\documentclass[twoside]{article}
\usepackage{amssymb,amsmath} % font used for R in Real numbers
\pagestyle{myheadings}

\markboth{\hfil A global factorization theorem \hfil
EJDE--2002/77} {EJDE--2002/77\hfil Derchyi Wu \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations}, 
Vol. {\bf 2002}(2002), No. 77, pp. 1--13. \newline 
ISSN: 1072-6691. URL:
http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline 
ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
  A global factorization theorem for the ZS-AKNS system
 %
\thanks{ {\em Mathematics Subject Classifications:} 53C99.
\hfil\break\indent {\em Key words:} ZS-AKNS system, inverse
scattering. \hfil\break\indent 
\copyright 2002 Southwest Texas State University. \hfil\break\indent 
Submitted July 12, 2002. Published September 4, 2002.} }
\date{}
%
\author{Derchyi Wu}
\maketitle

\begin{abstract}
  We prove a global Birkhoff factorization theorem for general
  loops with finite poles in the ZS-AKNS hierarchy
  (Zakharov-Shabat-Ablowitz-Kaup-Newell-Segur).
  We use the inverse scattering method.
\end{abstract}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\numberwithin{equation}{section}

\section{Introduction}

Factorization theory and Lie algebra decomposition play a key role
in algebraic geometry and representation theory associated with
integrable systems; see for example
 \cite{Sa,DrSo,SW,RS,Se1,Se2,FNR1,FNR2,HSS,SaSz,DGS}.
However, many algebraic or geometric decomposition theorems are
valid only locally and in general the scattering data properties
of their derived solution space can not, or can only, be partially
characterized.

Nevertheless, in the case of the ZS-AKNS system
(Zakharov-Shabat-Ablowitz-Kaup-Newell-Segur), as defined by
$su(2)$- or $su(n)$-reality condition (i.e., the solution space
contained in $su(2)$ or $su(n)$), Faddeev and Takhtajan \cite{FT},
Uhlenbeck and Terng \cite{TU1,TU2} derived a global decomposition
theorem and characterize full scattering data for solutions.

Using the inverse scattering theory \cite{BC1,BC2,SZ}, a
completely different approach, we obtain a global Birkhoff
factorization theorem and find the global group loop action in the
$su(2)$ (reprove the results of \cite {TU1,TU2}) and $su(1,1)$
cases \cite{W}. More precisely, in the $su(2)$ case, the
scattering theory of \cite {BC1,BC2,SZ} immediately implies a
global Birkhoff factorization theorem for each renormalized
eigenfuction of the ZS-AKNS system (see (2.1) below). We then show
a global factorization theorem for general loop in $D_-$ and
characterize $D_-$. The $D_-$ is actually characterized by that
each loop in $D_-$ is a multiple of a certain renormalized
eigenfuction. Where the multiple factor is a diagonal matrix in
$D_-$.

Under this framework, we generalize the methods in \cite{W} and
obtain the following generalized theorem.


\begin{theorem} \label{thm1}
 For $g(z)$ in a dense and open subset of $ D_-$, there exists an
unique global factorization
 $$ g(z)^{-1}e^{xzJ}=E(x,z)M(x,z)^{-1},
 $$
for $\forall x\in \mathbb{R}$, $\forall z\in \mathbb{C}\backslash
(\mathbb{R}\cup P)$,
 with $E(x,\cdot)\in G_+$, $M(x,\cdot)\in D_-$, and $P$ poles of $g$.
Where $G_+=\{\,g:\mathbb{C}\rightarrow SL(2,\mathbb{C}),
 \text{ holomorphic }\}$, and $D_-$ consist of those $g$
satisfying
\begin{enumerate}
\item $g:\mathbb{C}\backslash \mathbb{R}\rightarrow SL(2,\mathbb{C})$,
$g$ is meromorphic
\item $g$  has smooth boundary values $g_\pm$

\item $g$ tends to 1 at $\infty$

\item $g_+g_-^{-1}-I \in\mathcal{S}$  and decays rapidly at $\infty$

\item $\sum_{\mathbb{C}^-}\min\{\tilde a,\tilde
b\}-\sum_{\mathbb{C}^+}\min\{\tilde c,\tilde d\}=0$
\begin{align*}
\sum_{z\in\mathbb{C}^+}\min\{\tilde a,\tilde b\} -
\sum_{z\in\mathbb{C}^-}\min\{\tilde c,\tilde d\} =&\frac
1{2\pi}\int_{\mathbb{R}}d\,\arg(\tilde A_-\tilde D_+-\tilde
B_-\tilde C_+)\\ &+\frac 1{2\pi}\int_{\mathbb{R}}d\,\arg(\tilde
A_+ \tilde D_--\tilde B_+\tilde C_-),
\end{align*}
with $g=\begin{pmatrix} \tilde A & \tilde B\\ \tilde C&\tilde
D\end{pmatrix}$,  and $\tilde a$, $\tilde b$, $\tilde c$, $\tilde
d$ as exponents  of the leading terms in the power series
expansion of $\tilde A$, $\tilde B$, $\tilde C$, and $\tilde D$.
\end{enumerate}
\end{theorem}

\begin{theorem} \label{thm2}
The scattering data for loop $g$ in Theroem \ref{thm1}
 can be explicitly computed.
\end{theorem}

Note that $D_-$ is not a group owing to condition 5). Therefore,
the methods of Terng and Uhlenbeck, loop group factorization
theorem \cite {PS}, are unable to derive the factorization formula
in this case.

\section{Necessary Conditions for Factorization}

The ZS-AKNS system has the form $$
\frac{d}{dx}\psi(x,z)=\psi(x,z)(\,z J + q(x)\,), $$
 with $x\in \mathbb{R}$, $ z\in \mathbb{C}$, $\psi(x,z)\in SL(n,{\mathbb{C}})$,
and $q\in \mathcal{Q} $. Where $J=\begin{pmatrix} i &0\\ 0
&-i\end{pmatrix}$, $\mathcal{Q}=\{q\in \mathcal{S}(\mathbb{R};
M_2(\mathbb{C})\,): q_{11}=q_{22}=0\}$, the space of $2\times 2$
off-diagonal matrices whose entries belong to the Schwarz class.
The associated renormalized eigenfunction $m(x, z)$ satisfies
\begin{equation*}
\begin{gathered}
\frac {d}{dx}m(x,z)+z[ J , m(x, z)]=m(x,z) q(x),\\ \lim_{x\to
-\infty} m(x, z)=I,
\end{gathered} \tag{2.1}
\end{equation*}
with $m(\cdot, z)$ absolutely continuous and bounded. Therefore,
denote the scattering data \cite{BC1,BC2,S} for generic $q\in
\mathcal{Q} $ as $\{V, z_j$, $U^\pm\}$, where
\begin{equation*}
m_+(x,\xi)=e^{-x\xi J}V(\xi)e^{x\xi J}m_-(x,\xi),\tag{2.2}
\end{equation*}
$\{z_j\}$ are the totality of poles of $m$ in $\mathbb{C}^\pm$,
and $U^\pm$ are the upper (lower) triangular factors of
$m(x,z)=e^{-xzJ}(1+U^\pm(z))e^{xzJ}\eta^\pm(x,z)$ in
$\{x\le0\}\times\mathbb{C}^\pm$. Here, $\eta^\pm(x,z)$ is a
solution of (2.1), holomorphic in $z$ and approaching $1$ as
$z\to\infty$. Moreover, for generic $q$, the scattering data $\{V,
z_j, U^\pm\}$ satisfies:
\begin{enumerate}
\item[a)] Algebraic constraints:
$U^\pm$ are strictly upper (lower) triangular, $d_k^+(V)\ne 0$,
and $d_k^-(V)=1$, with $d_k^+(d_k^-(V))$ as upper (lower)
principal $k$-th minors of $V$,

\item[b)] Analytic constraints:
$V-I\in\mathcal{S}(\mathbb{R}; M_2(\mathbb{C}))$, $U^\pm$ is
rational in $z\in \mathbb{C}_\pm$, holomorphic for $z\in
\mathbb{C}_\mp$, and approaching zero as $z\to\infty$,

\item[c)] Topological constraints (winding number constraints):
\begin{equation*}
 P_j^+  - P_{j+1}^+ +P_{j}^--P_{j-1}^-
=\frac 1{2\pi}\int _{-\infty}^\infty d\arg
\frac{d^+_jV}{d^+_{j-1}V}(\xi), \tag{2.3}
\end{equation*}
 as $j=1,2$, with $P_j^+$ being the number of poles in the
$j$-th column of $U^+$,  and $P_j^-$ the number of poles in the
$j$-th column of $U^-$.
\end{enumerate}

\begin{lemma} \label{lm1}
Suppose that $m(x,z)$ is a normalized eigenfunction of the ZS-AKNS
system (2.1). Let $P\subset \mathbb{C}\backslash \mathbb{R}$ be
the set of poles of $m(x,\cdot)$. Then, there exists a global
factorization $$ m(0,z)^{-1}e^{xzJ}=E(x,z) m(x,z)^{-1}, $$ for all
$x\in \mathbb{R}$, all $z\in \mathbb{C}\backslash (\mathbb{R} \cup
P)$,  and $E(x,\cdot)\in G_+$.
\end{lemma}

One can prove this lemma by factorization formula
$m(x,z)=e^{-xzJ}(1+U^\pm(z))e^{xzJ}\eta^\pm(x,z)$ in $\{x\le
0\}\times\mathbb{C}^\pm$, and (2.2).

In the $su(2)$ case, to characterize each loop $g=\begin{pmatrix}
\tilde A&\tilde B \\ \tilde C& \tilde D\end{pmatrix}$ in $D_-$, we
first simplify $g$ by factoring out a diagonal factor such
 that all poles of $\tilde A$, $\tilde B$ lie in $C^-$, and all of the zeros
of $\tilde A$ and $\tilde B$ belong to $\overline C^+$ (see
\cite[lemma 4.3]{W}). Therefore, it becomes easier to detect the
zeros and poles of the diagonal multiple $\sigma$ such that
$\sigma g$ is some renormalized eigenfunction. Without the $su(2)$
symmetry, we generalize the factoring out process to the
following.

\paragraph{Definition}
 We define a transformation $\tau:SL(2,\mathbb{C})\to SL(2,\mathbb{C})$,
 that maps
$g=\begin{pmatrix} \tilde A&\tilde B \\ \tilde C& \tilde
D\end{pmatrix} $ to $\tau (g)=\begin{pmatrix} \tilde X&0\\0&\tilde
Y\end{pmatrix} \begin{pmatrix} \tilde A&\tilde B \\ \tilde C&
\tilde D\end{pmatrix}$. Here
\begin{align*}
\tilde X=&\frac{(z- t_1^+)\dots (z-t_\alpha^+)(z-\bar q_1^-)\dots
(z-\bar q_\beta^-)(z-\bar r_1^-)} {(z-\bar t_1^+)\dots(z-\bar
t_\alpha^+)(z- q_1^-)\dots (z- q_\beta^-)(z- r_1^-)}
\\
&\times \frac{\dots (z-\bar r_\gamma^-)(z-s_1^+)\dots
(z-s_\delta^+)} {\dots (z- r_\gamma^-)(z-\bar s_1^+)\dots(z-\bar
s_\delta^+)},
\\
\tilde Y=&\frac {(z-\bar p_1^+)\dots (z-\bar p_\alpha^+)(z-
q_1^-)\dots (z- q_\beta^-)(z- r_1^-)} {(z-p_1^+)\dots
(z-p_\alpha^+)(z-\bar q_1^-)\dots (z-\bar q_\beta^-)(z-\bar
r_1^-)}
\\
&\times\frac{\dots (z- r_\gamma^-)(z-\bar s_1^+)\dots (z-\bar
s_\delta^+)} {\dots (z-\bar r_\gamma^-)(z-s_1^+)\dots
(z-s_\delta^+)},
\end{align*}
and
\begin{gather*}
\{t_1^+,\dots,t_\alpha^+\}= \{\text{ poles of $\tilde A$ in
$\mathbb{C}^+$ }\}\underset{\max}{\cup}\{\text{ poles of $\tilde
B$ in $\mathbb{C}^+$ }\},\\ \{q_1^-,\dots,q_\beta^-\}=\{\text{
zeros of $\tilde A$ in $\mathbb{C}^-$
}\}\underset{\max}{\cup}\{\text{ zeros of $\tilde B$ in
$\mathbb{C}^-$ }\},\\ \{r_1^-,\dots,r_\gamma^-\}=\{\text{ poles of
$\tilde C$ in $\mathbb{C}^-$ }\}\underset{\max}{\cup}\{\text{
poles of $\tilde D$ in $\mathbb{C}^-$ }\},\\
\{s_1^+,\dots,s_\delta^+\}=\{\text{ zeros of $\tilde C$ in
$\mathbb{C}^+$ }\}\underset{\max}{\cup}\{\text{ zeros of $\tilde
D$ in $\mathbb{C}^+$ }\}.
\end{gather*}

In this paper we define $\underset{\max }{\cup}$,
$\underset{\min}{\cup}$, and $\underset{\mathrm{mult}}{\subset}$
to be $$\{\underset {n_1} {\underbrace{z_1\dots z_1}}\dots
\underset {n_k} {\underbrace{z_k\dots z_k}}\}\underset{\max
}\cup\{\underset {m_1} {\underbrace{z_1\dots z_1}}\dots\underset
{m_k} {\underbrace{z_k\dots z_k}}\}=\{\underset {\max (n_1,m_1)}
{\underbrace{z_1\dots z_1}}\dots\underset {\max (n_k,m_k)}
{\underbrace{z_k\dots z_k}}\}, $$ $$\{\underset
{n_1}{\underbrace{z_1\dots  z_1}}\dots\underset {n_k}
{\underbrace{z_k\dots z_k}}\}\underset{\min}\cup\{\underset {m_1}
{\underbrace{z_1\dots z_1}}\dots \underset {m_k}
{\underbrace{z_k\dots z_k}}\}=\{\underset {\min(n_1,m_1)}
{\underbrace{z_1\dots z_1}}\dots\underset {\min(n_k,m_k)}
{\underbrace{z_k\dots z_k}}\}, $$ $$\{\underset
{n_1}{\underbrace{z_1\dots  z_1}}\dots\underset {n_k}
{\underbrace{z_k\dots
z_k}}\}\underset{\mathrm{mult}}{\subset}\{\underset {m_1}
{\underbrace{z_1\dots z_1}}\dots \underset {m_k}
{\underbrace{z_k\dots z_k}}\},\text{ if }n_i\le m_i,\,\,\forall
i\in\{1,\dots,k\}. $$

\begin{theorem} \label{thm3}  Suppose that $g(z)\in SL(2,\mathbb{C})$,
and $\sigma g=m(0,z)$, with $\sigma$ a diagonal matrix, and
$m(x,z)$ as a normalized eigenfunction of (2.1). Then, the formula
of $\sigma$ and scattering data $\{V,z_j, U^\pm\}$ of $m(x,z)$,
 can be explicitly computed.
\end{theorem}

The proof of Theorem \ref{thm3} follows from the following lemmas.
First, let $\tau(g)=\begin{pmatrix} A& B\\ C& D\end{pmatrix}$,
$\sigma=\begin{pmatrix} X&0\\0& Y\end{pmatrix}$,
$U^+=\begin{pmatrix} 0&u_{12}\\0&0\end{pmatrix}$, and
$U^-=\begin{pmatrix} 0&0\\u_{21}&0\end{pmatrix}$. Then by (2.3),
and the factorization property of scattering theory
\cite{BC1,BC2,S}, we obtain
\begin{enumerate}
\item[a)] $XY=1$, $AD-BC=1$,

\item[b)] $X_-Y_+(A_-D_+-B_-C_+)=1$, $4X_+Y_-(A_+D_--B_+C_-)\ne 0$

\item[c)] $\frac CX$, and $\frac DX$ are holomorphic in $\mathbb{C}^+$,
$XA-u_{12}\frac CX$ is holomorphic in $\mathbb{C}^+$,
$XB-u_{12}\frac DX$ is holomorphic in $\mathbb{C}^+$,

\item[d)] $XA$ and $XB$ are holomorphic in $\mathbb{C}^-$,
$\frac CX-u_{21}XA$ is holomorphic in $\mathbb{C}^-$ $\frac
DX-u_{21}XB$ is holomorphic in $\mathbb{C}^-$,

\item[e)]
\begin{equation*}
\begin{aligned}-(&\text{number of poles of }u_{12})
+(\text{number of poles of }u_{21})\\ &=\frac
1{2\pi}\int_{\mathbb{R}} d\arg (X_+Y_-(A_+D_--B_+C_-)).
\end{aligned}\tag{2.4}
\end{equation*}
\end{enumerate}

\begin{lemma} \label{lm2}
Let $\{p_1^+,\dots,p_m^+\}$ be poles of $X$, and
$\{z_1^-,\dots,z_k^-\}$ zeros of $X$. Then,
\begin{gather*}
m=k,\\ \{\text{ poles of $A$ }\}\underset{\max}{\cup}\{\text{
poles of $ B$ }\}
\underset{\mathrm{mult}}{\subset}\{z_1^-,\dots,z_k^-\},\\ \{\text{
poles of $C$ }\}\underset{\max}{\cup}\{\text{ poles of $D$ }\}
\underset{\mathrm{mult}}{\subset}\{p_1^+,\dots,p_k^+\},\\
X(z)=\frac {(z- z_1^-)\dots (z-z_k^-)}{(z-
p_1^+)\dots(z-p_k^+)}\exp \big\{\frac 1{2\pi
i}\int_{\mathbb{R}}\frac {\log(A_-D_+-B_-C_+)}{t-z}dt\big\}.
\end{gather*}
\end{lemma}

\paragraph{Proof.} Let $\{z_1^+,\dots,z_h^+\}$ be zeros of $X$ in
$\mathbb{C}^+$, $\{\hat p_1^+,\dots,\hat p_{\hat m}^+\}$  be poles
of $X$ in $\overline{\mathbb{C}}^+$, $\{\hat z_1^-,\dots,\hat
z_{\hat k}^-\}$ be zeros of $X$ in $\overline{\mathbb{C}}^-$, $\{
p_1^-,\dots, p_n^-\}$ be poles of $X$ in $\mathbb{C}^-$
 By the first formula of (2.4-c) and (2.4-d), we obtain
\begin{gather*}
\{\text{poles of $ A$ in $\mathbb{C}^-$}
\}\underset{\max}{\cup}\{\text{ poles of $ B$ in $\mathbb{C}^-$}\}
\underset{\mathrm{mult}}{\subset}\{\hat z_1^-,\dots,\hat z_{\hat
k}^-\},\\ \{\text{ zeros of $ A$ in $\mathbb{C}^-$
}\}\underset{\min}{\cap}\{\text{ zeros of $ B$ in
$\mathbb{C}^-$}\}
\underset{\mathrm{mult}}\supset\{p_1^-,\dots,p_n^-\},\\
\{\text{zeros of $ C$ in $\mathbb{C}^+$}
\}\underset{\min}{\cap}\{\text{ zeros of $ D$ in $\mathbb{C}^+$}\}
\underset{\mathrm{mult}}\supset\{z_1^+,\dots,z_h^+\},\\
\{\text{poles of $ C$ in $\mathbb{C}^+$}\}
\underset{\max}{\cup}\{\text{ poles of $ D$ in $\mathbb{C}^+$  }\}
\underset{\mathrm{mult}}{\subset}\{\hat p_1^+,\dots,\hat p_{\hat
m}^+\}.
\end{gather*}
Because of the definition of $\tau$, we know that all zeros of $A$
and $B$ lie in $\overline{\mathbb{C}}^+$, and all zeros of $C$ and
$D$ lie in $\overline{\mathbb{C}}^-$. Thus, $\{ p_1^-,\dots,
p_n^-\}=\{ z_1^+,\dots, z_h^+\}=\phi$, $\{ p_1^+,\dots, p_m^+\}=\{
\hat p_1^+,\dots, \hat p_{\hat m}^+\}$ and $\{ z_1^-,\dots,
z_k^-\}=\{ \hat z_1^-,\dots, \hat z_{\hat k}^-\}$. Applying
(2.4-b) and the Riemann Hilbert theorem, we obtain the formula for
$X$, $$X(z)=z^{m-k}\frac {(z- z_1^-)\dots (z-z_k^-)}{(z-
p_1^+)\dots(z-p_m^+)}\exp\big\{\frac 1{2\pi
i}\int_{\mathbb{R}}\frac {\log(A_-D_+-B_-C_+)}{t-z}dt\big\}. $$
 Finally, since $g$ has smooth
boundary values on $\mathbb{R}$, $\tau(g)$, $g$ and $m(0,\cdot)$
to $1$ as $z\to\infty$, we derive $m=k$. \hfill$\square$

\begin{lemma} \label{lm3} We have
\begin{align*}
\{&p_1^+,\dots,p_k^+\}\\ &\underset{\mathrm{mult}}{\subset}
\{\text{poles of $ C$}\} \underset{\max }\cup\{\text{poles of $
D$}\}\underset{\max }\cup\{\text{zeros of $ A$}\}\underset{\max
}\cup\{\text{zeros of $ B$}\},\\ \{&z_1^-,\dots,z_k^-\}\\
&\underset{\mathrm{mult}}{\subset} \{\text{poles of $ A$}\}
\underset{\max }\cup\{\text{poles of $ B$}\}\underset{\max
}\cup\{\text{zeros of $ C$}\}\underset{\max }\cup\{\text{zeros of
$ D$}\}.
\end{align*}
\end{lemma}

\paragraph{Proof.}
 First, we introduce  $a$, $b$,
$c$, $d$, $\kappa$, $\hat a$, $\hat b$, $\hat c$, $\hat d$, and
$\hat \kappa$ by
 \begin{gather*}
A(z)=A_a( z_0)(z- z_0)^a+A_{a+1}(z_0)(z-z_0)^{a+1}+\dots,\\
B(z)=B_b( z_0)(z- z_0)^b+B_{b+1}(z_0)(z- z_0)^{b+1}+\dots,\\
C(z)=C_c( z_0)(z- z_0)^c+C_{c+1}(z_0)(z-z_0)^{c+1}+\dots,\\
D(z)=D_d( z_0)(z- z_0)^d+D_{d+1}(z_0)(z- z_0)^{d+1}+\dots, \\
u_{12}(z)={u_{12,u}}( z_0)(z-z_0)^u+{u_{12,u+1}}(z_0)(z-
z_0)^{u+1}+\dots,\\ (z-p_1^+)\dots(z- p_k^+)=f_\kappa( z_0)(z-
z_0)^\kappa+f_{\kappa+1}( z_0) (z- z_0)^{\kappa+1}+\dots,
\end{gather*}
for $z_0\in\mathbb{C}^+$. And
\begin{gather*}
A(z)=\hat A_{\hat a}( \omega_0)(z- \omega_0)^{\hat a}+\hat A_{\hat
a+1}( \omega_0)(z-\omega_0)^{\hat a+1}+\dots,\\ B(z)=\hat B_{\hat
b}( \omega_0)(z- \omega_0)^{\hat b}+\hat B_{\hat b+1}(
\omega_0)(z- \omega_0)^{\hat b+1}+\dots,\\ C(z)=\hat C_{\hat c}(
\omega_0)(z- \omega_0)^{\hat c}+\hat C_{\hat c+1}(
\omega_0)(z-\omega_0)^{\hat c+1}+\dots,\\ D(z)=\hat D_{\hat d}(
\omega_0)(z- \omega_0)^{\hat d}+\hat D_{\hat d+1}( \omega_0)(z-
\omega_0)^{\hat d+1}+\dots, \\ u_{21}(z)={u_{21,\hat u}}(
\omega_0)(z- \omega_0)^{\hat u}+{u_{12,\hat u+1}}(\omega_0)(z-
\omega_0)^{\hat u+1}+\dots,\\ (z- z_1^-)\dots(z- z_k^-)=\hat
f_{\hat \kappa}( \omega_0)(z- \omega_0)^{\hat\kappa}+\hat
f_{\hat\kappa+1}( \omega_0) (z- \omega_0)^{\hat\kappa+1}+\dots,
\end{gather*}
 for $\omega_0\in\mathbb{C}^-$.

If $z_0\in \{p_1^+,\dots,p_k^+\}\underset{\mathrm{mult}}
\backslash(\{\text{ zeros of $ A$ }\}\underset{\max }\cup\{\text{
zeros of $ B$ }\})$, then $a-\kappa<0$, and $b-\kappa<0$.
Therefore, by (2.4-c) we obtain
\begin{gather*}
a-\kappa=u+\kappa+c,\\ b-\kappa=u+\kappa+d, A_aD_d-B_bC_c=0.
\end{gather*}
Then with (2.4-a), we have $$a+d=b+c<0.$$ Thus $c<0$, $d<0$, and
the first formula is proved. Additionally, the same argument can
be applied to demonstrate the second formula. \hfill $\square$


\begin{lemma} \label{lm4} We have
\begin{gather*}
\{p_1^+,\dots,p_k^+\}\underset{\mathrm{mult}} = \{\text{ poles of
$ C$ }\} \underset{\max }\cup\{\text{ poles of $ D$ }\},\\
\{z_1^-,\dots,z_k^-\}\underset{\mathrm{mult}} = \{\text{ poles of
$ A$ }\} \underset{\max }\cup\{\text{ poles of $ B$ }\}.
\end{gather*}
\end{lemma}

\paragraph{Proof.} It is sufficient to show that
$$\kappa=-\min\{c,d\}\quad\text{and}\quad \hat\kappa=-\min\{\hat
a,\hat b\}. $$
 We will prove the
first formula of this lemma. The same argument can be adapted for
demonstrating the other formula. Without loss of generality, let
$c=\min \{c,d\}$. Hence, $\kappa\ge -c$ by Lemma \ref{lm3}. Since
$AD-BC=1$ ((2.4-a)), either of the following cases i) $a+d=b+c\le
0$, ii) $a+d=0$, $b+c>0$, iii) $a+d>0$, $b+c=0$.

In case i), we obtain $\kappa=-c$ by Lemma \ref{lm3} directly. In
case ii), we derive
 $a-\kappa\le a+c \le a+d=0$. Thus, either $a=\kappa$ or $a<\kappa$. If
 $a=\kappa$, then $\kappa=-d\le-c$. Hence, $\kappa=-c$ by Lemma \ref{lm3}.
 If $a<\kappa$, then $a-\kappa=u+\kappa+c$. This implies
 $u+\kappa+d=-\kappa-c\le 0$. It suffices to examine the case
 $u+\kappa+d=-\kappa-c< 0$. However, it implies $a-\kappa=u+\kappa+c$,
 and $b-\kappa=u+\kappa+d$. That is $a+d=b+c$, which is a contradiction.

Case iii) can be proved with the same argument as that of case
ii). \hfill $\square$


\begin{lemma} \label{lm5}
The associated scattering data , denoted by $\{V, z_j, U^\pm\}$,
in Theorem \ref{thm3} can be explicitly computed in terms of $g$.
\end{lemma}

\paragraph{Proof.} It suffices to determine the scattering data in terms
of $\tau(g)$, that is, in terms of $A$, $B$, $C$, and $D$. First
of all, by (2.2) and (2.3), we obtain
\begin{align*}
V(\xi)=&\,\sigma_+\tau(g)_+ \tau(g)_-^{-1}\sigma_-^{-1}\\
=&\begin{pmatrix} X_+Y_-(A_+D_--B_+C_-)&-X_+X_-(A_+B_--A_-B_+)\\
Y_+Y_-(C_+D_--C_-D_+)&1\end{pmatrix}.
\end{align*}
Let $z_0\in\mathbb{C}^+$ be a pole of $u_{12}$. If
$z_0\notin\{p_1^+,\dots, p_k^+\}$, then we would have $0\le a-u$
and $u+\kappa+c<0$. This contradicts with the second formula of
(2.4-c). Therefore, we conclude that
$u_{12}(z)=p(z)(z-p_1^+)^{u_1}\dots(z-p_k^+)^{u_k}$, with $u_j\le
0$, and $p(z)$ is a polynomial of degree $\deg
p<-(u_1+\dots+u_k)$. Moreover, under similar argument with Lemma
\ref{lm4}, we can prove that
\begin{equation*}
 u_i=\min\{a, b\}+\min\{c,d\}.\tag{2.5}
\end{equation*}
Since the degree of polynomial $p$ is strictly less than
$-(u_1+\dots+u_k)$, to determine polynomial $p$, it is equivalent
to determine the $u_1+\dots+u_k$ number of coefficients, of
negative terms of expansion $$ u_{12}(z)=u_{i,u_i}(z-p_i^+)^{u_i}+
u_{i,u_i+1}(z-p^+_i)^{u_i+1} +\dots+u_{i,-1}(z-p^+_i)^{-1}+
\text{h.o.t.} $$ at $p^+_i$, for $p^+_i\in \{p^+_1, \dots,
p^+_k\}$. Here, h.o.t. denotes the higher order terms. However,
the $u_i$ number of coefficients can be successively computed by
equating the $u_i$ number of negative terms of either the second
or third formula of (2.4-c), up to $a-\kappa<0$ or $b-\kappa<0$.

Similarly we can determine $u_{21}$ using the second formula of
Lemmas \ref{lm2}-\ref{lm4} and adapting the above argument. Note
that
\begin{equation*}
\hat u_i=\min\{\hat a,\hat  b\}+\min\{\hat c,\hat d\}\tag{2.5'}
\end{equation*}
which completes the proof of this lemma. \hfill$\square$

Now the proof of Theorem \ref{thm3} follows from Lemmas
\ref{lm2}-\ref{lm5}. Next we use condition (2.4-e) to characterize
$g$ in Theorem \ref{thm3}, through the the following two lemmas.

\begin{lemma} \label{lm6}
We have
\begin{align*}
\frac 1{2\pi}&\int_{\mathbb{R}}d\,\arg(A_-D_+-B_-C_+)+\frac
1{2\pi}\int_{\mathbb{R}}d\,\arg(A_+D_--B_+C_-) \\
=&\#\bigl(\{\text{zeros of }A\} \underset{\min}\cap \{\text{zeros
of }B\}\bigr) -\#\bigl(\{\text{zeros of
}C\}\underset{\min}\cap\{\text{zeros of }D\}\bigr).
\end{align*}
\end{lemma}

\paragraph{Proof.} Since $AD-BC=1$, we have $\min\{a,
b\}+\min\{c, d\}\le 0$ in $\mathbb{C}^+$, and $\min\{\hat a, \hat
b\}+\min\{\hat c, \hat d\}\le 0$ in $\mathbb{C}^-$. Therefore, by
Lemma \ref{lm5}, (2.5) and (2.5'), we obtain
\begin{gather*}
\#(\text{pole of
}u_{12})=-(u_1+\dots+u_m)=-\sum_{z\in\mathbb{C}^+}(\min\{a,
b\}+\min\{c, d\}),\\
 \#(\text{pole of }u_{21})=-(\hat
u_1+\dots+\hat u_k)=-\sum_{z\in\mathbb{C}^-}(\min\{\hat a, \hat
b\}+\min\{\hat c, \hat d\}).
\end{gather*}
 Using conditions (2.4-b), (2.4-e), and the above formula,
\begin{align*}
\frac 1{2\pi}&\int_{\mathbb{R}}d\,\arg(A_-D_+-B_-C_+)+\frac
1{2\pi}\int_{\mathbb{R}}d\,\arg(A_+D_--B_+C_-)\\
=&\#\bigl(\{\text{zeros of }A\}\underset{\min}{\cap}\{\text{zeros
of }B\}\bigr) +\#\bigl(\{\text{poles of
}A\}\underset{\max}{\cup}\{\text{poles of }B\}\bigr)\\
&-\#\bigl(\{\text{zeros of }C\}\underset{\min}{\cap}\{\text{zeros
of }D\}\bigr)-\#\bigl(\{\text{poles of
}C\}\underset{\max}{\cup}\{\text{poles of}D\}\bigr).
\end{align*}
The lemma is proved by noting that $$\#\bigl(\{\text{poles of
}A\}\underset{\max}{\cup}\{\text{poles of
}B\}\bigr)=\#\bigl(\{\text{poles of }
C\}\underset{\max}{\cup}\{\text{poles of }D\}\bigr)=k.$$ cf.
Lemmas \ref{lm2} and \ref{lm4}. \hfill$\square$


\begin{lemma} \label{lm7}
The loop $g$ in Theorem \ref{thm3} satisfies condition 5) of
Theorem \ref{thm1}.
\end{lemma}

\paragraph{Proof.} By the definition of $\tau$ and $\tilde A\tilde
D-\tilde B\tilde C=1$, we obtain
\begin{equation*}\begin{aligned}
\{\text{zeros of }A\}\underset{\min}{\cap} \{\text{zeros of
}B\}&=\{\text{zeros of }\tilde X\tilde A\text{ in $\mathbb{C}^+$}
\} \underset{\min}{\cap}\{\text{zeros of }\tilde X \tilde  B\text{
in $\mathbb{C}^+$} \}\\
&=\alpha+\beta+\gamma+\delta+\sum_{\mathbb{C}^+}\min\{\tilde
a,\tilde b\},
 \\ \{\text{zeros of }C\}\underset{\min}{\cap}
\{\text{zeros of }D\} &=\{\text{zeros of }\tilde Y\tilde C\text{
in $\mathbb{C}^-$} \}\underset{\min}{\cap}\{\text{zeros of }\tilde
Y \tilde D\text{ in $\mathbb{C}^-$} \}\\
&=\alpha+\beta+\gamma+\delta+\sum_{\mathbb{C}^- }\min\{\tilde
c,\tilde d\}.
\end{aligned} \tag{2.6}
\end{equation*}
 Thus, we prove that $g$ satisfies the second formula of 5) of
Theorem \ref{thm1}. Moreover, note that
\begin{equation*}\begin{aligned}
 \{\text{poles of }A\}\underset{\max}{\cup}
\{\text{poles of }B\}&=\{\text{poles of }\tilde X\tilde A\text{ in
$\mathbb{C}^-$} \}\underset{\max}{\cup}\{\text{poles of }\tilde X
\tilde B\text{ in $\mathbb{C}^-$} \}\\
&=\alpha+\beta+\gamma+\delta-\sum_{\mathbb{C}^- }\min\{\tilde
a,\tilde b\}, \\ \{\text{poles of }C\}\underset{\max}{\cup}
\{\text{poles of }D\}&=\{\text{poles of }\tilde Y\tilde C\text{ in
$\mathbb{C}^+$} \}\underset{\max}{\cup}\{\text{poles of }\tilde Y
\tilde D\text{ in $\mathbb{C}^+$} \}\\
&=\alpha+\beta+\gamma+\delta-\sum_{\mathbb{C}^+ }\min\{\tilde
c,\tilde d\} .
\end{aligned}\tag{2.6'}
\end{equation*}
Thereby, the first formula of condition 5) is proved by Lemmas
\ref{lm2} or \ref{lm4}. \hfill$\square$

\section{Proof Theorems \ref{thm1} and \ref{thm2}}

\paragraph{Step 1. Extraction of Scattering Data}
 For loop $g(z)\in D_-$, we denote
\begin{gather*}
\tau(g)=\begin{pmatrix} A& B\\ C& D\end{pmatrix},\quad
\sigma=\begin{pmatrix} X&0\\0& Y\end{pmatrix},\\
U^+=\begin{pmatrix} 0&u_{12}\\0&0\end{pmatrix}, \quad
U^-=\begin{pmatrix} 0&0\\u_{21}&0\end{pmatrix}.
\end{gather*}
Therefore (2.6') and condition 5) of $D_-$ implies the existence
of a $k$, such that
\begin{align*}
 \{p_1^+,\dots,p_k^+\}\underset{\mathrm{mult}} =
\{\text{ poles of $ C$ }\} \underset{\max }\cup\{\text{ poles of $
D$ }\},\\ \{z_1^-,\dots,z_k^-\}\underset{\mathrm{mult}} = \{\text{
poles of $A$ }\} \underset{\max }\cup\{\text{ poles of $B$}\}.
\end{align*}
Set
\begin{gather*}
X(z)=\frac {(z- z_1^-)\dots (z-z_k^-)}{(z-
p_1^+)\dots(z-p_k^+)}\exp\{\frac 1{2\pi i}\int_{\mathbb{R}}\frac
{\log(A_-D_+-B_-C_+)}{t-z}dt\},\\ Y(z)=\frac {(z-
p_1^+)\dots(z-p_k^+)}{(z- z_1^-)\dots (z-z_k^-)}\exp\{-\frac
1{2\pi i}\int_{\mathbb{R}}\frac {\log(A_-D_+-B_-C_+)}{t-z}dt\}.
\end{gather*}
Let
\begin{align*}
V(\xi)=&\begin{pmatrix}
X_+Y_-(A_+D_--B_+C_-)&-X_+X_-(A_+B_--A_-B_+)\\
Y_+Y_-(C_+D_--C_-D_+)&X_-Y_+(A_-D_+-B_-C_+)\end{pmatrix}\\=&\begin{pmatrix}
V_{11}&V_{12}\\V_{21}&V_{22}\end{pmatrix}=\sigma_+\tau(g)_+
\tau(g)_-^{-1}\sigma_-^{-1}.
\end{align*}
By the formula of $X$, and $Y$, and condition 3)-5) of $D_-$, one
can verify that $V-I\in \mathcal{S}(\mathbb{R},
SL(2,\mathbb{C}))$, $V_{11}\ne 0$, and $V_{22}=1$.


Now define
\begin{gather*}
u_{12}(z)=p(z)(z-p_1^+)^{u_1}\dots(z-p_k^+)^{u_k},\\
u_{21}(z)=\hat p(z)(z-z_1^-)^{\hat u_1}\dots(z-z_k^-)^{\hat u_k},\
\end{gather*}
with $0\ge u_i=\min\{ a,b\}+\min\{ c, d\}$, $ 0\ge\hat
u_i=\min\{\hat a,\hat b\}+\min\{\hat c,\hat d\}$, $a$, $b$, $c$,
$d$ as exponents of the leading terms in the power series
expansion of $A$, $B$, $C$, $D$ at points in $\text{\bf C}^+$,
$\hat a$, $\hat b$, $\hat c$, $\hat d$ as exponents of the leading
terms in the power series expansion of $A$, $B$, $C$, $D$ at
points in $\mathbb{C}^-$, and
 $p(z)$, $\hat p(z)$ as polynomials of degrees strictly less than
$-(u_1+\dots+u_k)$, $-(\hat u_1+\dots+\hat u_k)$ respectively.
Note that it implies either $XA$ or $XB$ has a pole at $p_i^+$ if
$u_i<0$. At these points, we can determine the $u_i$ number of
negative terms of $$ u_{12}(z)=u_{i, u_i}(z-p_i^+)^{u_i}+ u_{i,
u_i+1}(z-p^+_i)^{u_i+1}+\dots+u_{i,-1}(z-p^+_i)^{-1}+
\text{h.o.t.} $$ at $p^+_i$, in order for both $XA-u_{12} C/X$ and
$XB-u_{12} D/X$ to be holomorphic in $\mathbb{C}^+$. Thus,
$u_{12}$ is uniquely defined. Respectively, we can define $\hat
p$, and hence $u_{21}$, by asserting the $\hat u_i$ number of
negative terms of $$ u_{21}(z)=\hat u_{i, \hat u_i}(z-z_i^-)^{\hat
u_i}+ \hat u_{i, \hat u_i+1}(z-z^-_i)^{\hat u_i+1}+\dots+\hat
u_{i,-1}(z-z^-_i)^{-1}+ \text{h.o.t.} $$ in order for both $C/X
-u_{21}XA$ and $D/X-u_{21}XB$ to be holomorphic in $\mathbb{C}^-$.

One can verify the topological constraint (2.3-c) of $\{V,
U^\pm\}$ by:
\begin{align*}
 -\#(&\text{poles of
$u_{12}$})+\#(\text{poles of $u_{21}$})\\&=(u_1+\dots+u_k)-(\hat
u_1+\dots+\hat u_k)\\ &=\sum_{z\in\text{\bf
C}^+}(\min\{a,b\}+\min\{c,d\})-\sum_{z\in\text{\bf
C}^-}(\min\{\hat a,\hat b\}+\min\{\hat c,\hat d\})\\
=&\#\bigl(\{\text{zeros of }A\}\underset{\min}{\cap}\{\text{zeros
of }B\}\bigr) +\#\bigl(\{\text{poles of
}A\}\underset{\max}{\cup}\{\text{poles of }B\}\bigr)\\
&-\#\bigl(\{\text{zeros of }C\}\underset{\min}{\cap}\{\text{zeros
of }D\}\bigr)-\#\bigl(\{\text{poles of
}C\}\underset{\max}{\cup}\{\text{poles of }D\}\bigr)\
\end{align*}
By (2.6), (2.6'), condition 5) in $D_-$, and $V_{22}=1$, result in
\begin{align*}
 &\sum_{z\in\text{\bf
C}^+}\bigl (\min\{\tilde a,\tilde b\} +\min\{\tilde c,\tilde
d\}\bigr)- \sum_{z\in\mathbb{C}^-}\bigl (\min\{\tilde a,\tilde b\}
+\min\{\tilde c,\tilde d\}\bigr)\\ &=\frac
1{2\pi}\int_{\mathbb{R}}d\,\arg(\tilde A_-\tilde D_+-\tilde
B_-\tilde C_+)+\frac 1{2\pi}\int_{\mathbb{R}}d\,\arg(\tilde
A_+\tilde D_--\tilde B_+\tilde C_-)\\ &=\frac
1{2\pi}\int_{\mathbb{R}}d\,\arg\,\tilde X_-\tilde Y_+(\tilde
A_-\tilde D_+-\tilde B_-\tilde C_+)+\frac 1{2\pi}\int_{\text{\bf
R}}d\,\arg\,\tilde X_+\tilde Y_-(\tilde A_+\tilde D_--\tilde
B_+\tilde C_-)\\ &=\frac 1{2\pi}\int_{\mathbb{R}}d\,\arg( A_- D_+-
B_- C_+)+\frac 1{2\pi}\int_{\mathbb{R}}d\,\arg( A_+ D_-- B_+C_-)\\
&=\frac 1{2\pi}\int_{\mathbb{R}}d\,\arg\, X_-Y_+( A_- D_+- B_-
C_+)+\frac 1{2\pi}\int_{\mathbb{R}}d\,\arg\, X_+Y_-( A_+ D_-- B_+
C_-)\\&=\frac 1{2\pi}\int_{\mathbb{R}}d\,\arg V_{11}.
\end{align*}
In summary, we have defined $V$, $U^\pm$, and proved that
$\{V,U^\pm\}$ satisfies (2.3), to become a formal scattering data
\cite{BC1,BC2}. Therefore, the inverse scattering theory
\cite{BC1,BC2} implies generically, the existence of a normalized
eigenfunction, denoted as $ m(x,z)$, such that its associated
scattering data is $\{V,U^\pm\}$.

\paragraph{Step 2. Proof of $\sigma\tau(g)=m(0,x)$}
By the results of {\it Step 1}, there exists $\eta^\pm(z)$,
holomorphic in $\mathbb{C}^\pm$, such that $$ \sigma
\tau(g)=\begin{pmatrix} 1&u_{12}\\0&1\end{pmatrix}\eta^+\text{ in
$\mathbb{C}^+$, }\sigma \tau(g)=\begin{pmatrix}
1&0\\u_{21}&1\end{pmatrix}\eta^-\text{ in $\mathbb{C}^-$. } $$
Thereby,
\begin{equation*}
\eta_+\eta_-^{-1}=\begin{pmatrix}1&-u_{12}\\0&1\end{pmatrix}
\sigma_+\tau(g)_+\tau(g)_-^{-1}\sigma_-^{-1}\begin{pmatrix}
1&0\\u_{21}&1\end{pmatrix}.\tag{3.1}
\end{equation*}
On the other hand, by inverse scattering theory, there exists
$\tilde \eta^\pm(z)$, holomorphic in $\mathbb{C}^\pm$, such that
$$m(0,z)=\begin{pmatrix}
1&u_{12}\\0&1\end{pmatrix}\tilde\eta^+\text{ in $\mathbb{C}^+$,
}m(0,z)=\begin{pmatrix} 1&0\\u_{21}&1\end{pmatrix}\tilde\eta^-
\text{ in $\mathbb{C}^-$.} $$ Hence,
\begin{equation*}
\tilde\eta_+\tilde\eta_-^{-1}=\begin{pmatrix}
1&-u_{12}\\0&1\end{pmatrix} V\begin{pmatrix}
1&0\\u_{21}&1\end{pmatrix}.\tag{3.1'}
\end{equation*}
Therefore defining $\eta(z)=\eta^\pm(z)$, and $\tilde
\eta(z)=\tilde \eta^\pm(z)$ when $z\in\mathbb{C}^\pm$, and
applying (3.1), (3.1'), $\eta\to 1$, $\tilde\eta\to 1$ as
$z\to\infty$, and the holomorphicy of $\eta^\pm$, $\tilde\eta^\pm$
in $\mathbb{C}^\pm$, we obtain $\eta=\tilde \eta$. Hence,
$\sigma\tau(g)=m(0,x)$.


\paragraph{Step 3. Proof of factorization of $g$}
Applying the results of {\it Step 2} and Lemma \ref{lm1}, we
derive
\begin{align*}
g(z)^{-1}e^{xzJ} =&\tau(g)^{-1}e^{xzJ}\begin{pmatrix} \tilde
X&0\\0&\tilde Y\end{pmatrix}\\
=&m(0,z)^{-1}e^{xzJ}\sigma\begin{pmatrix} \tilde X&0\\0&\tilde
Y\end{pmatrix}\\
=&E_{\tau(g)}(x,z)m(x,z)^{-1}\sigma\begin{pmatrix} \tilde
X&0\\0&\tilde Y\end{pmatrix}\\ =&E_{\tau(g)}(x,z)M_g(x,z)^{-1},
\end{align*}
for all $x\in \mathbb{R}$, all $z\in \mathbb{C}\backslash
(\mathbb{R}\cup P)$, with $E(x,\cdot)\in G_+$, and $P$ as poles of
$g$. With
 $E_{\tau(g)}(0,z)=1$,  and the property that integer-valued
 continuous functions are constant, it readily demonstrates that
 $M(x,\cdot)\in D_-$.\hfill $\square$

\begin{thebibliography}{00} \frenchspacing

\bibitem{BC1} Beals, R., Coifman, R. R.,
\textit{Scattering and inverse scattering for first order
systems}, Commun. Pure. Appl. Math., Vol. 37, (1984), 39-90.

\bibitem{BC2} Beals, R., Coifman, R. R.,
\textit{Inverse scattering and evolution equations},
 Commun. Pure. Appl. Math., Vol. 38 (1985), 29-42.


\bibitem{DGS}  Dorfmeister, J.,Gradl, J., and Szmigielski, J.,
 \textit{Systems of PDE obtained from the factorization in loop
groups}, Acta App. Math., Vol. 53 (1998), 1-58.

\bibitem{DrSo} Drinfel'd, V. G., and Sokolov, V. V.,
 \textit{Equations of Korteweg-de Vries type and simple Lie algebras},
Dokl. Akad. Nauk SSSR, Vl. 258 (1981), 11-16.

\bibitem{FNR1} Flaschka, H., Newell, A. C., and Ratiu, T.,
\textit{Kac-Moody Lie algebras and soliton equations, II. Lax
equations associated with $A_1^{(1)}$}, Physica Vol. 9D (1983),
300-323.

\bibitem{FNR2} Flaschka, H., Newell, A. C., and Ratiu, T.,
\textit{Kac-Moody Lie algebras and soliton equations, III.
Stationary equations associated with $A_1^{(1)}$}, Physica, Vol.
9D (1983), 324-332.

\bibitem{FT} Faddeev, L. D., and Takhtajan, L. A.,
\textit{Hamiltonian methods in the theory of solitons}, Springer
series in Soviet Math. (1987).

\bibitem{HSS} Haak, G., Schmidt, M., and Schrader, R.,
\textit{Group-theoretic formulation of the Segal-Wilson approach
to integrable systems with applications}, Rev. in Math. Phys.,
Vol. 4 (1992), 451-499.

\bibitem{PS} Pressley, A., and Segal, G. B.,
 \textit{Loop groups}, 1986, Oxford Science Publ., Clarendon Press,
Oxford.

 \bibitem{RS} Reyman, A. G., and
Semenov-Tian-Shansky, M. A., \textit{Current algebras and
non-linear partial differential equations}, Sov. Math., Dokl. Vol.
21 (1980),  630-634.

\bibitem{S} Sattinger, D. H., \textit{Flat connections and
scattering theory on the line},  SIAM J. Math. Anal. Vol.21(1990),
729-756.

\bibitem{Sa} Sato, M. \textit{Soliton equations as dynamical
systems on infinite dimensional Grassmann manifolds}, RIMS
Kokyuroku , vol. 439(1981), 30-40.

\bibitem{SaSz}  Sattinger, D. H., Szmigielski, J.
\textit{Factorization and the dressingmethod for the Gelfand-Dikii
hierarchy}, Physica D Vol. 64 (1993), 1-34.

\bibitem{Se1} Semenov-Tian-Shansky, M. A., \textit{Dressing
transformations and Poisson group actions}, Publ. RIMS Kyoto
Univ., Vol. 21 (1985), 1237-1260.

\bibitem{Se2} Semenov-Tian-Shansky, M. A.m
\textit{Classical $r$-matrices, Lax equations, Poisson Lie groups,
and dressing transformations}, Lecture Notes in Physics,
Springer-Verlag, Vol. 280 (1986), 174-214.

\bibitem{SW} Segal, G. and Wilson, G.,
\textit{Loop groups and equations of KDV type}, Publ. Math. IHES ,
Vol. 61 (1985), 5-65.

\bibitem{SZ} Sattinger, D. H., and Zurkowski, V. D.,
\textit{Gauge theory of B$\ddot a$cklund transformations, II},
Phys., D. Vol. 26 (1987), 225-250.

\bibitem{TU1} Terng, C. L., and Uhlenbeck, K.,
 \textit{Poisson group actions and scattering theory for
 integrable systems},
Survey in Differential Geometry survey volume, integrable systems
(1998), Cambridge, M A, international Press.

\bibitem{TU2} Terng, C. L., and Uhlenbeck, K.,
 \textit{B$\ddot a$cklund transformations and loop group actions},
Comm. Pure Appl. Math., Vol. 53 (2000), 1-75.

\bibitem{W} Wu, D. C., \textit{Global Birkhoff factorization
on loop groups of the ZS-AKNS flows}, Inverse Problems, Vol. 18
(2002), 95-109.

\end{thebibliography}

\noindent\textsc{Derchyi Wu }\\ 
Institute of Mathematics, Academia Sinica\\ 
Taipei, Taiwan, ROC\\ 
e-mail: mawudc@ccvax.sinica.edu.tw
\end{document}