)\\
&=& \left( \ba{cc}
\dot{\rho}_e - \rho_e(g_{2,r+1} -g_{1,r+1}) \\
\dot{\rho}_o + \rho_o(g_{2,r+1} -g_{1,r+1}^+) \ea \right) =0.
\label{ulkmhie}
\eea
As in the Toda context (\ref{todahi}), varying $r\in\N_0$ yields the Kac-van Moerbeke
hierarchy ($\km$ hierarchy) which we denote by
\begin{equation} \label{kmhie}
\km_r(\rho) =0, \quad r\in\N_0.
\end{equation}

Again the Lax equation (\ref{laxkm}) implies

\begin{thm} \label{thmexupkm}
Let $\rho$ satisfy (H.\ref{hrho}) and $\km(\rho)=0$. Then the Lax equation
(\ref{laxkm}) implies the existence of a unitary propagator $V_r(t,s)$
such that we have
\begin{equation}
D(t) = V_r(t,s) D(s) V_r(t,s)^{-1}, \qquad (t,s)\in \R^2.
\end{equation}
Thus all operators $D(t)$, $t \in\R$ are unitarily equivalent.
\end{thm}

And as in Theorem~\ref{thmexistandunique} we infer

\begin{thm} \label{thmexistanduniquekm}
Suppose $\rho_0 \in \ell^\infty(\Z)$. Then there exists a unique integral curve
$t \mapsto \rho(t)$ in $C^\infty(\R,\ell^\infty(\Z))$ of the Kac-van Moerbeke
equations, that is, $\km_r(\rho) = 0$, such that $\rho(0) = \rho_0$.
\end{thm}

As a simple consequence of Theorem~\ref{thmexupkm} we have
\begin{equation}
\frac{d}{dt} D(t)^2 - [Q_{2r+2}(t), D(t)^2] =0
\end{equation}
and observing
\begin{equation}
D(t)^2 = \left(\ba{cc} H_1(t) & 0\\ 0 & H_2(t) \ea\right)
\end{equation}
yields the implication
\begin{equation} \label{kmimpltl}
\km_r(\rho) =0 \Rightarrow \tl_r (a_k, b_k)=0, \quad k=1,2,
\end{equation}
that is, given a solution $\rho$ of the $\km_r$ equation (\ref{kmhie}), one obtains two
solutions, $(a_1, b_1)$ and $(a_2, b_2)$, of the $\tl_r$ equations (\ref{todahi}) related
to each other by the Miura-type (\cite{mi}) transformations (\ref{defaot}), (\ref{defatt}).
Note that due to (H.\ref{hrho}), $(a_1,b_1)$ and $(a_2,b_2)$ both fulfill (H.\ref{habt}).

In addition, we can define
\begin{equation}
\phi_1(n,t) = -\frac{\rho_e(n,t)}{\rho_o(n,t)}, \qquad \phi_2(n,t) =
- \frac{\rho_o(n,t)}{\rho_e(n+1,t)}.
\end{equation}
This implies
\begin{equation}
a_k(n,t) \phi_k(n,t) + \frac{a_k(n-1,t)}{\phi_k(n-1,t)} + b_k(n,t) =0,
\end{equation}
and
\bea \nn
&&\frac{d}{dt} \ln\phi_k(n,t) = -2 a_k(n,t) ( g_{k,r}(n,t) \phi_k(n,t)
+ g_{k,r}(n+1,t) \phi_k(n,t)^{-1} ) \\ \nn &&\quad {}-
2 b_k(n+1,t) g_{k,r}(n+1,t) + (g_{k,r+1}(n+1,t) - g_{k,r+1}(n,t)) \\
&&\quad {}- (h_{k,r}(n+1,t) - h_{k,r}(n,t)).
\eea
Hence we infer
\begin{equation}
H_k(t) u_k(n,t) = 0, \qquad \frac{d}{dt} u_k(n,t) = P_{k,2r+2}(t) u_k(n,t)
\end{equation}
(in the weak sense, i.e., $u_k$ is not necessarily square summable), where
\bea \nn
u_k(n,t) &=& \exp \Big( \int_{t_0}^t ( 2a_k(n_0,x)
g_{k,r}(n_0,x) \phi_k(n_0,x) - h_{k,r}(n_0,x)\\ \label{defuj}
&& {} + g_{k,r+1}(n_0,x)) dx\Big)
\left\{ \ba{c@{\quad\mbox{for }}l}
\prod\limits_{m=n_0}^{n-1} \phi_k(m,t) & n > n_0 \\ 1 & n=n_0\\
\prod\limits_{m=n}^{n_0-1} \phi_k(m,t)^{-1} & n < n_0 \ea \right. .
\eea
Furthermore, explicitly writing out (\ref{laxkm}) shows that if
\begin{equation} \label{fuequj}
H_k(t) u_k(z,n,t) = z u_k(z,n,t), \qquad \frac{d}{dt} u_k(z,n,t) = P_{k,2r+2}(t) u_k(z,n,t)
\end{equation}
holds (weakly) for $u_1(z,n,t)$ (resp.\ $u_2(z,n,t)$) then it also holds for
$u_2(z,n,t)= A u_1(z,n,t)$ (resp.\ $u_1(z,n,t)= A^* u_2(z,n,t)$).

Summarizing:

\begin{thm} \label{thmrltkmu}
Suppose $\rho$ satisfies (H.\ref{hrho}) and $\km_r(\rho)=0$. Then
$(a_k,b_k)$, $k=1,2$ satisfies (H.\ref{habt}) and $\tl_r(a_k,b_k)=0$, $k=1,2$.
In addition, if $u_1(z,n,t)$ (resp.\ $u_2(z,n,t)$) is a weak solution of (\ref{fuequj})
then so is $u_2(z,n,t)= A u_1(z,n,t)$ (resp.\ $u_1(z,n,t)= A^* u_2(z,n,t)$).
\end{thm}




\section{Time evolution of solutions of the Jacobi equation}



The objective of this section is to investigate weak solutions of the system
(\ref{syshp}). As a first step we  try to calculate the time evolution of the
fundamental matrix $\Phi(z,n,t)$ corresponding to the difference equation
$\tau(t) u = z u$, that is,
\begin{equation}
\Phi(z,n,t) = \left( \ba{cc}  c(z,n,t) & s(z,n,t) \\
c(z,n+1,t) & s(z,n+1,t) \ea \right)
\end{equation}
is the matrix valued solution of $\tau(t) \Phi = z \Phi$ corresponding to the
initial condition $\Phi(z,0,t)=\id$. We assume that $a(t),b(t)$ satisfy
$\tl_r(a,b)=0$. First observe, that (\ref{laxp}) implies
\begin{equation}
(H(t)-z)(\frac{d}{dt} - P_{2r+2}(t)) \Phi(z,.,t) = 0.
\end{equation}
But this means
\begin{equation}
(\frac{d}{dt} - P_{2r+2}(t)) \Phi(z,.,t) = \Phi(z,.,t) C_r(z,t),
\end{equation}
for a certain matrix $C_r(z,t)$. If we evaluate the above expression at $n=0$
using $\Phi(z,0,t)=\id$ we obtain
\bea \nn
&& C_r(z,t) = P_{2r+2}(t) \Phi(z,0,t)\\ && \quad = 
\left(\ba{c@{\:\:}c} -H_{r+1}(z,0,t) & 2a(0,t) G_r(z,0,t) \\
-2a(0,t) G_r(z,1,t) & 2(z-b(1,t)) G_r(z,1,t) - H_{r+1}(z,1,t)
\ea\right).
\eea
and hence
\begin{equation} \label{dotPhi}
\dot{\Phi}(z,n,t) = P_{2r+2}(t) \Phi(z,n,t) + \Phi(z,n,t) C_r(z,t).
\end{equation}

This result enables us to prove

\begin{lemma} \label{lemHB}
Assume (H.\ref{habt}) and suppose $\tl_r(a,b)=0$. Let $u_0(z,n)$ be a weak 
solution of $H(0) u_0 = z u_0$.  Then the system
\begin{equation} \label{systemHB}
H(t) u(z,n,t) = z u(z,n,t), \quad \frac{d}{dt} u(z,n,t) = P_{2r+2}(t) u(z,n,t),
\end{equation}
has a unique weak solution fulfilling the initial condition
\begin{equation}
u(z,n,0)=u_0(z,n).
\end{equation}
If $u_0(z,n)$ is continuous (resp.\ holomorphic) with respect to $z$ then so is
$u(z,n,t)$.

Furthermore, if $u_{1,2}(z,n,t)$ both solve (\ref{systemHB}) then
\begin{equation} \label{systemHBwr}
W_n(u_1(z,t),u_2(z,t)) = a(n,t) \Big( u_1(z,n,t) u_2(z,n+1,t) - u_1(z,n+1,t)
u_2(z,n,t)\Big),
\end{equation}
depends neither on $n$ nor on $t$.
\end{lemma}

\begin{proof}
Clearly, any solution $u(z,n,t)$ of the system (\ref{systemHB}) can be written  
as
\begin{equation}
u(z,n,t) = u(z,0,t) c(z,n,t) + u(z,1,t) s(z,n,t),
\end{equation}
and from (\ref{dotPhi}) we infer that (\ref{systemHB}) is equivalent to
\begin{equation}
\left(\ba{c} \dot{u}(z,0,t) \\ \dot{u}(z,1,t) \ea\right) = - C_r(z,t)
\left(\ba{c} u(z,0,t) \\ u(z,1,t) \ea\right), \quad \left(\ba{c} u(z,0,0) \\
u(z,1,0) \ea\right) = \left(\ba{c} u_0(z,0) \\ u_0(z,1) \ea\right),
\end{equation}
which proves the first assertion. The second is a straightforward calculation
using (\ref{btqptFG}) and $\dot{a} = a ( H_{r+1}^+ +H_{r+1}- 2(z-b^+) G_r^+ )$.
\end{proof}

In the special case $r=0$ this result was first given in \cite{TKvM}, Lemma~2.4.
Next, let us verify some additional properties of solutions of (\ref{systemHB}). The
following result was first observed in \cite{gtjc} for the special case $r=0$,
$\lam<\sig(H)$.

\begin{lemma} \label{lemfpm}
Let $u_{\pm,0}(z,n)$ be a solution of $H(0) u = z u$, $z\in\C$ which is square summable
near $\pm\infty$. Then the solution $u_\pm(z,n,t)$ of the system (\ref{systemHB}) with
initial data $u_{\pm,0}(z,n)$ is square summable near $\pm\infty$ for all $t$.
\end{lemma}

\begin{proof}
We only prove the $-$ case (the $+$ case follows from reflection) and drop $z$ for
notational simplicity. By Lemma~\ref{lemHB} we have a solution $u(n,t)$ of
(\ref{systemHB}) with initial condition
$u(n,0)=u_{+,0}(n)$ and hence
\begin{equation}
S(n,t)  = S(n,0) + 2 \int_0^t \re \sum_{j=-n}^0 \ol{u(j,s)} P_{2r+2}(s) u(j,s) ds,
\end{equation}
where $S(n,t) =\sum_{j=-n}^0 |u(j,t)|^2$. Next, by boundedness of $a(t), b(t)$ we can find a 
constant $C>0$ such that $4 |H_{r+1}(n,t)| \le C$ and $8 |a(n,t) G_r(n,t)| \le C$. Using
(\ref{btqptFG}) and Cauchy's inequality implies
\begin{equation}
|\sum_{j=-n}^0 \ol{u(j,s)} P_{2r+2}(s) u(j,s)| \le \frac{C}{2} \Big(|u(1,s)|^2 +S(n,s) \Big).
\end{equation}
Invoking Gronwall's inequality shows
\begin{equation}
S(n,t) \le \Big(S(n,0) + C \int_0^t |u(1,s)|^2 \mathrm{e}^{-Cs} ds\Big) \mathrm{e}^{Ct}
\end{equation}
and letting $n\to\infty$ completes the proof.
\end{proof}

We finish this section by investigating positive solutions of (\ref{systemHB}).

\begin{lemma} \label{lemfpos}
Suppose $\lam\le\sig(H(0))$ and $a(n,t)<0$. Then $u_0(\lam,n)>0$ implies
that the solution $u(\lam,n,t)$ of (\ref{systemHB}) with initial condition
$u_0(\lam,n)$ is positive.
\end{lemma}

\begin{proof}
Shifting $H(t) \to H(t)-\lam$ we can assume $\lam=0$. Now
use $u_0(0,n)>0$ to define $\rho_0(n)$ by $\rho_{0,o}(n) =
-\sqrt{-a(n,0) u_0(0,n) / u_0(0,n+1)}$ and $\rho_{0,e}(n) = \sqrt{-a(n,0) u_0(0,n+1) /
u_0(0,n)}$. By Theorem~\ref{thmexistanduniquekm} we have a corresponding
solution $\rho(n,t)$ of the $\km$ hierarchy and hence (by (\ref{kmimpltl})) two solutions
$a_j(n,t),b_j(n,t)$ of the $\tl$ hierarchy. Since $a_1(n,0)=a(n,0)$ and $b_1(n,0)=b(n,0)$
we infer $a_1(n,t)=a(n,t)$ and $b_1(n,t)=b(n,t)$ by uniqueness
(Theorem~\ref{thmexistandunique}). Finally, we conclude $u(0,n,t) = u_0(0,n_0) u_1(n,t)
>0$ (with $u_1(n,t)$ as in (\ref{defuj})) again by uniqueness (Theorem~\ref{lemHB}).
\end{proof}

In the special case $r=0$ and under the additional assumption $b(n,0)/a(n,0) = o(|n|)$
as $n\to\pm\infty$, this result has first been proven in \cite{TKvM}, Lemma~2.6. We remark
that for $\lam<\sig(H(0))$ it also follows from Lemma~\ref{lemfpm} and \cite{tosc}, Lemma~A.2.

Recall that positive solutions of $\tau(t)u=\lam u$ ($a(n,t)<0$) can be characterized in
terms of minimal (also principal or recessive) positive solutions (\cite{bd}, \cite{crit},
\cite{pat}). Suppose $H(t)-\lam\ge 0$, then the minimal positive solution $u_+(\lam,n,t)$
near $+\infty$ is determined by
\begin{equation}
\frac{u_+(\lam,n,t)}{u(\lam,n,t)} < \frac{u_+(\lam,0,t)}{u(\lam,0,t)},\: n\in\N, \quad
\lim_{n\to+\infty} \frac{u_+(\lam,n,t)}{u(\lam,n,t)} =0
\end{equation}
for any linearly independent solution $u(\lam,n,t)$ with $u(\lam,n,t)>0$, $n\in\N$.
Similarly for $u_-(\lam,n,t)$, the minimal solution near $-\infty$.

Two cases may occur
\begin{enumerate}
\item[(i).] $u_-(\lam,n,t)$, $u_+(\lam,n,t)$ are linearly dependent and there is only one
(up to constant multiples) positive solution. ($H(t)-\lam$ is critical.)\\
\item[(ii).] $u_-(\lam,n,t)$, $u_+(\lam,n,t)$ are linearly dependent and
\begin{equation}
u_\sig(\lam,n,t) = \frac{1+\sig}{2} u_+(\lam,n,t) + \frac{1-\sig}{2} u_-(\lam,n,t),
\end{equation}
is positive if and only if $\sig\in[-1,1]$. ($H(t)-\lam$ is subcritical.)
\end{enumerate}
In case (ii) one can easily show that for two positive solutions
$u_j(\lam,n,t)$, $j=1,2$ we have
\begin{equation} \label{charminsolt}
u_\sig(\lam,n,t) = \frac{1+\sig}{2} u_1(\lam,n,t) + \frac{1-\sig}{2}
u_2(\lam,n,t)>0 \quad \Leftrightarrow\quad \sig\in[-1,1],
\end{equation}
if and only if $u_{1,2}$ equal $u_\pm$ up to constant (w.r.t.\ $n$) multiples.


\begin{lemma}  \label{lemfmin}
Let $\lam\le\sig(H(0))$ and $a(n,t)<0$. Suppose $u(\lam,n,t)$ solves
(\ref{systemHB}) and is a minimal positive solution for one $t=t_0$, then this holds for
all $t\in\R$. In particular, $H(t)-\lam$ is critical (resp. subcritical) for all $t\in\R$ if
and only if it is critical for one $t=t_0$.
\end{lemma}

\begin{proof}
Since linear independence and positivity is preserved by the system (\ref{systemHB}) (by
(\ref{systemHBwr}) and Lemma~\ref{lemfpos}) $H(t)-\lam$ is critical (resp. subcritical)
for all $t\in\R$ if and only if it is critical for one $t=t_0$. If $H(t)-\lam$ is subcritical
we note that the characterization (\ref{charminsolt}) of minimal solutions is independent
of
$t$. Hence it could only happen that $u_+(\lam,n,t)$ and $u_-(\lam,n,t)$ change place
during time evolution. But this would imply $u_+(\lam,n,t)$ and $u_-(\lam,n,t)$ are
linearly dependent at some intermediate time contradicting $H(t)-\lam$ subcritical.
\end{proof}

That minimal solutions remain minimal is suggested by the notation
chosen in \cite{TKvM}, Theorem~2.9 (however, no proof is given).
The remaining assertion for the special case $r=0$ with the additional assumption
$b(n,0)/a(n,0) = o(|n|)$ as $n\to\pm\infty$ corresponds to \cite{TKvM}, Lemma~2.10.
Again, for $\lam<\sig(H)$ the lemma already follows from Lemma~\ref{lemfpm}.

In particular, this shows that the choice in (\ref{ulamsig}) below is exhaustive.





\section{$N$-soliton solutions on an arbitrary background}




In Theorem~\ref{thmrltkmu} we saw, that from one solution $\rho$ of 
$\km_r(\rho)=0$ we can get two solutions $(a_1,b_1)$, $(a_2,b_2)$ of
$\tl_r(a,b)=0$. In this section we want to invert this process. 

Suppose $(a,b)$ satisfies (H.\ref{habt}), $a(n,t)<0$ and
$\tl_r(a,b)=0$. Suppose $\lam_1 \le\sig(H(0))$ and let $u_\pm(\lam_1,n,t)>0$ be the
minimal positive solutions of (\ref{systemHB}) found in Lemma~\ref{lemfmin} and set
\begin{equation}
u_{\sig_1}(\lam_1,n,t) = \frac{1+\sig_1}{2} u_+(\lam_1,n,t) + \frac{1-\sig_1}{2}
u_-(\lam_1,n,t).
\end{equation}
Note that the dependence on $\sig_1$ will drop out in what follows if $u_+(\lam_1,n,t)$
and $u_-(\lam_1,n,t)$ are linearly dependent (for one and hence for all $t$). Now define
\begin{equation}
\rho_{\sig_1,o}(n,t) = -\sqrt{-\frac{a(n,t)}{\phi_{\sig_1}(\lam_1,n,t)}},\qquad
\rho_{\sig_1,e}(n,t) = \sqrt{-a(n,t) \phi_{\sig_1}(\lam_1,n,t)},
\end{equation}
where $\phi_{\sig_1}(\lam_1,n,t)= u_{\sig_1}(\lam_1,n+1,t)/u_{\sig_1}(\lam_1,n,t)$.

Then, proceeding as in the proof of Lemma~\ref{lemfpos} shows that the
sequence
\begin{equation} \label{defrhosigo}
\rho_{\sig_1}(n,t) = \left\{ \ba{c@{\text{ for }}l} \rho_{\sig_1,e}(m,t) & n=2m \\
\rho_{\sig_1,o}(m,t) & n=2m+1\ea \right.,
\end{equation}
fulfills (H.\ref{hrho}) and $\km_r(\rho)=0$. Hence by (\ref{kmimpltl})
\begin{equation}
a_{\sig_1}(n,t) =\rho_{\sig_1,e}(n+1,t) \rho_{\sig_1,o}(n,t), \qquad b_{\sig_1}(n,t) =
\rho_{\sig_1,e}(n,t)^2 + \rho_{\sig_1,o}(n,t)^2
\end{equation}
satisfy $\tl_r(a_{\sig_1},b_{\sig_1})=0$.

We summarize this result in our first main theorem.

\begin{thm} \label{thmsolsc}
Suppose $(a,b)$ satisfies (H.\ref{habt}) and
$\tl_r(a,b)=0$. Pick $\lam_1 \le\sig(H(0))$, $\sig_1\in[-1,1]$ and
let $u_\pm(\lam_1,n,t)$ be the minimal positive solutions of (\ref{systemHB}).
Then the sequences
\bea
a_{\sig_1}(n,t) &=& -\sqrt{\frac{a(n,t) a(n+1,t) u_{\sig_1}(\lam_1,n,t)
u_{\sig_1}(\lam_1,n+2,t)}{u_{\sig_1}(\lam_1,n+1,t)^2}}, \\
b_{\sig_1}(n,t) &=& b(n,t) - \partial^* \frac{a(n,t) u_{\sig_1}(\lam_1,n,t)
}{u_{\sig_1}(\lam_1,n+1,t)}
\eea
with
\begin{equation} \label{ulamsig}
u_{\sig_1}(\lam_1,n,t) = \frac{1+\sig_1}{2} u_+(\lam_1,n,t) + \frac{1-\sig_1}{2}
u_-(\lam_1,n,t),
\end{equation}
satisfy (H.\ref{habt}) and $\tl_r(a_{\sig_1},b_{\sig_1})=0$. Here $\partial^* f(n) = f(n-1)
- f(n)$. In addition,
\begin{equation}
\frac{a(n,t) ( u_{\sig_1}(\lam_1,n,t) u(z,n+1,t) - u_{\sig_1}(\lam_1,n+1,t)
u(z,n,t) )}{\sqrt{-a(n,t) u_{\sig_1}(\lam_1,n,t) u_{\sig_1}(\lam_1,n+1,t)}}
\end{equation}
satisfies $H_{\sig_1} u = z u$ and $d/dt\, u = P_{\sig_1,2r+2} u$ (weakly) (in obvious
notation) and $\rho_{\sig_1}(n,t)$ defined as in (\ref{defrhosigo}) satisfies
(H.\ref{hrho}) and $\km_r(\rho)=0$.
\end{thm}

The special case $r=0$ was first proven in \cite{TKvM}, Theorem~2.9. The general case
is stated in \cite{bght}, Theorem~7.2 without proof.

\begin{rem}  \label{remsolsc}
(i). Alternatively, one could give a direct algebraic proof of the above theorem using
$H_{\sig_1}^{j+1} = A_{\sig_1} H^j A_{\sig_1}^*$ to express the quantities $g_{\sig_1,j},
h_{\sig_1,j}$ in terms of $g_j, h_j$.\\
(ii). We have omitted the requirement $a(n,t)<0$ since the formulas for
$a_{\sig_1},b_{\sig_1}$ are actually independent of the sign of $a(n,t)$. In addition,
we could even allow $\lam_1\ge \sig(H(0))$. However, $\rho_{\sig_1,e}(n,t)$ and
$\rho_{\sig_1,o}(n,t)$ would be purely imaginary in this case.
\end{rem}

Iterating this procedure (cf.\ \cite{gtjc}, Theorem~3.1) gives

\begin{thm}
Let $a(t), b(t)$ satisfy (H.\ref{habt}) and $\tl_r(a,b)=0$. Let $H(t)$ be
the corresponding Jacobi operators and choose ($N \in\N$)
\begin{equation}
\lam_N < \dots < \lam_2 < \lam_1 \le\sig(H(0)), \quad \sig_\ell \in [-1,1],
\quad 1\le \ell \le N, \: N \in \N.
\end{equation}
Suppose $u_\pm(\lam,n,t)$, are the principal solutions of (\ref{systemHB}). Then
the sequences
\bea \nn
a_{\sig_1,\dots,\sig_N}(n,t) &=& - \sqrt{a(n,t) a(n+N,t)}\\
&& \times \frac{ \sqrt{
C_n(U_{\sig_1,\dots,\sig_N}(t)) C_{n+2}(U_{\sig_1,\dots,\sig_N}(t))
}}{C_{n+1}(U_{\sig_1,\dots,\sig_N}(t))},\\
b_{\sig_1,\dots,\sig_N}(n,t) &=& b(n,t) + \partial^* a(n,t)
\frac{D_n(U_{\sig_1,\dots,\sig_N}(t)) }{C_{n+1}(U_{\sig_1,\dots,\sig_N}(t))}
\eea
satisfy $\tl_r(a_{\sig_1,\dots,\sig_N},b_{\sig_1,\dots,\sig_N})=0$.
Here $C_n$ denotes the $n$-dimensional Casoratian
\bea
C_n(u_1,\dots,u_N) &=& \det\left( u_i(n+j-1) \right)_{1\le i,j\le N},\\
D_n(u_1,\dots,u_N) &=& \det\left( \ba{c@{\quad}l} u_i(n), & j=1\\
u_i(n+j), & j>1 \ea\right)_{1\le i,j\le N}
\eea
and
$(U_{\sig_1,\dots,\sig_N}(t)) = (u_{\sig_1}^1(t),\dots,u_{\sig_N}^N(t))$ with
\begin{equation}
u_{\sig_\ell}^\ell(n,t) = \frac{1+\sig_\ell}{2} u_+(\lam_\ell,n,t) +
(-1)^{\ell+1} \frac{1-\sig_\ell}{2} u_-(\lam_\ell,n,t).
\end{equation}
Defining
\bea \nn
&& \rho_{\sig_1,\dots,\sig_N,o}(n,t) =\\
&&\quad -\sqrt{-a(n,t) \frac{C_{n+2}(U_{\sig_1,\dots,\sig_{N-1}}(t))
C_n(U_{\sig_1,\dots,\sig_N}(t))} {C_{n+1}(U_{\sig_1,\dots,\sig_{N-1}}(t)) 
C_{n+1}(U_{\sig_1,\dots,\sig_N}(t))}},\\ \nn
&& \rho_{\sig_1,\dots,\sig_N,e}(n,t) =\\ && \quad \sqrt{-a(n+N-1,t)
\frac{C_n(U_{\sig_1,\dots,\sig_{N-1}}(t))
C_{n+1}(U_{\sig_1,\dots,\sig_N}(t))}
{C_{n+1}(U_{\sig_1,\dots,\sig_{N-1}}(t))  C_n(U_{\sig_1,\dots,\sig_N}(t))}},
\eea
the corresponding sequence $\rho_{\sig_1,\dots,\sig_N}(n)$ solves
$\km_r(\rho_{\sig_1,\dots,\sig_N})=0$.
\end{thm}

\begin{rem}
The formula for $b_{\sig_1,\dots,\sig_N}(n,t)$ is new. It can be obtained from
the one given in \cite{gtjc} (equation (3.6)) by observing that this formula holds with
$\lam_N$ replaced by arbitrary $z\in\C$ and performing the limit $z\to\infty$.
\end{rem}

Clearly, if we drop the requirement $\lam \le\sig(H(0))$ the solution
$u_{\sig_1}(\lam_1,n,t)$ used to perform the factorization will no longer be
positive. Hence the sequences $a_{\sig_1}(n,t)$, $b_{\sig_1}(n,t)$ can be
complex valued and singular.  Nevertheless there are two situations
where a second factorization step produces again real-valued non-singular
solutions.

Firstly we perform two steps with $\lam_{1,2}$ in the same spectral gap
of $H(0)$ (see \cite{tddm} for a detailed spectral analysis of this method).

\begin{thm} \label{thmsolddm}
Suppose $(a,b)$ satisfies (H.\ref{habt}) and
$\tl_r(a,b)=0$. Pick $\lam_{1,2}$, $\sig_{1,2}\in\{\pm 1\}$ and let
$\lam_{1,2}$ lie in the same spectral gap of $H(0)$ (\/$(\lam_1,\sig_1) \ne
(\lam_2,-\sig_2)$ to make sure we get something new). Then the sequences
\bea
a_{\sig_1,\sig_2}(n,t) &=& a(n,t)\sqrt{\frac{W_{\sig_1,\sig_2}(n-1,t)
W_{\sig_1,\sig_2}(n+1,t)}{W_{\sig_1,\sig_2}(n,t)^2}},\\
b_{\sig_1,\sig_2}(n,t) &=& b(n,t) - \partial^* \frac{a(n,t) u_{\sig_1}(\lam_1,n,t)
u_{\sig_2}(\lam_2,n+1,t)}{W_{\sig_1,\sig_2}(n,t)},
\eea
are real-valued non-singular solutions of $\tl_r(a_{\sig_1,\sig_2},b_{\sig_1,\sig_2})=0$.
Here
\begin{equation}
W_{\sig_1,\sig_2}(n,t) = \left\{ \ba{l@{,\quad}l} \frac{W_n(u_{\sig_1}(\lam_1,t),
u_{\sig_2}(\lam_2,t))}{\lam_2-\lam_1} & \lam_1\ne\lam_2\\
\sum\limits_{m=\sig_1\infty}^n u_{\sig_1}(\lam_1,m,t)^2 & (\lam_1,\sig_1) =
(\lam_2,\sig_2)\ea \right. ,
\end{equation}
where $\sum_{m=+\infty}^n = -\sum_{m=n+1}^{\infty}$.

In addition, the sequence
\begin{equation}
\frac{W_{\sig_1,\sig_2}(n,t) u(z,n,t) - \frac{1}{z-\lam_1} u_{\sig_2}(\lam_2,n,t)
W_n(u_{\sig_1}(\lam_1,t),u(z,t))}{\sqrt{W_{\sig_1,\sig_2}(n-1,t)W_{\sig_1,\sig_2}(n,t)}},
\end{equation}
satisfies $H_{\sig_1,\sig_2}(t) u = z u$, $d/dt \, u = P_{\sig_1,\sig_2,2r+2}(t) u$
(weakly).
\end{thm}

\begin{proof}
Theorem~4.6 of \cite{tosc} implies $W_{\sig_1,\sig_2}(n,t) W_{\sig_1,\sig_2}(n+1,t)>
0$ and hence the sequences $a_{\sig_1,\sig_2}(t), b_{\sig_1,\sig_2}(t)$ satisfy
(H.\ref{habt}) (see also \cite{tddm}). The rest follows from the previous theorem
(with $N=2$) as follows. Replace $\lam_1$ by $z\in(\lam_1-\eps,\lam_1+\eps)$ and
observe that $a_{\sig_1,\sig_2}(n,t)$, $b_{\sig_1,\sig_2}(n,t)$ and
$\dot{a}_{\sig_1,\sig_2}(n,t)$, $\dot{b}_{\sig_1,\sig_2}(n,t)$ are meromorphic with
respect to $z$. From the algebraic structure we have simply performed
two single commutation steps. Hence, provided Theorem~\ref{thmsolsc} applies to this
more general setting of meromorphic solutions, we can conclude that our claims hold
except for a discrete set with respect to $z$ where the intermediate operators are
ill-defined due to singularities of the coefficients. However, the proof of
Theorem~\ref{thmsolsc} uses these intermediate operators and in order to see that
Theorem~\ref{thmsolsc} still holds, one has to resort to the direct algebraic proof
outlined in Remark~\ref{remsolsc}(i). Continuity with
respect to $z$ takes care of the remaining points.
\end{proof}

To the best of our knowledge Theorem~\ref{thmsolddm} is novel even in the case of the
first Toda equation $r=0$. Secondly, we consider again two commutation steps but now
with $\lam_1=\lam_2$.

\begin{thm} \label{thmsoldc}
Suppose $(a,b)$ satisfies (H.\ref{habt}) and
$\tl_r(a,b)=0$. Pick $\lam_1$ in a spectral gap of $H(0)$ and
$\gam_1\in[-\|u_-(\lam_1)\|^{-2},\infty)\cup\{\infty\}$. Then the sequences
\bea
a_{\gam_1}(n,t) &=& a(n,t) \frac{\sqrt{c_{\gam_1}(\lam_1,n-1,t)
c_{\gam_1}(\lam_1,n+1,t)}}{c_{\gam_1}(\lam_1,n,t)},\\ b_{\gam_1}(n,t) &=& b(n,t) -
\partial^* \frac{a(n,t) u_-(\lam_1,n,t) u_-(\lam_1,n+1,t)}{c_{\gam_1}(\lam_1,n,t)}.
\eea
satisfy $\tl(a_{\gam_1},b_{\gam_1})=0$, where
\begin{equation}
c_{\gam_1}(\lam_1,n,t) = \frac{1}{\gam_1} + \sum_{m=-\infty}^n u_-(\lam_1,m,t)^2.
\end{equation}
In addition, the sequence
\begin{equation}
\frac{c_{\gam_1}(\lam_1,n,t) u(z,n,t) - \frac{1}{z-\lam_1} u_-(\lam_1,n,t)
W_n(u_-(\lam_1,t),u(z,t))}{\sqrt{c_{\gam_1}(\lam_1,n-1,t) c_{\gam_1}(\lam_1,n,t)}}, 
\end{equation}
satisfies $H_{\gam_1}(t) u = z u$, $d/dt \, u = P_{\gam_1,2r+2}(t) u$
(weakly).
\end{thm}

\begin{proof}
Following \cite{gtjc}, p256 we can obtain the double commutation method from
two single commutation steps. We pick $\sig_1=-1$ for the first factorization. 
Considering $A_{\sig_1} u_-(z,n+1,t)/(z-\lam_1)$ and performing
the limit $z\to\lam_1$ shows that
\begin{equation}
v(\lam_1,n,t) = \frac{c_{\gam_1}(\lam_1,n,t)}{\sqrt{-a(n,t) u_-(\lam_1,n,t)
u_-(\lam_1,n+1,t)}}
\end{equation}
is a solution of the new (singular) operator which can be used to perform a second 
factorization. The resulting operator is associated with $a_{\gam_1}, b_{\gam_1}$.
Now argue as before.
\end{proof}

As already mentioned before, the special case $r=0$ was first given in \cite{gtjc}.
Again we point out that one can also prove this theorem directly as follows.
Without restriction we choose $\lam_1=0$. Then one computes
\bea \nn
\frac{d}{dt} c_{\gam_1}(0,n,t) \!\!&=&\!\! 2a(n,t)^2 \Big( g_{r-1}(n+1,t) u_-(0,n,t)^2 +
g_{r-1}(n,t) u_-(0,n+1,t)^2 \Big)\\
&& {} + 2 h_{r-1}(n,t) a(n,t) u_-(0,n,t) u_-(0,n+1,t)
\eea
and it remains to relate $g_{\gam_1,j}, h_{\gam_1,j}$ and $g_j, h_j$. Since
these quantities arise as coefficients of the Neumann expansion of the respective
Green functions it suffices to relate the Green functions of $H_{\gam_1}$ and
$H$. This can be done using \cite{gtjc}, Lemma~4.6 (compare \cite{gtjc}, (2.40)).

Iterating this procedure (cf.\ \cite{gtjc}, Theorem~6.1) gives

\begin{thm}
Let $a(n,t), b(n,t)$ satisfy (H.\ref{habt}) and $\tl_r(a,b)=0$. and let $H(t)$ be
the corresponding Jacobi operators. Let $\lam_j\in\rho(H(0))$,
$\gam_j\in[-\|u_-(\lam_j)\|^{-2},\infty)\cup\{\infty\}$, $1 \le j \le N$ and
\begin{equation}
\frac{d}{dt} u_-(\lam_j,n,t) = P_{2r+2}(t) u_-(\lam_j,n,t).
\end{equation}
We define the following  matrices $(1 \le \ell \le N)$
\begin{equation}
C^N(n,t) =  \left( \frac{\delta_{ij}}{\gam_i} + \sum_{m=-\infty}^n
u_-(\lam_i,m,t) u_-(\lam_j,m,t) \right)_{1 \le i,j \le N},
\end{equation}
\begin{equation}
D^N(n,t) = \left( \ba{c@{\quad}l}
C^N(n,t)_{i,j}, & {\scriptstyle i,j \le N}\\
u_-(\lam_j,n-1,t), & {\scriptstyle j \le N, i=N+1}\\
u_-(\lam_i,n,t), & {\scriptstyle i \le N, j=N+1}\\
0, & {\scriptstyle i=j=N+1}
\ea \right)_{1 \le i,j \le N+1}.
\end{equation}
Then the sequences
\bea
a_{\gam_1,\dots,\gam_N}(n,t) 
&=& a(n,t) \frac{\sqrt{\det C^N(n-1,t) \det C^N(n+1,t)}}{\det C^N(n,t)}, \\
b_{\gam_1,\dots,\gam_N}(n,t) &=& b(n,t) - \partial^* a(n,t) \frac{\det
D^N(n+1,t)}{\det C^N(n,t)}
\eea
satisfy $\tl_r(a_{\gam_1,\dots,\gam_N},b_{\gam_1,\dots,\gam_N})=0$.
\end{thm}

\begin{rem}
(i). The formulas of \cite{gtjc} have been slightly rephrased to include the limit
$\gam_j=\infty$. In addition,  the formula for $b_{\gam_1,\dots,\gam_N}(n,t)$
is new. It can be obtained from the one given in \cite{gtjc} (equation (6.9)) by observing
that this formula holds with $\lam_N$ replaced by arbitrary $z\in\C$ and performing the
limit
$z\to\infty$.\\
(ii). The limit $N\to\infty$ can be performed as in \cite{rw}.
\end{rem}

We conclude this section with an example; the $N$-soliton solutions of the
$\tl$ and
$\km$ hierarchies.

We take the constant solution of the Toda hierarchy
\begin{equation}
a_0(n,t) = \frac{1}{2}, \qquad b_0(n,t) = 0,
\end{equation}
as our background. If $H_0$, $P_{0,2r+2}$  denotes the associated Lax pair we obtain
\begin{equation}
H_0(t) u_{0,\pm}(z,n,t) = z u_{0,\pm}(z,n,t), \quad \frac{d}{dt} u_{0,\pm}(z,n,t) =
P_{0,2r+2}(t) u_{0,\pm}(z,n,t),
\end{equation}
where
\begin{equation}
u_{0,\pm}(z,n,t) = k^{\pm n} \exp\Big( \pm\frac{\alpha_r(k) t}{2} \Big), \quad
k=z-\sqrt{z^2-1}, |k|\le 1 
\end{equation}
and
\begin{equation}
\alpha_r(k) = 2( k G_{0,r}(z) - H_{r+1,0}(z)) = (k - k^{-1}) G_{0,r}(z).
\end{equation}
Explicitly we have
\bea \nn 
\alpha_0(k) &=& k - k^{-1}, \\ \nn 
\alpha_1(k) &=& \frac{k^2 - k^{-2}}{2} +  c_1 (k - k^{-1}),\\
&&\mbox{etc.}
\eea

Then the $N$-soliton solution of the Toda hierarchy is given by
\bea \nn
a_{0,\sig_1,\dots,\sig_N}(n,t) &=& \frac{ \sqrt{
C_n(U_{\sig_1,\dots,\sig_N}(t)) C_{n+2}(U_{\sig_1,\dots,\sig_N}(t))
}}{2 C_{n+1}(U_{\sig_1,\dots,\sig_N}(t))},\\
b_{0,\sig_1,\dots,\sig_N}(n,t) &=& \partial^*
\frac{D_n(U_{\sig_1,\dots,\sig_N}(t)) }{2 C_{n+1}(U_{\sig_1,\dots,\sig_N}(t))},
\eea
where $(U_{0,\sig_1,\dots,\sig_N}(t)) = (u_{\sig_1}^1(t),\dots,u_{\sig_N}^N(t))$ with
\begin{equation}
u_{0,\sig_j}^j(n,t) = k_j^n + (-1)^{\ell+1} \frac{1-\sig_j}{1+\sig_j} \exp(\alpha_r(k_j)t)
k_\ell^{-n}, \quad k_j = \lam_j-\sqrt{\lam_j^2-1}.
\end{equation}
The corresponding $N$-soliton solution $\rho_{\sig_1,\dots,\sig_N}(n)$
of the Kac--van Moerbeke hierarchy reads
\bea \nn
&& \rho_{0,\sig_1,\dots,\sig_N,o}(n,t) =\\
&&\quad -\sqrt{-\frac{C_{n+2}(U_{0,\sig_1,\dots,\sig_{N-1}}(t))
C_n(U_{0,\sig_1,\dots,\sig_N}(t))} {2 C_{n+1}(U_{0,\sig_1,\dots,\sig_{N-1}}(t)) 
C_{n+1}(U_{0,\sig_1,\dots,\sig_N}(t))}},\\ \nn &&
\rho_{0,\sig_1,\dots,\sig_N,e}(n,t)
=\\ && \quad \sqrt{-\frac{C_n(U_{0,\sig_1,\dots,\sig_{N-1}}(t))
C_{n+1}(U_{0,\sig_1,\dots,\sig_N}(t))} {2C_{n+1}(U_{0,\sig_1,\dots,\sig_{N-1}}(t)) 
C_n(U_{0,\sig_1,\dots,\sig_N}(t))}}.
\eea
Introducing the time dependent norming constants
\begin{equation}
\gam_j(t) = \gam_j \exp(-\alpha_r(k_j)t)
\end{equation}
we obtain the following alternate expression for the $N$-soliton solution of
the Toda hierarchy
\bea
a_{0,\gam_1,\dots,\gam_N}(n,t) 
&=& \frac{\sqrt{\det C^N_0(n-1,t) \det C^N_0(n+1,t)}}{2\det C^N_0(n,t)}, \\
b_{0,\gam_1,\dots,\gam_N}(n,t) &=& - \partial^* \frac{\det
D^N_0(n+1,t)}{2 \det C^N_0(n,t)},
\eea
where
\begin{equation}
C^N_0(n,t) =  \left( \frac{\delta_{ij}}{\gam_j(t)} + \frac{(k_i k_j)^{-n}}{1-k_i
k_j}
\right)_{1 \le i,j \le N},
\end{equation}
\begin{equation}
D^N_0(n,t) = \left( \ba{c@{\quad}l}
C^N_0(n,t)_{i,j}, & {\scriptstyle i,j \le N}\\
k_j^{1-n}, & {\scriptstyle j \le N, i=N+1}\\
k_i^{-n}, & {\scriptstyle i \le N, j=N+1}\\
0, & {\scriptstyle i=j=N+1} \ea \right)_{1 \le i,j \le N+1}.
\end{equation}
The sequences $a_{0,\gam_1,\dots,\gam_N}$, $b_{0,\gam_1,\dots,\gam_N}$
coincide with $a_{0,\sig_1,\dots,\sig_N}$, $b_{0,\sig_1,\dots,\sig_N}$
provided (cf.\ \cite{TKvM}, Lemma~3.2 or \cite{gtjc}, Remark~6.5)
\begin{equation}
\gam_j = \left( \frac{1-\sig_j}{1+\sig_j}
\right)^{-1} |k_j|^{-1-N} \frac{\prod_{\ell=1}^N
 |1 - k_j k_\ell|}{\prod_{
\genfrac{}{}{0pt}{}{\ell=1}{\ell \ne j}}^N |k_j-k_\ell|}, 
\quad 1 \le j \le N.
\end{equation}
We remark that these formulas can also be obtained by the inverse scattering
transform (cf.\ e.g., \cite{fl2}, \cite{tist}, \cite{ta}, Section 3.6).


\section*{Acknowledgments}

I thank F. Gesztesy for bringing \cite{dlt} to my attention and the referees
for their valuable remarks.



\begin{thebibliography}{XXXX}
\bibitem{amr} R. Abraham, J.E. Marsden, and T. Ratiu, {\it Manifolds, Tensor
Analysis, and Applications}, $2^{\text{nd}}$ edition, Springer, New York, 1983.
\bibitem{bd} M. Benammar and W.D. Evans, {\em On the Friedrichs extension of
semi-bounded difference operators}, Math. Proc. Camb. Phil. Soc. {\bf 116}, 167-177
(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{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{fl2} H. Flaschka, {\em On the Toda lattice. II}, Progr. Theoret. Phys.
{\bf 51}, 703--716 (1974).
\bibitem{gtjc} F. Gesztesy and G. Teschl, {\em  Commutation
methods for Jacobi operators}, J. Diff. Eqs. {\bf 128}, 252--299 (1996).
\bibitem{crit} F. Gesztesy and Z. Zhao, {\em Critical and subcritical
Jacobi operators defined as Friedrichs extensions}, J. Diff. Eq. {\bf 103},
68--93 (1993).
\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{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{mi} R.~M.~Miura, {\it Korteweg-de Vries equation and generalizations.
I.~A remarkable explicit nonlinear transformation}, J. Math. Phys.
{\bf 9}, 1202--1204 (1968).
\bibitem{pat} W. T. Patula, {\em Growth, oscillation and 
comparison theorems for second order linear difference equations}, SIAM J. Math. 
Anal. {\bf 6}, 1272--1279 (1979).
\bibitem{rs2} M. Reed and B. Simon, {\em Methods of Modern Mathematical Physics II.
Fourier Analysis, Self-Adjointness}, Academic Press, San Diego, 1975.
\bibitem{rw} W. Renger, {\em Toda soliton limits on general background}, in
Differential equations, asymptotic analysis, and mathematical physics,
(ed. M. Demuth), 287-291, Math. Res. 100, Akademie Verlag, (1997).
\bibitem{tist} G. Teschl, {\em Inverse scattering transform for the Toda hierarchy},
Math. Nach. {\bf 202}, 163--171 (1999).
\bibitem{tddm} G. Teschl, {\em Spectral Deformations of Jacobi Operators},
J. Reine und Angew. Math. {\bf 491}, 16-47 (1997).
\bibitem{tosc} G. Teschl, {\em Oscillation theory and renormalized oscillation
theory for Jacobi operators}, J. Diff. Eqs. {\bf 129}, 532--558 (1996).
\bibitem{ta} M. Toda, {\em Theory of Nonlinear Lattices}, $2^{\text{nd}}$ enl.
edition, Springer, Berlin, 1989.
\end{thebibliography}

\end{document}
