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KdV, inverse scattering, finite-gap background, steplike
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\DeclareMathOperator{\wronsk}{\textup{\textsf{W}}}



\numberwithin{equation}{section}

\begin{document}

\title[On the KdV Equation with Steplike Finite-Gap Initial Data]{On the Cauchy Problem for the
Korteweg--de Vries Equation with Steplike Finite-Gap Initial Data II. Perturbations with Finite Moments}



\author[I. Egorova]{Iryna Egorova}
\address{B. Verkin Institute for Low Temperature Physics\\
47 Lenin Avenue\\61103 Kharkiv\\Ukraine}
\email{\mailto{iraegorova@gmail.com}}

\author[G. Teschl]{Gerald Teschl}
\address{Faculty of Mathematics\\
Nordbergstrasse 15\\ 1090 Wien\\ Austria\\ and\\ International
Erwin Schr\"odinger
Institute for Mathematical Physics\\ Boltzmanngasse 9\\ 1090 Wien\\ Austria}
\email{\mailto{Gerald.Teschl@univie.ac.at}}
\urladdr{\url{http://www.mat.univie.ac.at/~gerald/}}

\thanks{Research supported by the Austrian Science Fund (FWF)
 under Grant No.\ Y330.}
%\thanks{.... (to appear)}

\keywords{KdV, inverse scattering, finite-gap background, steplike}
\subjclass[2000]{Primary 35Q53, 37K15; Secondary 37K20, 81U40}


\begin{abstract}
We solve the Cauchy problem for the Korteweg--de Vries equation with
steplike finite-gap initial conditions under the assumption
that the perturbations have a given number of derivatives and moments finite.
\end{abstract}

\maketitle


\section{Introduction}



The purpose of the present paper is to investigate the Cauchy problem for the
Korteweg--de Vries (KdV) equation
\beq\label{KdV}
q_t(x,t) = -q_{xxx}(x,t) + 6 q(x,t) q_x(x,t), \qquad q(x,0)= q(x),
\eeq
(where subscripts denote partial derivatives as usual) for the case of steplike
initial conditions $q(x)$. More precisely, we will assume
that $q(x)$ is asymptotically close to (in general) different finite-gap potentials
$p_\pm(x)$ in the sense that
\beq\label{2.111}
\pm \int_0^{\pm \infty} \left|
\frac{d^n}{dx^n}\big( q(x) - p_\pm(x)\big) \right| (1+|x|^{m_0})dx <\infty,
\quad 0\leq n\leq n_0,
\eeq
for some positive integers $m_0, n_0$. If \eqref{2.111} holds for all $m_0, n_0$
we will call it a Schwartz-type perturbation. We will denote the spectra of
the one-dimensional finite-gap Schr\"{o}dinger operators $L_\pm = - \pa_x^2 + p_\pm$
associated with the potentials $p_\pm(x)$ by
\beq
\sigma_\pm = [E_0^\pm, E_1^\pm]\cup\dots\cup[E_{2j-2}^\pm,
E_{2j-1}^\pm]\cup\dots\cup[E_{2r_\pm}^\pm,\infty).
\eeq
The various possible locations of the two spectra is illustrated in the  following example.
%-------------------%
\vskip 0.2cm\noindent
{\bf Example.}
Let $L_+$ be the two-band operator with spectrum
$\sigma_+=[E_1, E_2]\cup[E_4, +\infty)$ and $L_-$ the
three band operator with spectrum $\sigma_-=[E_1, E_2]\cup[E_3,
E_4]\cup [E_5,+\infty)$, where $E_1<E_2<\cdots<E_5$.
%-------------------%
\begin{figure}[ht]
\begin{picture}(11,1.2)
\put(1,0.2){$\sigma_-$} \put(1,0.7){$\sigma_+$}

\put(1,0.5){\line(1,0){8}} \put(3,0.4){\rule{9mm}{2mm}}
\put(2.9,0){$\scriptstyle E_1$} \put(3.8,0){$\scriptstyle E_2$}
\put(5,0.4){\rule{11mm}{1mm}} \put(4.9,0){$\scriptstyle E_3$}
\put(6,0){$\scriptstyle E_4$}
\put(6.1,0.5){\rule{39mm}{1mm}}
\put(7.7,0.4){\rule{23mm}{1mm}} \put(7.6,0){$\scriptstyle E_5$}
\end{picture}
\caption{Typical locations of $\sigma_-$ and
$\sigma_+$.}\label{figsi}
\end{figure}

In order to solve the Cauchy problem for the KdV equation \eqref{KdV} with initial data satisfying
\eqref{2.111} for suitable $m_0,n_0$, our main ingredient will be the inverse
scattering transform pioneered by Gardner, Green, Kruskal, and Miura \cite{GGKM}.
In the case where $p_\pm(x) \equiv 0$ this is of course classical and we refer to
the monographs by Eckhaus and Van Harten \cite{EVH}, Marchenko \cite{M},
Novikov, Manakov, Pitaevskii, and Zakharov \cite{NMPZ}, or Faddeev and Takhtajan
\cite{FT}. However, when one investigates the case of steplike constant potentials,
$p_\pm(x)=c_\pm$, the current knowledge decreases drastically! The foundations
for scattering theory are completely understood and were given in Buslaev and Fomin \cite{BF},
Davies and Simon \cite{DS}, Cohen and Kappeler \cite{CK0}, Gesztesy \cite{G}, and
Aktosun \cite{A}.

The corresponding Cauchy problem for the KdV equation was first investigated by Cohen
in \cite{C} where she solves the case when $q(x)$ is the Heaviside step function. Later on
Kappeler \cite{Kap} showed how to handle general initial conditions, with only a fixed number
of moments finite, based on some advances in scattering theory in Cohen and Kappeler \cite{CK1}
(the time evolution of the scattering data for the entire KdV hierarchy were computed
recently by Khasanov and Urazboev \cite{KU}).
However, while his result is impressive from a technical point of view, it still does not
give a satisfactory answer since it only determines the decay properties of the solution
near one side, whereas concerning the decay properties at the other side only very mild
information is given. In particular, even if one starts with a Schwartz-type initial condition,
the results in \cite{Kap} will not guarantee that the solution stays within this
class. The reason for this is that \cite{Kap} (as well as \cite{C}) does not use the full
inverse scattering machinery but only a half-sided approach. For further results where
the initial condition is supported on a half-line see Rybkin \cite{ry} and
the references therein.

To shed some additional light on this we recall that the Marchenko kernel $F_\pm(x,y,t)$
(cf.\ \eqref{Fhat}) consists of three summands $F_\pm(x,y,t)= F_{\pm,D}(x,y,t) + F_{\pm,H}(x,y,t)
+ F_{\pm,R}(x,y,t)$, the first one being a sum over all eigenvalues, the second one an integral over
$\si_\mp\backslash\si_\pm$, and the last one an integral over $\si_\pm$. The crucial part is
to show decay properties of $F_\pm(x,y,t)$ (and its partial derivatives). The first term
$F_{\pm,D}(x,y,t)$ is as nice as one can wish for and can thus be ignored. To get the
necessary decay for the remaining two terms one needs to use integration by parts.
In the classical case (and more general at
points $E_1$ and $E_2$ in our example), the corresponding boundary terms arising during
integration by parts will vanish and the required decay follows. However in a steplike
situation (at points $E_4$ and also $E_5$ for the $-$ case), this is no longer true.
Moreover, at a point like $E_5$ the integrand of $F_{+,R}$ has a non-differentiable singularity
which prevents an immediate integration by parts. By just working with the other kernel
one can evade this obstacle at the price of loosing the information about decay of the solution
at this side. Clearly these problems evaporate if points like $E_4$ and $E_5$ in our
example are absent, and this was the case established in \cite{EGT} (there we also establish
some necessary technical ingredients which we will use freely here). Note that while this
restriction (which says that the respective spectral bands either coincide
or are disjoint) excludes the steplike constant case, it includes the case of short-range
perturbations of arbitrary finite-gap solutions.

It is the aim of the present paper to overcome these problems. To this end, rather than looking
at the terms $F_{\pm,H}(x,y,t)$ and $F_{\pm,R}(x,y,t)$ individually, we will in fact show that the
boundary terms mutually cancel. While this sounds like a pretty straightforward strategy,
this cancellation is by no means obvious and is nothing short of a little miracle.
Interestingly, points like $E_4$ and $E_5$ (the first one being absent in the
steplike constant case) turn out to require somewhat different miracles, with the first
being more involved. In particular, our result finally settles the aforementioned open problem of
steplike constant Schwartz-type perturbations as a special case. Moreover, based on
the recent advances in inverse scattering theory with steplike finite-gap backgrounds in
\cite{BET} (cf.\ also \cite{GNP}) and our preparations in \cite{EGT} we are able to handle
not only steplike constant but arbitrary steplike finite-gap backgrounds. For more on the
history of this problem and additional literature we refer to \cite{EGT}.

Next, let us state our main result.
Denote by $C^n(\R)$ the set of functions $x\in\R \mapsto q(x)\in \R$ which have $n$ continuous
derivatives with respect to $x$  and by $C^n_k(\R^2)$ the set of functions $(x,t)\in\R^2 \mapsto q(x,t)\in \R$
which have $n$ continuous derivatives with respect to $x$ and $k$ continuous derivatives with respect to $t$.

\begin{theorem}\label{theor1}
Let $p_\pm(x,t)$ be two real-valued finite-gap solutions of the KdV equation
corresponding to arbitrary finite-gap initial data $p_\pm(x)=p_\pm(x,0)$.
Let $m_0\geq 8$ and $n_0\geq m_0+5$ be fixed natural numbers.

Suppose, that $q(x)\in C^{n_0}(\R)$ is a real-valued function such that
\eqref{2.111} holds. Then there exists a unique classical solution $q(x,t)\in C^{n_0 - m_0-2}_1(\R^2)$
of the initial-value problem for the KdV equation \eqref{KdV} satisfying
\beq \label{1.71}
\pm \int_0^{\pm \infty} \left| \frac{\pa^n}{\pa x^n} \big( q(x,t) -
p_\pm(x,t)\big) \right| (1+|x|^{\floor{\frac{m_0}{2}} - 2})dx
<\infty, \quad n \leq n_0 - m_0 -2,
\eeq
for all $t\in\R$.
\end{theorem}

In particular, this theorem shows that the KdV equation
has a solution within the class of steplike Schwartz-type perturbations of finite-gap potentials:

\begin{corollary}
Let $p_\pm(x,t)$ be two real-valued finite-gap solutions of the KdV equation
corresponding to arbitrary finite-gap initial data $p_\pm(x)=p_\pm(x,0)$.
In addition, suppose, that $q(x)$ is a steplike Schwartz-type perturbations of $p_\pm(x)$.
Then the solution $q(x,t)$ of the initial-value problem for the KdV equation \eqref{KdV}
is a steplike Schwartz-type perturbations of $p_\pm(x,t)$ for all $t\in\R$.
\end{corollary}




\section{The inverse scattering transform for the KdV equation with steplike
finite-gap initial data}




In \cite{EGT} we have established the inverse scattering transform for the KdV
equation in the case of Schwartz-type perturbations. In this section we will
review the necessary steps and identify the required changes for the
present more general situation. These changes will then be preformed in
the next section.

To set the stage let
\beq
L_\pm(t) =-\frac{d^2}{d x^2} +p_\pm(x,t)
\eeq
be two one-dimensional Schr\"{o}dinger operators, corresponding to
two finite-gap solutions $p_\pm(x,t)$ of the KdV equation that are associated
with the spectra
\beq\label{1.61}
\sigma_\pm = [E_0^\pm, E_1^\pm]\cup\dots\cup[E_{2j-2}^\pm,
E_{2j-1}^\pm]\cup\dots\cup[E_{2r_\pm}^\pm,\infty)
\eeq
and the Dirichlet divisors
\beq\label{divisor}
\left\{\big(\mu_1^\pm(t),\si_1^\pm(t)\big), \dots, \big(\mu_{r_\pm}^\pm(t),
\si_{r_\pm}^\pm(t)\big)\right\},
\eeq
respectively. Here we assume without loss of
generality that all gaps are open, that is, $E_{2j-1}^\pm<E_{2j}^\pm$ for
$j=1,2,...,r_\pm$. We will abbreviate $\mu_j^\pm(0)=\mu_j^\pm$,
$\si_j^\pm(0)=\si_j^\pm$.

Let us cut the complex plane along the spectrum $\sigma_\pm$ and
denote the upper and lower sides of the cuts by $\sipmu$ and
$\sipml$. The corresponding points on these cuts will be denoted by
$\lau$ and $\lal$, respectively. In particular, this means
\beq
f(\lau) := \lim_{\varepsilon\downarrow0} f(\la+\I\varepsilon),
\qquad f(\lal) := \lim_{\varepsilon\downarrow0}
f(\la-\I\varepsilon), \qquad \la\in\sigma_\pm.
\eeq
Set
\beq\label{1.0}
Y_\pm(\la)=-\prod_{j=0}^{2r_\pm} (\la-E_j^\pm),
\eeq
and
introduce the functions
\beq\label{1.88}
g_\pm(\la,t)=
-\frac{\prod_{j=1}^{r_\pm}(\la - \mu_j^\pm(t))}{2 Y_\pm^{1/2}(\la)},
\eeq
where the branch of the square root is chosen such that
\beq\label{1.8}
\frac{1}{\I} g_\pm(\lau,t) = \Im(g_\pm(\lau,t))  >0
\quad \mbox{for}\quad \la\in\sigma_\pm,\quad t\in\R_+.
\eeq
Denote by
\beq\label{psin}
\psi_\pm(\la,x,t)=c_\pm(\la,x,t)+ m_\pm(\la,t)s_\pm(\la,x,t)
\eeq
the  Weyl solutions of the equations
\beq\label{eqqp}
L_\pm(t)y=\la y,
\eeq
normalized according to $\psi_\pm(\la,0,t)=1$ and such, that $\psi_\pm(\la,\cdot,t)\in
L^2(\R_\pm)$ for $\la\in\C \setminus\si_\pm$. Here $m_\pm(t)$ are
the Weyl functions and $c_\pm(\la,x,t)$ and $s_\pm(\la,x,t)$ are
solutions of \eqref{eqqp}, satisfying the initial conditions
\beq
c_\pm(\la,0,t)=s_\pm^\prime(\la,0,t)=1, \quad s_\pm(\la,0,t)=
c_\pm^\prime(\la,0,t)=0.
\eeq
The functions $\psi_\pm$ admit the well-known representation
\beq\label{1.23}
\psi_\pm(\la,x,t)=u_\pm(\la,x,t)\E^{\pm\I\theta_\pm(\la)x},
\quad\la\in\C\setminus\si_\pm,
\eeq
where  $\theta_\pm(\la)$ are the
quasimoments and the functions $u_\pm(\la,x,t)$ are quasiperiodic
with respect to $x$ with the same basic frequencies as the
potentials $p_\pm(x,t)$. The quasimoments are holomorphic for
$\la\in\C\setminus\si_\pm$ and normalized according to
\beq\label{1.24}
\frac{d\theta_\pm}{d\la}>0 \quad
\mbox{as}\quad\la\in\sipmu,\qquad \theta_\pm(E_0^\pm)=0.
\eeq
This
normalization implies
\beq\label{1.25}
\frac{d\theta_\pm}{d\la}=\frac{\I\prod_{j=1}^{r_\pm}(\la -
\zeta_j^\pm)}{ Y_\pm^{1/2}(\la)},\qquad \zeta_j^\pm\in(E_{2j-1}^\pm,
E_{2j}^\pm),
\eeq
and therefore, the quasimoments are real-valued on
$\sipmul$. Note, that in the case where $p_\pm(x,t)\equiv0$ we have
$\theta_\pm(\la)=\sqrt{\la}$, $u_\pm(\la,x,t)\equiv 1$ and
$m_\pm(\la,t)=\pm\I\sqrt\la$. In the general finite-gap cases the
two Weyl $m$-functions associated with $L_\pm$ are given by
(\cite[eq.~(1.165)]{GH})
\beq\label{1.29}
m_\pm(\la,t)=\frac{H_\pm(\la,t)\pm
Y_\pm^{1/2}(\la)}{\prod_{j=1}^{r_\pm}(\la - \mu_j^\pm(t))}, \quad
\breve m_\pm(\la,t)=\frac{H_\pm(\la,t)\mp
Y_\pm^{1/2}(\la)}{\prod_{j=1}^{r_\pm}(\la - \mu_j^\pm(t))}.
\eeq
Here $H_\pm(\la,t)$ are  polynomials with respect $\la$ of
$\deg(H_\pm)\leq r_\pm-1$ with real-valued coefficients and smooth
with respect to $t$. Moreover,
\beq\label{ischez}
H_\pm(\mu_j^\pm(t),t)=0\quad\mbox{for}\quad
\mu_j^\pm(t)\in\pa\si_\pm.
\eeq
 Associated with the second Weyl
$m$-function $\breve m_\pm(\la,t)$ is the second Weyl solution
\begin{align}\nn
\breve\psi_\pm(\la,x,t) &= c_\pm(\la,x,t)+\breve m_\pm(\la,t) s_\pm(\la,x,t)\\\label{1.30}
&=  \breve u_\pm(\la,x,t)\E^{\mp\I\theta_\pm(\la)x},
\quad\la\in\C\setminus\si_\pm,
\end{align}
such that $\breve\psi_\pm(\la,\cdot,t)\in L^2(\R_\mp)$ for $\la\in\C
\setminus\si_\pm$. The Wronski determinant, $\wronsk(f,g)(x)=f(x)g'(x)-f'(x)g(x)$, of the functions $\psi_\pm$ and
$\breve\psi_\pm$ is given by
\beq\label{wrpsi}
\wronsk(\psi_\pm(\la,.,t),\breve\psi_\pm(\la,.,t)) = \pm g_\pm(\la,t)^{-1}.
\eeq
Introduce the Lax operators corresponding to the finite-gap solutions $p_\pm(x,t)$,
\begin{align}\label{Lop}
L_\pm(t) &= -\pa_x^2 + p_\pm(x,t),\\
P_\pm(t) &=  -4\pa_x^3 + 6p_\pm(x,t)\pa_x +3 \pa_x p_\pm(x,t).
\end{align}
Then the following result is valid (\cite{BBEIM}, \cite{GH})
\begin{lemma}\label{lemweyl1}
The functions
\beq\label{1.37}
\hat\psi_\pm(\la,x,t) = \E^{\alpha_\pm(\la,t)} \psi_\pm(\la,x,t),
\eeq
where
\beq\label{1.38}
\alpha_\pm(\la,t) := \int_0^t \left(2(p_\pm(0,s) + 2\la)
m_\pm(\la,s) - \frac{\pa p_\pm(0,s)}{\pa x}\right)ds,
\eeq
satisfy the system of equations
\begin{align}\label{LP1}
L_\pm(t)\hat\psi_\pm &= \la\hat\psi_\pm,\\ \label{LP2}
\frac{\pa\hat\psi_\pm}{\pa t} &= P_\pm(t)\hat\psi_\pm.
\end{align}
\end{lemma}

Set
\begin{align}\label{Mset}
M_\pm(t) &=\{ \mu^\pm_j(t) \mid \mu^\pm_j(t) \in
(E_{2j-1}^\pm,E_{2j}^\pm) \text{ and } m_\pm(\la,t) \text{ has a
simple pole}\},\\ \nn \hat M_\pm(t) &=\{ \mu^\pm_j(t) \mid
\mu^\pm_j(t) \in \{E_{2j-1}^\pm, E_{2j}^\pm\} \},
\end{align}
and introduce the functions
\begin{align} \nn
\delta_\pm(\la,t) &:= \prod_{\mu^\pm_j(t) \in M_\pm(t)}(\la-\mu^\pm_j(t)),\\
\label{S2.6}
\hat \delta_\pm(\la,t) &:= \prod_{\mu^\pm_j(t) \in M_\pm(t)}
(\la-\mu_j^\pm(t)) \prod_{\mu^\pm_j(t) \in \hat M_\pm(t)} \sqrt{\la -
 \mu^\pm_j(t)},
\end{align}
where $\prod =1$ if the index set is empty.
These functions will allow us to remove the singularities of the Weyl solutions
$\psi_\pm(\la,x,t)$ whenever necessary.

Next we collect now some facts from scattering theory for Schr\"odinger operators
with smooth steplike finite-gap potentials (cf.\ \cite{BET}, \cite{EGT}).
To shorten notations we omit the dependence on $t$ throughout this
discussion.

Let $n_1\geq 0$ and $m_1\geq 2$ be given natural numbers and let
 $q(x)\in C^{n_1}(\R)$ be a real-valued  function such that
\beq\label{S.2}
\pm \int_0^{\pm \infty}\left|\frac{d^n}{d x^n}\big(q(x) -
p_\pm(x)\big)\right| (1+|x|^{m_1})dx <\infty,
 \quad  \forall\, 0\leq n\leq n_1.
\eeq
Consider the {\em perturbed} operator
\beq\label{S.12}
L :=- \frac{d^2}{dx^2} +q(x)
\eeq
with a potential $q(x)$, satisfying \eqref{S.2}. The spectrum of $L$
consists of a purely absolutely continuous part
$\sigma:=\sigma_+\cup\sigma_-$ plus a finite number of eigenvalues $
\sigma_d=\{\la_1,\dots,\la_p\} $ situated in the gaps,
$\sigma_d\subset\R\setminus\sigma$.  The set
$\sigma^{(2)}:=\sigma_+\cap\sigma_-$ is the spectrum of multiplicity
two for the operator $L$ and $\sigma_+^{(1)}\cup\sigma_-^{(1)}$ with
$\sigma_\pm^{(1)}= \clos(\sigma_\pm\setminus\sigma_\mp)$ is the
spectrum multiplicity one.

%-------------------%
The Jost solutions of the spectral equation
\beq\label{S.4}
\left(-\frac{d^2}{dx^2}+q(x)\right) \phi(x)= \la \phi(x),\quad \la\in \C,
\eeq
are defined by the requirement that they asymptotically look like the
Weyl solutions of the background operators as $x\to\pm\infty$,




\begin{lemma}\label{lemJost5}
Assume \eqref{S.2}. Then there exist solutions
$\phi_\pm(\la, x)$, $\la \in \C$, of \eqref{S.4} satisfying
\beq
\phi_\pm(\la,x) = \psi_\pm(\la,x) (1 + o(1)), \qquad x\to\pm\infty.
\eeq
The Jost solutions $\phi_\pm(\la, .)$ are meromorphic with respect to $\la\in\C\backslash\si_\pm$
with the same poles as $\psi_\pm(\la, .)$ and $\hat\delta(\la)\phi_\pm(\la, .)$ are continuous up to the
boundary $\siu_\pm\cup\sil_\pm$. Moreover, $\hat\delta(\la)\phi_\pm(\la, .)$ are $m_1$ times differentiable
with respect to $\la \in \inte(\siu_\pm\cup\sil_\pm)$ and $m_1-1$ times continuously differentiable with
respect to the local variable $\sqrt{\la - E}$ near $E\in\pa\si_\pm$.
\end{lemma}

\begin{proof}
Set
\beq\label{J}
J_\pm(\la,x,y)=\frac{\psi_\pm(\la,y)\breve\psi_\pm(\la,x) -
\psi_\pm(\la,x)\breve\psi_\pm(\la,y)}{\wronsk(\psi_\pm(\la),\breve\psi_\pm(\la))}.
\eeq
Then the Jost solutions of \eqref{S.4} formally satisfy the following integral equation
\beq
\phi_\pm(\la,x) = \psi_\pm(\la,x) -
\int_x^{\pm\infty} J_\pm(\la,x,y) (q(y) - p_\pm(y)) \phi_\pm(\la,y)dy.
\eeq
To remove the singularities of $\psi_\pm(\la,x)$ near $\la\in M_\pm \cup \hat M_\pm$
one can multiply the whole equation by $\hat\delta_\pm(\la)$.

Similarly, the $x$ derivatives satisfy
\[
\frac{\pa}{\pa x} \phi_\pm(\la,x) = \frac{\pa}{\pa x} \psi_\pm(\la,x) -
\int_x^{\pm\infty} \left( \frac{\pa}{\pa x} J_\pm(\la,x,y) \right) (q(y) - p_\pm(y)) \phi_\pm(\la,y)dy.
\]
Hence existence of the Jost solutions together with their derivatives follows by proving
existence of solutions for these integral equations. This follows by the method of
successive iterations in the usual manner. Observe that since at points $\la\in\pa\si_\pm$
the second solution grows linearly, the above kernel can only be estimated by $C|x-y|$
near such points.
\end {proof}

We will also need the asymptotic behavior of the Jost solutions as $\la\to\infty$.
To this end we recall the well-know expansion
\beq\label{exppsi}
\psi_\pm(\la,x)=\exp\left(\pm\I \sqrt{\la} x
+\int_0^x\left(\sum_{j=1}^n \frac{\kappa^\pm_j(y)}
{(\pm 2\I \sqrt{\la})^j} +\frac{\tilde\kappa_{n,\pm}(\sqrt{\la},y)}
{(\pm 2\I \sqrt{\la})^n}\right)dy\right),
\eeq
up to any order $n$, where
\beq\label{expcpsi}
\kappa^\pm_1(x)=p_\pm(x),\quad
\kappa^\pm_{j+1}(x)=- \frac{\pa}{\pa x}\kappa^\pm_j(x)-\sum_{i=1}^{j-1}
\kappa^\pm_{j-i}(x)\kappa^\pm_i(x),
\eeq
and the error term satisfies
\beq
\frac{\pa^l }{\pa k^l}\tilde\kappa_{n,\pm}(k,x)=o(1), \quad  l=0,1,\dots
\eeq
for fixed $x$ as $k\to \infty$.

\begin{lemma}\label{lemasymphi}
Assume \eqref{S.2}. Then the Jost solutions have an asymptotic expansion
\begin{equation}
\phi_\pm(\la,x) =   \psi_\pm(\la,x) \left( 1 + \frac{\phi_{\pm,1}(x)}{\la^{1/2}} + \dots + \frac{\phi_{\pm,n_1+1}(x)}{\la^{(n_1+1)/2}} +
o\big(\la^{-(n_1+1)/2} \big)\right)
\end{equation}
which can be differentiated $m_1$ times with respect to $\la^{1/2}$. An analogous
expansion holds for $\frac{\pa}{\pa x} \phi_\pm(\la,x)$.
\end{lemma}

\begin{proof}
To obtain the asymptotic expansion consider $\tilde{\phi}_\pm(\la,x)= \frac{\phi_\pm(\la,x)}{\psi_\pm(\la,x)}$
which satisfy
\beq
\tilde{\phi}_\pm(\la,x) = 1 -
\int_x^{\pm\infty} \tilde{J}_\pm(\la,x,y) (q(y) - p_\pm(y)) \tilde{\phi}_\pm(\la,y)dy,
\eeq
where $\tilde{J}_\pm(\la,x,y) = J_\pm(\la,x,y) \frac{\psi_\pm(\la,y)}{\psi_\pm(\la,x)}$. Next recall
\eqref{1.23}, \eqref{1.30}, and \eqref{wrpsi} which imply
\[
\tilde{J}_\pm(\la,x,y) = \pm g_\pm(\la) \left( u_\pm(\la,y)^2 \frac{\breve u_\pm(\la,x)}{u_\pm(\la,x)}
\E^{\pm 2\I\theta_\pm(\la)(x-y)} - \breve u_\pm(\la,y) u_\pm(\la,y)\right),
\]
where $u_\pm(\la,x)$, $\breve u_\pm(\la,x)$ are quasiperiodic with respect to $x$ and have
convergent expansions around $\infty$ with respect to $\theta_\pm(\la)^{-1}$. Now use the fact that
\[
\int_0^\infty \E^{2 \I \theta(\la) y} f(\la,y) dy = \sum_{j=1}^n \frac{f_j}{\theta(\la)^j} + o(\theta(\la)^{-n})
\]
provided $f(\la,x)$ is $n$ times differentiable with respect to $x$ and the first $n-1$ derivatives
have an asymptotic expansion with respect to $\theta(\la)^{-1}$ of order $n$ and the $n$'th
derivative satisfies $\lim_{\la\to\infty} \frac{\pa^n}{\pa x^n} f(\la,x) = g(x)$ in $L^1(0,\infty)$.
This follows by $n$ partial integrations together with the Riemann-Lebesgue lemma (cf.\ 
also \cite[Theorem~3.2]{MT}).

The claims for the derivatives follow by considering the corresponding integral equations as
in the previous lemma.
\end{proof}

\begin{corollary}\label{corasymphi}
Assume \eqref{S.2}. Then the Weyl $m$-functions $m_{q,\pm}(\la,x)= \frac{\phi_\pm'(\la,x)}{\phi_\pm(\la,x)}$
have an asymptotic expansion
\beq
m_{q,\pm}(\la,x) =  \pm \I \sqrt{\la} + \sum_{j=1}^{n_1} \frac{\kappa_j(x)}{(\pm 2\I\sqrt{\la})^j} + o(\la^{-n_1/2}),
\eeq
which can be differentiated $m_1$ times with respect to $\la^{1/2}$. The coefficients $\kappa_j(x)$ are
given by \eqref{expcpsi} with $q(x)$ in place of $p_\pm(x)$.
\end{corollary}

\begin{proof}
Existence of the expansion follows from the previous lemma and the expansion coefficients follow
by comparing coefficients in the Riccati equation
\[
\frac{\pa}{\pa x} m_{q,\pm}(\la,x) + m_{q,\pm}(\la,x)^2 + \la - q(x) =0. 
\]
\end{proof}

The Jost solutions can be represented with the help of the transformation operators as
\beq\label{S2.2}
\phi_\pm(\la,x) =\psi_\pm(\la,x)\pm\int_{x}^{\pm\infty} K_\pm(x,y)
\psi_\pm(\la,y) dy,
\eeq
where $K_\pm(x,y)$ are real-valued functions that satisfy
\beq\label{A.5}
 K_\pm(x,x)=\pm\frac{1}{2}\int_x^{\pm\infty} (q(y)-p_\pm(y))dy.
\eeq
Moreover, as a consequence of \cite[(A.15)]{BET}, the following
estimate is valid
\beq\label{S2.3}
\left|\frac{\pa^{n+l}}{\pa x^n\pa
y^l} K_\pm(x,y)\right|\leq C_\pm(x)\left(Q_\pm(x+y)
+\sum_{j=0}^{n+l-1} \left|\frac{\pa^j}{\pa x^j}\big(q(\frac{x+y}{2})
-p_\pm(\frac{x+y}{2})\big)\right|\right),
\eeq
for $\pm y>\pm x$, where $C_\pm(x)=C_{n,l,\pm}(x)$ are continuous positive
functions decaying as $x\to\pm\infty$ and
\beq\label{S2.333}
Q_\pm(x):=
\pm\int_{\frac{x}{2}}^{\pm\infty} \big|q(y) - p_\pm(y)\big| dy.
\eeq
Formula \eqref{S2.2} shows, that the Jost solutions inherit all
singularities of the background  Weyl functions $m_\pm(\la)$ and
Weyl solutions $\psi_\pm(\la)$. In particular, as a direct
consequence of formulas \eqref{psin}, \eqref{1.25}, \eqref{1.29}, \eqref{ischez},
\eqref{S2.2}, and Lemma~\ref{lemJost5}
we have the following

\begin{lemma}\label{lemMhat}
Let $E\in\pa\si_\pm$ and $\varepsilon>0$ be such that
$[E-\varepsilon, E+\varepsilon]\cap\pa\si_\pm=\{E\}$ and
$\varepsilon \dist(\mu_j^\pm,E)$ if $\mu_j^\pm\neq E$.

\noindent
{\bf(i)} Let $\mu_j^\pm= E$.
Introduce the functions
\[
\phi_{\pm,E}(\la,x):=
\I\left(\theta_\pm(\la)-\theta_\pm(E)\right)\phi_\pm(\la,x),\
g_{\pm,E}(\la):=
\left|\theta_\pm(\la)-\theta_\pm(E)\right|^{-2}g_\pm(\la)
\]
for $\la [E-\varepsilon, E+\varepsilon]$. Then the function $\phi_{\pm,E}(\la,x)$ admits
the representation
\beq\label{struct3}
\phi_{\pm,E}(\la,x)=c_{\pm,E}(\la,x) +\I
\left(\theta_\pm(\la)-\theta_\pm(E)\right)s_{\pm,E}(\la,x),
\eeq
where $c_{\pm,E}(\cdot,x), s_{\pm,E}(\cdot,x)\in C^{m_0-1}([E-\varepsilon, E+\varepsilon])$
together with their $x$ derivatives and $c_{\pm,E}(\cdot,x), s_{\pm,E}(\cdot,x)\in \mathbb R$.
Moreover,
\beq\label{realval}
\phi_{\pm,E}(\la,x)\in\mathbb{R},\qquad \la\in
[E-\varepsilon, E+\varepsilon]\setminus\si_\pm,
\eeq
and the following formula is valid
\beq\label{xiwronsk}
g_{\pm,E}(\la)^{-1}=\pm\wronsk(\phi_{\pm,E},
\overline{\phi_{\pm,E}})\qquad \la\in (E-\varepsilon,
E+\varepsilon)\cap\si_\pm.
\eeq
{\bf (ii)} Let $\mu_j^\pm\neq E$.
Then the function $\phi_\pm(\la,x)$  admits the same representation
\eqref{struct3} on the set $[E-\varepsilon_1, E+\varepsilon_1]$.
\end{lemma}

Next, set (recall \eqref{S2.6})
\beq\label{S2.12}
\tilde\phi_\pm(\la,x)=\delta_\pm(\la) \phi_\pm(\la,x)
\eeq
such that the functions $\tilde\phi_\pm(\la,x)$ have no poles in the interior
of the gaps of the spectrum $\si$. For every eigenvalue $\la_k$ we
introduce the corresponding norming constants
\beq \label{S2.14}
\left(\gamma_k^\pm\right)^{-2}=\int_{\R} \tilde\phi_\pm^2(\la_k,x)
dx.
\eeq
Furthermore, recall the scattering relations
\beq\label{S2.16}
T_\mp(\la) \phi_\pm(\la,x)
=\overline{\phi_\mp(\la,x)} + R_\mp(\la)\phi_\mp(\la,x),
\quad\la\in\simpul,
\eeq
where the transmission and reflection
coefficients are defined as usual, \beq\label{2.17} T_\pm(\la):=
\frac{\wronsk(\overline{\phi_\pm(\la)},
\phi_\pm(\la))}{\wronsk(\phi_\mp(\la), \phi_\pm(\la))},\qquad
R_\pm(\la):= -
\frac{\wronsk(\phi_\mp(\la),\overline{\phi_\pm(\la)})}
{\wronsk(\phi_\mp(\la), \phi_\pm(\la))}, \quad\la\in \sipmul. \eeq

%-------------------%
\begin{lemma}\label{lem2.3} Suppose, that \eqref{S.2} is fulfilled. Then
the scattering data
\begin{align}\nn
{\mathcal S} = \Big\{ & R_+(\la),\;T_+(\la),\;
\la\in\sigma_+^{\mathrm{u,l}}; \; R_-(\la),\;T_-(\la),\;
\la\in\sigma_-^{\mathrm{u,l}};\\\label{S4.6} &
\la_1,\dots,\la_p\in\R\setminus \sigma,\;
\gamma_1^\pm,\dots,\gamma_p^\pm\in\R_+\Big\}
\end{align}
have the following properties:
\begin{enumerate}[\bf I.]
\item
\begin{enumerate}[\bf(a)]
\item
$T_\pm(\lau) =\overline{T_\pm(\lal)}$ for $\la\in\sigma_\pm$.\\
$R_\pm(\lau) =\overline{R_\pm(\lal)}$ for $\la\in\sigma_\pm$.
\item
$\dfrac{T_\pm(\la)}{\overline{T_\pm(\la)}}= R_\pm(\la)$
for $\la\in\sigma_\pm^{(1)}$.
\item
$1 - |R_\pm(\la)|^2 =
\dfrac{g_\pm(\la)}{g_\mp(\la)} |T_\pm(\la)|^2$ for
$\la\in\sigma^{(2)}$.
\item
$\overline{R_\pm(\la)}T_\pm(\la) +
R_\mp(\la)\overline{T_\pm(\la)}=0$ for
$\la\in\sigma^{(2)}$.
\item
$T_\pm(\la) = 1 + O\Big(\frac{1}{\sqrt\la}\Big)$ for
$\la\to\infty$.
\item
$R_\pm(\la) = o\Big(\frac{1}{\left(\sqrt{\la}\right)^{n_1+1}}\Big)$ for
$\la\to\infty$.
\end{enumerate}
\item
The functions $T_\pm(\la)$ can be extended as meromorphic
functions into the domain $\C \setminus \sigma$ and satisfy
\beq\label{S2.18}
\frac{1}{T_+(\la) g_+(\la)} = \frac{1}{T_-(\la) g_-(\la)}=:-W(\la),
\eeq
where $W(\la)$ possesses the following properties:
\begin{enumerate}[\bf(a)]
\item
The function $\tilde W(\la)=\delta_+(\la)\delta_-(\la) W(\la)$ is
holomorphic in the domain $\C\setminus\sigma$, with simple
zeros at the points $\la_k$, where
\beq\label{S2.11}
\frac{d\tilde W}{d \la}(\la_k) = (\gamma_k^+\gamma_k^-)^{-1}.
\eeq
In addition, it satisfies
\beq\label{S2.9}
\overline{\tilde W(\lau)}=\tilde W(\lal), \quad
\la\in\sigma\quad \text{and}\quad \tilde W(\la)\in\R
\quad \text{for} \quad \la\in\R\setminus \sigma.
\eeq
\item
The function $\hat W(\la) = \hat\delta_+(\la) \hat\delta_-(\la)
W(\la)$ is continuous on the set $\C\setminus\sigma$ up to the
boundary $\siu\cup\sil$. Moreover, this function is $m_1-1$ times
differentiable with respect to $\la$ on the set
$\left(\siu\cup\sil\right)\setminus (\pa\si_-\cup\pa\si_+)$ and
$m_1-1$ times continuously differentiable with respect to the local
variable $\sqrt{\la - E}$ for $E\in\pa\si_-\cup\pa\si_+$. It can have
zeros on the set $\pa\sigma_-\cap\pa\si_+$ and does not vanish
at the other points of the set $\sigma$.  If $\hat W(E)=0$ for
$E\in\pa\sigma_-\cap\pa\si_+$, then $\hat W(\la) = \sqrt{\la -E}
(C(E)+o(1))$, $C(E)\ne 0$.
\end{enumerate}
\item
\begin{enumerate}[\bf(a)]
\item
The reflection coefficients $R_\pm(\la)$ are continuous functions on $\siu_\pm\cup\sil_\pm$.
They are $m_1$ times differentiable with respect to $\la$ on the sets
$\sipmul\setminus\{\pa\si_+\cup\pa\si_-\}$ and
$m_1-j$ times differentiable with respect to the coordinate $\sqrt{\la-E}$ with
$E\in\{\pa\si_+\cup\pa\si_-\}\cap\sipmul$, where $j=1$ if $\hat W(E)\neq 0$ and
$j=2$ if $\hat W(E)\neq 0$.
The asymptotics {\bf I.~(f)} hold for all derivatives as well.
\item
If $E\in\pa\si_\pm$ and $\hat W(E)\neq 0$, then
\beq\label{P.1}
R_\pm(E)=
\begin{cases}
-1 &\text{for } E\notin\hat M_\pm,\\
1 &\text{for } E\in\hat M_\pm.
\end{cases}
\eeq
\end{enumerate}
\end{enumerate}
\end{lemma}

\begin{proof}
Except for {\bf I.~(f)} and the corresponding statement for the derivatives in {\bf III.~(a)} everything follows
as in \cite[Lem.~4.1]{EGT}. To prove property {\bf I.~(f)} we have to check, that $\wronsk(\phi_\pm,\ov{\phi_\mp})=o(\la^{-n_1/2})$
together with all necessary derivatives with respect to $\sqrt\la$. But this follows from
\[
\wronsk(\phi_\pm,\ov{\phi_\mp}) = \phi_-(\la,x) \ov{\phi_+(\la,x)} ( m_{q,-}(\la,x) - \ov{m_{q,+}(\la,x)})
\]
since $\phi_-(\la,x) \ov{\phi_+(\la,x)}=O(1)$ by Lemma~\ref{lemasymphi} and $m_{q,-}(\la,x) - \ov{m_{q,+}(\la,x)}=o(\la^{-n_1/2})$
by Corollary~\ref{corasymphi}.
\end{proof}


Next recall the associated Gelfand--Levitan--Marchenko (GLM) equations
\beq\label{ME}
K_\pm(x,y) + F_\pm(x,y) \pm \int_x^{\pm\infty} K_\pm(x,\xi)
F_\pm(\xi,y)d\xi =0, \quad \pm y>\pm x,
\eeq
where\footnote{Here we have used the notation
$\oint_{\sigma_\pm}f(\la)d\la := \int_{\sipmu} f(\la)d\la -
\int_{\sipml} f(\la)d\la$.}
\begin{align}\label{4.2}
F_\pm(x,y) &= \frac{1}{2\pi\I}\oint_{\sigma_\pm}
R_\pm(\la) \psi_\pm(\la,x) \psi_\pm(\la,y)
g_\pm(\la)d\la + \\ \nn &\quad + \frac{1}{2\pi
\I}\int_{\sigma_\mp^{(1),\mathrm{u}}} |T_\mp(\la)|^2
\psi_\pm(\la,x) \psi_\pm(\la,y)g_\mp(\la)d\la\\ \nn
&\quad + \sum_{k=1}^p (\gamma_k^\pm)^2 \tilde\psi_\pm(\la_k,x)
\tilde\psi_\pm(\la_k,y).
\end{align}
These functions $F_\pm(x,y)$ satisfy the following properties:

\begin{enumerate}[\bf I.]\it
\addtocounter{enumi}{3}
\item
$F_\pm(x,y)\in C^{(n_1+1)}(\R^2)$.\\
There exists real-valued functions
$\tilde q_\pm(\cdot)\in C^{n_1}(\R_\pm)$ with
\[
\pm\int_0^{\pm\infty}(1+|x|^{m_1}) |
\tilde q_\pm^{(n)}(x)| dx < \infty, \quad  0\leq n\leq n_1,
\]
such that
\beq \label{4.4}
\left|\frac{\pa^{n+l}}{\pa x^n\pa y^l}
F_\pm (x,y)\right|\leq C\, Q_\pm(x+y),\ \  \mbox{as}\
x,y\to\pm\infty, \quad  n+l\leq n_1+1,
\eeq
where
\[
Q_\pm(x):=\pm\int_{\frac{x}{2}}^{\pm\infty}\left|\tilde
q_\pm(s)\right|d s.
\]
Moreover,
\beq \label{4.5}
\pm\int_0^{\pm\infty} \left|\frac{d^{n}}{dx^{n}} F_\pm
(x,x)\right|(1+ |x|^{m_1}) dx <\infty,\qquad 1\leq n\leq n_1+1.
\eeq
\end{enumerate}


As it is demonstrated in \cite{BET} and \cite{EGT}, the
properties \textbf{I--IV} are necessary and sufficient for a
set $\mathcal{S}$ to be the set of scattering data for operator $L$
with a potential $q(x)$ from the class \eqref{S.2}.

Now the procedure of solving of the inverse scattering problem is as follows:

Let $L_\pm$ be two one-dimensional finite-gap Schr\"odinger
operators associated with the potentials $p_\pm(x)$. Let
$\mathcal{S}$ be given data as in \eqref{S4.6} satisfying
\textbf{I--IV}. Define corresponding kernels $F_\pm(x,y)$ via
\eqref{4.2}. As it shown in \cite{BET}, under  condition
\textbf{IV} the GLM equations \eqref{ME} have unique smooth
real-valued solutions $K_\pm(x,y)$, satisfying  estimates of type
\eqref{4.4}. In particular,
\beq\label{5.101}
\pm\int_0^{\pm\infty} (1+|x|^{m_1})\left|\frac{d^n}{dx^n}
K_\pm(x,x)\right|
dx<\infty, \qquad 1\leq n\leq n_1+1.
\eeq
Now introduce the functions
\beq\label{5.1}
q_\pm(x) = p_\pm(x) \mp 2\frac{d}{dx}K_\pm(x,x)
\eeq
and note that the estimate \eqref{5.101} reads
\beq\label{5.2}
\pm \int_0^{\pm \infty}\left| \frac{d^n}{dx^n} \big( q_\pm(x) -
p_\pm(x)\big) \right| (1+|x|^{m_1}) d x <\infty ,\quad 0\leq n\leq n_1.
\eeq
Moreover, following \cite{BET} one obtains

\begin{theorem}[\cite{BET}]\label{theor4}
Let the data ${\mathcal S}$, defined as in \eqref{S4.6},
satisfy the properties \textbf{I--IV}. Then the functions $q_\pm(x)$ defined by \eqref{5.1}
satisfy \eqref{5.2} and coincide, $q_-(x)\equiv q_+(x)=:q(x)$. Moreover, the set ${\mathcal S}$
is the set of scattering data for the Schr\"odinger operator \eqref{S.12} with
potential $q(x)$ from the class \eqref{S.2}.
\end{theorem}


Our next step is to describe a formal scheme for solving the
initial-value problem for the KdV equation with initial data from
the class \eqref{2.111} with fixed $m_0\geq 8$ and $n_0\geq m_0+5$ by
the inverse scattering method.


Suppose the initial data $q(x)$ satisfies condition \eqref{2.111}
with some finite-gap potentials $p_\pm(x)$. Consider the
corresponding scattering data $\mathcal{S}=\mathcal{S}(0)$ which
obey conditions \textbf{I--IV} with $n_1=n_0$ and
$m_1=m_0$. Let $p_\pm(x,t)$ be the finite-gap solution of the KdV
equation with initial conditions $p_\pm(x)$ and let $m_\pm(\la,t)$,
$\breve m_\pm(\la,t)$ $\psi_\pm(\la,x,t)$,  $\alpha_\pm(\la,t)$ be
defined by \eqref{1.29}, \eqref{psin} and \eqref{1.38} as above. Set also
\beq\label{1.38new}
\breve\alpha_\pm(\la,t)=\int_0^t
\left(2(p_\pm(0,s) + 2\la) \breve m_\pm(\la,s) - \frac{\pa
p_\pm(0,s)}{\pa x}\right)ds.
\eeq
Introduce the set  $\mathcal S(t)$ by
\begin{align}\nn
{\mathcal S}(t) = \Big\{ & R_+(\la,t),\;T_+(\la,t),\;
\la\in\sigma_+^{\mathrm{u,l}}; \;
R_-(\la,t),\;T_-(\la,t),\;
\la\in\sigma_-^{\mathrm{u,l}};\\\label{S4.6t} &
\la_1,\dots,\la_p\in\R\setminus \sigma,\;
\gamma_1^\pm(t),\dots,\gamma_p^\pm(t)\in\R_+\Big\},
\end{align}
 where $\la_k(t)$, $R_\pm(\la,t)$, $T_\pm(\la,t)$ and
$\gamma_k^\pm(t)$ are defined by following formulas (\cite[Lem.~5.3]{EGT}):
\begin{align}
\label{refl} R_\pm(\la,t) &= R_\pm(\la,0)\E^{\alpha_\pm(\la,t)
-\breve{\alpha}_\pm(\la,t)}, \quad \la\in\si_\pm, \\ \label{trans}
T_\mp(\la,t) &= T_\mp(\la,0)\E^{\alpha_\pm(\la,t)
-\breve{\alpha}_\mp(\la,t)},\quad\la\in\mathbb C,\\ \label{norm}
\left(\gamma_k^\pm(t)\right)^2 &= \left(\gamma_k^\pm(0)\right)^2
\frac{\delta_\pm^2(\la_k,0)}{\delta_\pm^2(\la_k,t)}
\E^{2\alpha_\pm(\la_k,t)},
\end{align}
where $\alpha_\pm(\la,t)$, $\breve{\alpha}_\pm(\la,t)$,
$\delta_\pm(\la,t)$ are defined in \eqref{1.38}, \eqref{1.38new},
\eqref{S2.6} respectively.

In \cite{EGT} it is proved, that these data obey \textbf{I--III} with $g_\pm(\la,t)$,
defined by \eqref{1.88} and $\delta_\pm(\la)$, $\hat\delta_\pm(\la)$ defined
by \eqref{S2.6}.

Introduce the functions $F_\pm(x,y,t)$ via
\begin{align}\label{6.2}
F_\pm(x,y,t) =& \frac{1}{2\pi\I}\oint_{\si_\pm}
R_\pm(\la,t) \psi_\pm(\la,x,t) \psi_\pm(\la,y,t) g_\pm(\la,t)d\la +
\\ \nn & {} + \frac{1}{2\pi\I}\int_{\si_\mp^{(1),\mathrm{u}}}
 |T_\mp(\la,t)|^2
\psi_\pm(\la,x,t) \psi_\pm(\la,y,t)g_\mp(\la,t)d\la \\ \nn & {} +
\sum_{k=1}^p (\ga_k^\pm(t))^2 \tilde\psi_\pm(\la_k,x,t)
\tilde\psi_\pm(\la_k,y,t).
\end{align}
Suppose that we could prove that these functions satisfy
\beq\label{4.311}
\left|\frac{\pa^{n+l}}{\pa x^n\pa y^l}
F_\pm(x,y,t)\right|+\left|\frac{\pa^2} {\pa x\pa t}
F_\pm(x,y,t)\right|\leq \frac{C}{|x+y|^{m_1+2}}\quad
n+l\leq n_1+1,
\eeq
as $x,y\to\pm\infty$ for some $m_1\geq 2$, $n_1\geq 3$, and
$C=C(n_1,m_1,t)$. Then \eqref{4.311} implies that condition {\bf IV}
holds with $\tilde q_\pm(x)= (1+|x|^{m_1+3})^{-1}$ and thus Theorem~\ref{theor4}
ensures the unique solvability of the time dependent GLM equations
\beq\label{ME1}
K_\pm(x,y,t) + F_\pm(x,y,t) \pm \int_x^{\pm\infty}
K_\pm(x,\xi,t) F_\pm(\xi,y,t) d\xi =0, \quad \pm y>\pm x,
\eeq
and leads to the function
\beq\label{5.111}
q(x,t) = p_\pm(x,t) \mp 2 \frac{d}{dx}K_\pm(x,x,t).
\eeq
By construction it satisfies (cf.\ \eqref{5.2})
\beq\label{S.2t}
\pm \int_0^{\pm \infty} \left| \frac{\pa^n}{\pa x^n} \big( q(x,t) -
p_\pm(x,t)\big) \right| (1+|x|^{m_1})dx <\infty,\quad 0 \leq n\leq
n_1,
\eeq
and as in \cite{EGT} one concludes that \eqref{4.311} also implies differentiability with
respect to $t$ such that
\beq\label{Deriv t}
\pm \int_0^{\pm \infty} \left| \frac{\pa}{\pa t} \big( q(x,t) - p_\pm(x,t)\big) \right|
(1+|x|^{m_1})dx <\infty.
\eeq
Moreover, by verbatim following the arguments in Section~6 of \cite{EGT}
(see in particular Lemma~6.3 and Corollary~2.3) one establishes that $q(x,t)$
solves the associated initial-value problem of the KdV equation. Thus, to prove
Theorems~\ref{theor1} it is sufficient to prove the inequality \eqref{4.311}
with $m_1= \floor{\frac{m_0}{2}}-2$, $n_1=n_0 -m_0 -2$.



\section{Proof of the main result}



To obtain \eqref{4.311} we follow the approach, developed in
\cite{EGT}. First of all, recall that the functions $F_\pm(x,y,t)$
are given by
\begin{align}\label{Fhat}
F_\pm(x,y,t) =& \frac{1}{2\pi\I}\oint_{\si_\pm}
R_\pm(\la,0) \hat\psi_\pm(\la,x,t) \hat\psi_\pm(\la,y,t)
g_\pm(\la,0)d\la +
\\ \nn & {} + \frac{1}{2\pi\I}\int_{\si_\mp^{(1),\mathrm{u}}}
 |T_\mp(\la,0)|^2
\hat\psi_\pm(\la,x,t)\hat \psi_\pm(\la,y,t)g_\mp(\la,0)d\la \\ \nn &
{} + \sum_{k=1}^p (\ga_k^\pm(0))^2 \breve\psi_\pm(\la_k,x,t)
\breve\psi_\pm(\la_k,y,t),
\end{align}
where $\hat\psi_\pm(\la,x,t)$ are defined by \eqref{psin}, \eqref{1.37} and
\beq\label{hattil}
\breve\psi_\pm(\la,x,t):= \delta_\pm(\la,0)\hat\psi_\pm(\la,x,t).
\eeq
Furthermore, recall that the functions $\hat\psi_\pm(\la,x,t)$ inherit their
singularities from $\psi_\pm(\la,x,0)$, that is, they have simple poles on the set
$M_\pm(0)$ and square-root singularities on the set $\hat M_\pm(0)$.
Consequently, the functions \eqref{hattil} are bounded and smooth in small
vicinities of the points $\la_k$. Moreover, all integrands in \eqref{Fhat}
have only integrable singularities (cf.\ \cite[Sect.~5]{BET}) and thus
all three summands in \eqref{Fhat} are well-defined. Based on this
formula our aim is to study the decay of $F_\pm(x,y,t)$ as $x$, $y$
tend to $\pm\infty$, respectively.

First of all, we observe, that the third summand in \eqref{Fhat}
(corresponding to the discrete spectrum) decays exponentially as
$x+y\to\pm\infty$ together with all its derivatives, and, therefore,
satisfies \eqref{4.311} for all natural $m_1$ and $n_1$.  In the
second summand the function $\hat\psi_\pm(\la,x,t)\hat
\psi_\pm(\la,y,t)$ decays exponentially (together with all derivatives)
with respect to $(x+y)\to\pm\infty $ for $\la\notin\si_\pm$. Hence we have to
estimate this summand only in small vicinities of the points
$\si_\pm\cap\si_\mp^{(1)}$.

Our strategy is as follows: We make a change of variables
from $\la$ to the quasimoment variables $\theta_\pm$ in both integrals
in \eqref{Fhat} and use \eqref{1.23} to represent the integrands as
$\E^{\pm\theta_\pm (x+y)}\rho_\pm(\la(\theta_\pm),x,y,t)$, where the
functions $\rho_\pm$ are smooth and uniformly bounded with respect
to $x,y\in\R$, together with their derivatives. Moreover, since these
functions are differentiable with respect to  $\theta_\pm$
(and also bounded with respect to $x$ and $y$), we will
integrate by parts both integrals in \eqref{Fhat} as many times as
possible and then prove that the integrated terms either cancel
or vanish.

To investigate the possibility of integration by parts for the
first summand in \eqref{Fhat} we use \eqref{1.23}--\eqref{1.25} to
represent it as
\begin{align}\nn
F_{\pm,R}(x,y,t) &:= 2\Re \int_{\si_\pm^{\mathrm{u}}}
R_\pm(\la,t) \psi_\pm(\la,x,t)
\psi_\pm(\la,y,t) \frac{g_\pm(\la,t)}{2\pi\I}d\la\\ \label{Fc}
&=\Re\int_0^\infty \E^{\pm\I(x+y)\theta_\pm}
\rho_\pm(\theta_\pm,x,y,t) d\theta_\pm,
\end{align}
where
\beq\label{defrho}
\rho_\pm(\theta_\pm,x,y,t) := \frac{1}{2\pi} R_\pm(\la,0) u_\pm(\la,x,t) u_\pm(\la,y,t)
\E^{2\alpha_\pm(\la,t)}
\prod_{j=1}^{r_\pm}\frac{\la -\mu_j^\pm}{\la-\zeta_j^\pm},
\eeq
with $\la=\la(\theta_\pm)$.
Since the integrand in \eqref{Fc} is not
continuous at $\theta_\pm(E_{2k+1}^\pm)=\theta_\pm(E_{2k+2}^\pm)$, we regard this integral as
\beq\label{sumFc}
F_{\pm,R}(x,y,t)=
\Re \sum_{k=0}^{r_\pm +1} \int_{\theta_\pm(E_{2k}^\pm)}^
{\theta_\pm(E_{2k+1}^\pm)} \E^{\pm\I (x+y) \theta}
\rho_\pm(\theta,x,y,t) d\theta,
\eeq
where we set
\[
E_{2r_\pm+1}^\pm=E_{2r_\pm+2}^\pm=\tilde E>\max\{E_{2r_+}^+,
E_{2r_-}^-\},
\]
and $E_{2r_\pm+3}^\pm=+\infty$ for notational convenience.

Then the boundary terms arising during integration by parts (except for the last one, corresponding to
$+\infty$) will be
\beq\label{outint}
\Re \lim_{\la\to E}\frac{\E^{\pm\I\theta_\pm(E)(x+y)}\frac{\pa^s \rho_\pm(\theta_\pm,
x,y,t)}{\pa\theta_\pm^s}}{\left(\I (x+y)\right)^{s+1}},\quad
E\in\pa\si_\pm\cup\tilde E,\: s=0,1,\dots,m.
\eeq
The number $m$ of such integrations by parts is directly related to the smoothness
of $R_\pm(\la,0)$ and thus the to the values of $m_0$ and $n_0$ .

To estimate the boundary terms in \eqref{sumFc} we distinguish three cases:

\begin{enumerate}
\item[1)]
$E\in\pa\si_\pm\cap\pa\si$ (points $E_1$, $E_2$ in our example and
also point $E_3$ for $F_{-,R}(x,y,t)$);
\item[2)]
$E\in\pa\si_\pm\cap\inte(\si_\mp)$ (the point $E_5$ for $F_{-,R}(x,y,t)$);
\item[3)]
$E\in\pa\si_-^{(1)}\cap\pa\si_+^{(1)}$ (the point $E_4$).
\end{enumerate}

In the first case the boundary terms \eqref{outint} will vanish. In the second and the third
cases, however, these terms do not vanish, but we will prove, that they
cancel with a corresponding terms from the second summand in \eqref{Fhat}.
Finally, the two boundary terms stemming from our artificial boundary
point $\tilde E$ will cancel and hence do not need to be taken into account.

The following result taking care of 1) is an immediate consequence of the proof of
\cite[Lem.~6.2]{EGT}.

\begin{lemma} \label{lemestim3}
Let $E\in\pa\si_\pm\cap\pa\si$. Then the
following limits exists and take either real or pure imaginary values:
\beq\label{reflest} \lim_{\la\to
E,\,\la\in\si_\pm}\E^{\pm\I\theta_\pm(E)(x+y)}
\frac{\pa^s}{\pa\theta_\pm^s} \rho_\pm(\theta_\pm,x,y,t)\in \I^s\R,
\eeq
for $s=0,\dots, m_0-1$ if $\hat W(E)\ne 0$ and $s=0,\dots, m_0-2$
if $\hat W(E)= 0$.
\end{lemma}

This lemma shows, that the boundary terms \eqref{outint} vanish at the
points corresponding to case 1). Before turning to the cases 2) and 3)
let us first start by discussing smoothness of the integrand
$\rho_\pm(\theta,x,y,t)$ in \eqref{sumFc}.

Since except for $R_\pm(\la,0)$ all other parts of $\rho_\pm(\theta_\pm,x,y,t)$
are smooth with respect to $\la\in\inte(\si_\pm)$ it suffices to look
at $R_\pm(\la,0)$. By Lemma~\ref{lem2.3}, {\bf III.~(a)} the latter function
has $m_0$ derivatives with respect to $\la$ (and consequently also
with respect to $\theta_\pm$) as long as we stay in the interior of $\si_\pm$
and away from boundary points of $\si_\mp$. Hence all such points
do note pose any problems and the only problematic points are
those in $\pa\si_\mp \cap \inte(\si_\pm)$ (the point $E_5$ in our
example for $F_{+,R}(x,y,t)$). Hence we will address this issue first.

Let $E\in\pa\si_\mp \cap \inte(\si_\pm)$ be such a point.
As already pointed out, only $R_\pm(\la,0)$ matters and by
Lemma~\ref{lem2.3}, {\bf III.~(a)} we can locally write it as a
smooth function of $\sqrt{\la-E}$. Thus we obtain
\beq\label{sing}
\frac{\pa^s\rho_\pm(\theta_\pm,x,y,t)}{\pa\theta_\pm^s}=
O\left(\frac{1}{\sqrt{(\la-E)^{2s-1}}}\right)
\eeq
and since this singularity is non-integrable for $s\ge 2$ (more than one)
integration by parts is not an option near such points. Hence we
will split off the leading behavior near such a point. Then leading
term near each such point can be computed explicitly and the
remainder can be handled by integration by parts.

Since the last interval $(\tilde E,\infty)$ does not contain such points
we can restrict our attention to finite intervals. Moreover, for notational
convenience we will restrict ourselves to the case of $F_{+,R}$.

Abbreviate $\theta=\theta_+$ and denote by
\[
E_i\in \pa\si_-\cap\left(E_{2j}^+,E_{2j+1}^+\right),\quad i=1,\dots,N,
\]
our {\em bad} points. Let $\varepsilon>0$ and introduce the cutoff functions
\beq\label{func10}
B_i(\theta):=B(\frac{\theta-\theta(E_i)}{\varepsilon}),\quad i=1,\dots,N,
\eeq
where
\beq\label{funk1}
B(\xi)=\begin{cases}
\E^{-\xi^2}\left(1 - \xi^{2m_0}\right)^{m_0}, & \mbox{for }
|\xi|\leq 1,\\
0, & \mbox{else}.
\end{cases}
\eeq
We will choose $\varepsilon>0$ sufficiently small such that the supports of the
functions $B_i(\theta)$ do neither intersect nor do contain small
vicinities of the points $\theta(E_{2j}^+)$ and $\theta(E_{2j+1}^+)$.
Moreover, we have
\beq\label{funk2}
\aligned
\frac{d^s B_i}{d\theta^s}(\theta(E_i)\pm\varepsilon)=0,,\: s=0,\dots,m_0-1, \\
\frac{d^s B_i}{d\theta^s}(\theta(E_i))=0,\: s=1,\dots,2m_0+1.
\endaligned
\eeq
Now we can rewrite the $j$-th summand in \eqref{sumFc} (except for the last one) as
\begin{align*}
& \int_{\theta(E_{2j}^+)}^ {\theta(E_{2j+1}^+)}
\E^{\I (x+y) \theta} \rho_+(\theta,x,y,t) d\theta=\\
&\qquad = \int_{\theta(E_{2j}^+)}^{\theta(E_{2j+1}^+)}
\E^{\I (x+y) \theta} \left(1-\sum_{i=1}^N
B_i(\theta)\right)\rho_+(\theta,x,y,t) d\theta + \\
&\qquad\quad +\sum_{i=1}^N \int_{-\infty}^ {\infty}
\E^{\I (x+y) \theta}
B_i(\theta)\rho_+(\theta,x,y,t) d\theta.
\end{align*}
Due to \eqref{funk2} the first term can be integrated by parts $m_0$ times
and thus will be covered by Lemma~\ref{lemestim3}. For the second
term let us switch to the local variable $z=\sqrt{\theta - \theta(E_i)}$
and use a Taylor expansion for the integrand,
\[
\rho_+(\theta,x,y,t)=\rho_0^{(i)}(x,y,t) + \rho_1^{(i)}(x,y,t) z
+\dots+\rho_{m_0-1}^{(i)}(x,y,t) z^{m_0-2} + \beta_i(\theta),
\]
where $\beta_i(\theta) = O\left(z^{m_0-1}\right)$
has $\floor{\frac{m_0}{2}}$ integrable derivatives with respect to
$\theta$ in a small vicinity of the point $\theta(E_i)$. By construction
\[
\frac{\pa^s (B_i\beta_i)}{\pa\theta^s}
(\theta(E_i)\pm\varepsilon)=0,\quad
s=0,\dots,\floor{\frac{m_0}{2}},
\]
and thus we have
\beq\label{funk7}
\int_{-\infty}^ {\infty} \E^{\I (x+y) \theta} B_i(\theta)\beta_i(\theta) d\theta=
O\left( (x+y)^{-\floor{\frac{m_0}{2}}}\right).
\eeq
To compute the remaining terms observe
\begin{align*}
& \int_{-\infty}^ {\infty} \E^{\I (x+y) \theta}
B_i(\theta)\left(\sqrt{\theta - \theta(E_i)}\right)^\nu d\theta=\\
& \quad = (\varepsilon)^{\nu/2+1} \E^{\I(x+y)\theta(E_i) }
\int_{-1}^1 \E^{-\zeta^2+ \I \varepsilon(x+y) \zeta }
\left(1 - \zeta^{2m_0}\right)^{m_0}\zeta^{\nu/2}d\zeta,
\end{align*}
and note that we can extend the integral over $(1,1)$ to $(-\infty,\infty)$ since
\[
\int^{\pm\infty}_{\pm 1} \E^{-\zeta^2 + \I \varepsilon(x+y) \zeta}
\left(1 - \zeta^{2m_0}\right)^{m_0}\zeta^{\nu/2}d\zeta
= O\!\left( (x+y)^{-m_0-1}\right).
\]
Now we can simply expand
\[
\left(1 -
\zeta^{2m_0}\right)^{m_0}= 1 -m_0 \zeta^{2 m_0} + \cdots +(-1)^{m_0}
\zeta^{2m_0^2}
\]
and evaluate the integral by invoking the integral representation \cite[9.241]{GR}\footnote{It
also follows from 3.462 3, but this formula contains a sign error.}
for the parabolic cylinder functions $\mathcal D_\kappa(z)$ (cf.\ \cite{GR}, \cite{WW}),
which gives
\begin{align}\nn
&\int_{-\infty}^\infty \E^{\I \varepsilon (x+y) \zeta}
\E^{-\zeta^2} \zeta^{\kappa}d\zeta =\\ \label{kapp}
& \qquad =(-\I)^{\kappa} 2^{-\kappa/2} \sqrt\pi\exp\left(-\frac{\varepsilon^2
(x+y)^2}{8}\right)
\mathcal{D}_{\kappa}\left(\frac{\varepsilon(x+y)}{\sqrt 2}\right), \quad \Re(\kappa)>-1.
\end{align}
Since the parabolic cylinder functions have the following expansion \cite[9.246 1]{GR},
\[
\mathcal D_\kappa(z)\sim z^\kappa \E^{-\frac{z^2}{4}}\left(1 -
\frac{\kappa(\kappa -1)}{2 z^2} + \cdots\right), \quad |\arg(z)|<\frac{3\pi}{4},
\]
for large $z$,  the integral \eqref{kapp} decays exponentially as $(x+y)\to\infty$
for any $\kappa>0$. Combining these estimates
with Lemma~\ref{lemestim3} we obtain the following

\begin{lemma}\label{lemest4}
Let $E_{2j}^\pm, E_{2j+1}^\pm\in\pa\si_\pm\cap\pa\si$. Then
\beq\label{spec}
\frac{\partial^{n+l}}{\partial x^n\partial y^l}\Re
\int_{\theta_\pm(E_{2j}^\pm)}^{\theta_\pm(E_{2j+1}^\pm)} \E^{\pm\I
(x+y) \theta} \rho_\pm(\theta,x,y,t) d\theta = O\!\left(
(x+y)^{-\floor{\frac{m_0}{2}}}\right)
\eeq
as $x,y\to\pm\infty$ for all fixed $n,l=0,1,\dots$.
\end{lemma}

Note, that the condition $E_{2j}^\pm, E_{2j+1}^\pm\in\pa\si_\pm\cap\pa\si$ is only
used to take care of the boundary terms during integration by parts and can hence
be replaced by any other condition which takes care of these terms.

Now we come to Case 2) and study the behavior of the boundary terms
at the points $E\in\pa\si_\pm\cap\inte \si_\mp$. In this case formula
\eqref{reflest} remains valid only for $s=0$ and thus we need to take
the second summand in \eqref{Fhat} into account.

For notational convenience we will only consider the $+$ case
and suppose $E=E_{2j}^+$ without loss of generality. In this case
$\si^{(2)}$ is located to the right of $E$ and $\si_-^{(1)}$ to the
left. Moreover, without loss of generality we will assume that
the other boundary terms are already covered by the previous
considerations such that we do not have to worry about them.

Choose $\varepsilon>0$ so small that
\[
[\la(\theta_+(E) + \I\varepsilon), E]\subset \left((\xi_j^+, E]\cap
\si_-^{(1)}\right),\quad (E,\la(\theta_+(E +\varepsilon))]\subset
\inte \si^{(2)}.
\]
In these two small intervals introduce two new (positive) variables
\beq\label{ash}
h:=\frac{\theta_+ - \theta_+(E)}{\I},\quad k:=\theta_+ - \theta_+(E).
\eeq
We will compare the boundary terms at the point $E$ for the two integrals:
\beq\label{Rint}
\Re \int_{\theta(E)}^ {\theta(E+\varepsilon)} \E^{\I (x+y) \theta_+}
\rho_+(\theta_+,x,y,t) d\theta_+ =
\Re\int_0^\varepsilon R(k)\Psi(\la(k),x,y,t) \E^{\I k (x+y)} dk
\eeq
and
\begin{align}\nn
&\int_{\la(\theta(E) + \I\varepsilon)}^E |T_-(\la,0)|^2
\hat\psi_+(\la,x,t)\hat \psi_+(\la,y,t)\frac{g_-(\la,0)}{2\pi\I}d\la =\\ \label{T}
& \qquad = \int_\varepsilon^0 P(h)\Psi(\la(h),x,y,t) \E^{-h(x+y)} dh,
\end{align}
with
\beq\label{rho1}
\Psi(\la,x,y,t)=\frac{\E^{\I\theta(E)(x+y)}}{2\pi}
\prod_{j=1}^{r_+}\frac{\la -\mu_j^+}{\la -\zeta_j^+} \E^{-\I
(x+y) \theta} \hat\psi_+(\la,x,t) \hat\psi_+(\la,y,t),
\eeq
and
\begin{align}\label{defR}
R(k) := &R_+(\la,0),\\\nn
P(h) :=& \frac{-\I}{2g_+(\la,0)g_-(\la,0) |W(\la,0)|^2}\\\label{defP}
=& \frac{-\I}{2g_+(\la,0)g_-(\la,0)\wronsk_0(\phi_-,\phi_+)
\wronsk_0(\overline{\phi_-},\phi_+)},
\end{align}
where $\wronsk_0(\cdot,\cdot)=\wronsk(\cdot,\cdot)|_{t=0}$.
To obtain \eqref{defP} we used \eqref{S2.18} together with
$\overline{g_-(\la,0)} = -g_-(\la,0)$ if $\la\in\si_-$.

Integrating by parts integrals \eqref{Rint} and \eqref{T} with respect to $k$
and $h$ gives
\begin{align}\nn
& \int_{\varepsilon}^0 P(h)\Psi(\la(h)) \E^{-h(x-y)} dh =
-\sum_{j=0}^{m-1} \frac{1}{(x-y)^{j+1}}\frac{\pa^j (P \Psi)}{\pa h^j} (0) \\ \label{decompos}
& \qquad {} + \frac{1}{(x-y)^m}\int_\varepsilon^0 \frac{\pa^m (P\Psi)}{\pa h^m} \E^{-h(x-y)} dh
+ O(\E^{-\varepsilon(x-y)}) ,
\end{align}
\begin{align}\nn
& \Re\int_0^\varepsilon R(k)\Psi(\la(k)) \E^{\I k(x-y)} dk=
\Re \sum_{j=0}^{m-1} \frac{1}{(-\I (x-y))^{j+1}}\frac{\pa^j(R\Psi)}{\pa k^j}(0)\\ \label{decomposR}
& \qquad {} +  \Re \frac{1}{(-\I(x-y))^m}\int_0^\varepsilon\frac{\pa^{m}(R\Psi)}
{\pa k^m} \E^{\I k(x-y)} d k.
\end{align}
For the boundary terms to cancel each other we need to show
\beq\label{import}
\lim_{k\to 0}\Re \left(\I^{j+1} \frac{\pa^j(R\Psi)}{\pa k^j}(k)\right)=
\lim_{h\to 0}\frac{\pa^j(P\Psi)} {\pa h^j}(h), \quad j=0,\dots,m_0-1,
\eeq
where the left, right limit is taken from the side of the spectrum of multiplicity two, one, respectively.
Since $\Psi$ is smooth to any degree with respect to $k$ and $h$ near $E$,
\beq
\lim_{k\to 0} \left(\I^j \frac{\pa^j \Psi}{\pa k^j}(k)\right)=
\lim_{h\to 0}\frac{\pa^j \Psi} {\pa h^j}(h), \quad j=0,\dots,
\eeq
we observe that to prove \eqref{import} it is sufficient to prove

\begin{lemma}
Let $h, k, P(h), R(k)$ be defined by \eqref{ash}, \eqref{defR}, and \eqref{defP}. Then,
if $E\in\pa\si_\pm\cap\inte(\si_\mp)$,
\beq\label{import1}
\lim_{k\to 0} \Re\left(\I^{j+1}\,\frac{d^j R(k)}{d k^j}\right)=
\lim_{h\to 0} \frac{d^j P(h)}{d h^j}, \qquad j=0,\dots,m_0-1.
\eeq
\end{lemma}

\begin{proof}
To prove this formula, recall that  $\phi_-(\la)$,
$\ov{\phi_-(\la)}\in C^{m_0}(E-\varepsilon, E+\varepsilon)$ (and
similarly for the $x$ derivative) with respect to $\la$ since
$E\in\inte \si_-$. Therefore their derivatives with respect to
$\sqrt{\la - E}$ are smooth in a vicinity of $k=0$.  Without loss of
generality we suppose\footnote{Otherwise replace $\phi_+(\la,x,0)$
by $\phi_{+,E}(\la,x,0)$ and $g_+(\la,0)$ by $g_{+,E}(\la,0)$ (cf.\
Lemma~\ref{lemMhat}) in the subsequent considerations.},
that $E\neq \mu_j^+$, that is, the function $\phi_+(\la,x,0)$ as well as the
functions $g_+(\la,0)$ and $g_-(\la,0)$ (see \eqref{1.88}) are also
smooth with respect to $\sqrt{\la - E}$. Set
\[
\tilde{P}(k):=  \frac{-\I}{2g_+(\la,0)g_-(\la,0)\wronsk_0(\phi_-,\phi_+)
\wronsk_0(\overline{\phi_-},\phi_+)}, \quad \la>E,
\]
such that
\beq\label{mainn}
\lim_{k\to +0} \I^s \frac{d^s \tilde{P}(k)}{d k^s}= \lim_{h\to +0}\frac{d^s P}{dh^s}.
\eeq
Using $g_\pm(\la,0)^{-1}=\pm\wronsk_0(\phi_\pm,\overline{\phi_\pm})$ we
see
\beq\label{def P1}
\tilde{P}(k)= \frac{\I\,\wronsk_0(\phi_-,\ov{\phi_-})\wronsk_0(\phi_+,\ov{\phi_+})}
{2\wronsk_0(\phi_-,\phi_+)\wronsk_0(\ov{\phi_-},\phi_+)}
\eeq
and substituting
\[
\phi_+(\la,x,0)=\frac{\wronsk_0(\phi_+,\ov{\phi_-})}
{\wronsk_0(\phi_-,\ov{\phi_-})} \phi_-(\la,x,0)
 -\frac{\wronsk_0(\phi_+,\phi_-)}{\wronsk_0(\phi_-,\ov{\phi_-})}
 \ov{\phi_-(\la,x,0)}
\]
into the nominator we obtain
\[
\tilde{P}(k)=\frac{\I}{2}\left(-\frac{\wronsk_0(\phi_-,\ov{\phi_+})}
{\wronsk_0(\phi_-,\phi_+)}+
\frac{\wronsk_0(\ov{\phi_-},\ov{\phi_+})}{\wronsk_0(\ov{\phi_-},\phi_+)}
\right).
\]
Introducing the abbreviations
\beq\label{wrons1}
W(k):=\wronsk_0(\phi_-,\phi_+), \quad
V(k):=\wronsk_0(\phi_-,\ov{\phi_+}).
\eeq
we thus have
\beq\label{P1}
R(k)= - \frac{V(k)}{W(k)}, \qquad \tilde{P}(k) =
\frac{\I}{2}\left(-\frac {V(k)}{W(k)} +\frac{\ov{W(k)}}{\ov{V(k)}} \right).
\eeq
Next, for all $x$  and a small $\varepsilon >0$ we have
\[
\phi_-(\la,x,0), \frac{\pa}{\pa x}\phi_-(\la,x,0) \in
C^{m_0}(E-\varepsilon, E+\varepsilon),
\]
therefore according to Lemma \ref{lemMhat} {\bf (ii)} near $k=0$,
for positive $k$, we have a representation
\[
W(k) - V(k)= \I k f_1(k^2),\quad W(k) +V(k)=f_2(k^2),
\]
where $f_{1,2}(\cdot)\in C^{m_0-1}([0,\varepsilon_1))$. Differentiating
these relations we obtain
\beq\label{signs}
\lim_{k\to+0} \frac{\pa^s}{\pa k^s} V(k) =
(-1)^s\lim_{k\to+0} \frac{\pa^s}{\pa k^s} W(k),\qquad
s=0,\dots,m_0-1,
\eeq
and hence we see $V(k)= W_{m_0-1}(-k)+o(k^{m_0-1})$, where
$W_{m_0-1}(k)$ is the Taylor polynomial of degree $m_0-1$ for $W(k)$.
Now recall
\[
\wronsk_0(\phi_-,\phi_+)(E)\neq 0,
\]
which implies $R^{-1}(k)=R_{m_0-1}(-k) + o(k^{m_0-1})$, where $R_{m_0-1}(k)$
is the Taylor polynomial of degree $m_0-1$ for $R(k)$. Thus we finally obtain
\beq
\tilde{P}(k)=\frac{\I}{2}\left( R_{m_0-1}(k) - \ov{R_{m_0-1}(-k)}\right) +o(k^{m_0-1}),
\eeq
from which \eqref{import1} follows.
\end{proof}

This settles Case 2). To show Case 3), we can proceed as before and we need to
prove

\begin{lemma}
Let $h, k, P(h), R(k)$ be defined by \eqref{ash}, \eqref{defR}, and \eqref{defP}. Then,
if $E\in\pa\si_-^{(1)}\cap\pa\si_+^{(1)}$,
\beq
\lim_{k\to 0} \Re\left(\I^{j+1}\,\frac{d^j R(k)}{d k^j}\right)=
\lim_{h\to 0} \frac{d^j P(h)}{d h^j}, \qquad j=0,\dots,m_0-1.
\eeq
\end{lemma}

\begin{proof}
Note that now we cannot proceed as in Case 2) since now we no longer have
spectrum of multiplicity two to the right of $E$. In particular, we cannot use
$g_-(\la,0)^{-1}=-\wronsk_0(\phi_-,\overline{\phi_-})$ for $\la>E$, we do
not have the scattering relations at our disposal, and $\phi_-(\la)\not\in
C^{m_0}(E-\varepsilon, E+\varepsilon)$. Hence we need a different strategy

Let\footnote{Again, otherwise replace $\phi_+(\la,x,0)$
by $\phi_{+,E}(\la,x,0)$ (cf.\ Lemma~\ref{lemMhat}) in the subsequent considerations.}
$E\notin \hat M_-(0)\cup\hat M_+(0)$. Consider $\phi_\pm(\la,x,0)$ and note
that for sufficiently small $\varepsilon$ we can write (see Lemma~\ref{lemMhat})
\[
\phi_-(\la,x,0) = \begin{cases}
f^-_1(h^2,x) + \I h f^-_2(h^2,x) + o(h^{m_0-1}),
& E-\varepsilon<\la \le E,\\
f^-_1(-k^2,x) + k f^-_2(-k^2,x) + o(k^{m_0-1}), &
 E+\varepsilon>\la
\ge E,
\end{cases}
\]
where $f^-_1(z,x), f^-_2(z,x)$ are real-valued functions, which are
polynomials of degree $m_0-1$ with respect to $z$ and differentiable with
respect to $x$. Next, define
\[
\breve\phi_-(\la,x,0) = \begin{cases}
f^-_1(h^2,x) - \I h f^-_2(h^2,x), & \la \le E,\\
f^-_1(-k^2,x) - k  f^-_2(-k^2,x), & \la \ge E,
\end{cases}
\]
and note that we have $\ov{\phi_-(\la,x,0)} = \breve\phi_-(\la,x,0)
+o(h^{m_0-1})$ for $E-\varepsilon<\la \le E$.

Similarly, we can write
\[
\phi_+(\la,x,0) = \begin{cases}
f^+_1(h^2,x) + h f^+_2(h^2,x) + o(h^{m_0-1}), &E-\varepsilon< \la \le E,\\
f^+_1(-k^2,x) - \I k f^+_2(-k^2,x) + o(k^{m_0-1}), &
E+\varepsilon>\la \ge E,
\end{cases}
\]
and define
\[
\breve\phi_+(\la,x,0) = \begin{cases}
f^+_1(h^2,x) - h f^+_2(h^2,x) , & \la \le E,\\
f^+_1(-k^2,x) + \I kf^+_2(-k^2,x), & \la \ge E,
\end{cases}
\]
such that $\ov{\phi_+(\la,x,0)} = \breve\phi_+(\la,x,0)+o(k^{m_0-1})$
for $E+\varepsilon>\la \ge E$. In particular, note that
\[
\I^j \frac{\pa^j}{\pa h^j} \breve\phi_\pm(\la,x,0) =  \frac{\pa^j}{\pa k^j}
\breve\phi_\pm(\la,x,0), \qquad \la=E,\quad 0 \leq j \leq m_0-1.
\]
Moreover, we have
\beq \label{gipm}
g_\pm(\la,0) = \pm
\wronsk(\phi_\pm(\la,.,0),\breve\phi_\pm(\la,.,0)) +o\big((\la-E)^{(m_0-1)/2}
\big),\quad \la\in (E-\varepsilon,E+\varepsilon).
\eeq
Note that while the above Wronskian depends on $x$ ($\breve\phi_\pm$ do not solve \eqref{S.4}
in general), the leading order is independent of $x$. In particular, we take the value for, say, $x=0$
(and of course $t=0$) here and in all following Wronskians below.

Now consider the function (cf. \eqref{defP})
\[
P(\la) = \frac{-\I}{2g_+(\la,0)g_-(\la,0)\wronsk_0(\phi_-,\phi_+)
\wronsk_0(\overline{\phi_-},\phi_+)}, \quad \la < E,
\]
and set
\beq
\tilde{P}(\la) = \frac{\I}{2}
\frac{\wronsk(\phi_-,\breve\phi_-)\wronsk(\phi_+,\breve\phi_+)}
{\wronsk(\phi_-,\phi_+)\wronsk(\breve\phi_-,\phi_+)},
\quad \la \in (E-\varepsilon,E+\varepsilon),
\eeq
such that we have
\[
P(\la) = \tilde{P}(\la)+o\big((\la-E)^{(m_0-2)/2} \big),
\quad E-\varepsilon<\la\leq E
\]
In particular\footnote{Note that we loose one derivative if $\wronsk(\phi_+,\phi_-)(E)=0$ in which case
we have $\wronsk(\phi_+,\phi_-)=C k(1+o(1))$ by Lemma~\ref{lem2.3} {\bf II.~(b)}.}
\beq
\frac{\pa^j}{\pa h^j} P(\la) = \I^j \frac{\pa^j}{\pa k^j}
\tilde{P}(\la), \qquad \la=E,\quad 0 \leq j \leq m_0-2.
\eeq
Using the Pl\"ucker identity
\[
\wronsk(f_1,f_2) \wronsk(f_3,f_4) + \wronsk(f_1,f_3)
\wronsk(f_4,f_2) + \wronsk(f_1,f_4) \wronsk(f_2,f_3) =0
\]
with
$f_1=\phi_-$, $f_2=\breve\phi_-$, $f_3=\phi_+$, and
$f_4=\breve\phi_+$ we can rewrite we can write $\tilde{P}(\la)$
equivalently as
\[
\tilde{P}(\la) = \frac{\I}{2}\left(-\frac {V(k)}{W(k)}
 +\frac{\breve W(k)}{\breve V(k)} \right),
\]
where
\[
W(k):= \wronsk(\phi_-,\phi_+), \quad V(k):=
\wronsk(\phi_-,\breve\phi_+),
\]
and
\[
\breve W(k):=
\wronsk(\breve \phi_-,\breve\phi_+), \quad \breve V(k):=
\wronsk(\breve\phi_-,\phi_+).
\]
Moreover, using the definitions one checks
\[
\frac{\breve W(k)}{\breve V(k)} =
\ov{\left(\frac{V(-k)}{W(-k)}\right)} + o(k^{m_0-1}).
\]
Now, since we have $V(k)= \wronsk_0(\phi_-,\ov{\phi_+}) + o(k^{m_0-1})$,
we obtain
\beq R(k) =
-\frac {V(k)}{W(k)} + o(k^{m_0-2}).
\eeq
This implies
\beq
\Re\left(\I^{j+1}\,\frac{\pa^j }{d k^j} R(0)\right)= \I^j
\frac{\pa^j}{\pa k^j} \tilde{P}(0) =
\frac{\pa^j P}{\pa h^j}(0), \qquad j=0,\dots,m_0-2,
\eeq
and we are done.
\end{proof}

Finally we discuss the possibility of integration by parts in the last (unbounded)
integrand of \eqref{sumFc}. More precisely, we discuss the boundary
terms corresponding to the point $E_{2r_++3}^+=+\infty$ (again the
considerations are the same for the $+$ and $-$ cases, and we study only
the $+$ case). First of all we recall the well-known asymptotics
\[
m_+(\la)=\I\sqrt{\la}(1+o(1)),\quad \theta(\la)=\sqrt{\la}(1+o(1))
\]
as $\la\to\infty$. Moreover, (recall $\alpha_+(\la,t)=4\I(\sqrt\la)^3t (1+o(1))$)
we also have
\[
\frac{\pa^s}{\pa\theta^s} u_+(\la,x,t)=O(1), \qquad
\frac{\pa^s}{\pa\theta^s} \E^{\alpha_+(\la,t)} =O(t \theta^{2s})
\]
and as in the previous cases, the only interesting part is again the reflection
coefficient $R_+(\la,0)$ for which we have
\beq
\frac{\pa^s}{\pa\theta^s} R_+(\la,0)= O(\theta^{-n_0-1}),
\qquad s=0,\dots, m_0,
\eeq
as $\la\to\infty$ by Lemma~\ref{lem2.3} {\bf III.~(a)}. Hence we conclude
\[
\frac{\pa^s}{\pa\theta_+^s} \rho_+(\la(\theta),x,y,t) =O(\theta^{2s-n_0-1}),
\quad \mbox{as } \la\to\infty,
\]
uniformly with respect to $x,y\in\R$ and $t\in[0,T]$ for any $T>0$. As a consequence
we can perform $m \leq \floor{\frac{n_0}{2}}$ partial integrations such that the boundary
terms at $\infty$ vanish.

In summary, supposing \eqref{2.111}, the maximum number of
integration by parts is determined by Lemma~\ref{lemest4}, that is
by the points $\inte \si_+\cap\pa\si_-$, and is given by
$\floor{\frac{m_0}{2}}$. So, if we exclude Case 3), we have an
almost complete picture. However, up to this point we have only
looked at $F_{R,+}(x,y,t)$ and did not consider derivatives with
respect to $x,y,t$. Luckily, since $R_+(\la,0)$ is evidently
independent of these variables and all other parts of the integrand
\eqref{sumFc} can be differentiated as often as we please, these
derivatives will not affect our analysis except for the last summand
in \eqref{sumFc}. Moreover, for this last summand all one has to
take into account that a partial derivative with respect to $x$ or
$y$ adds $O(\theta_+)$ (from $\E^ {\I\theta_+(x+y)}$) and a partial
derivative with respect to $t$ adds $O(\theta_+^3)$ (from
$\E^{\alpha_+(\la,t)}$). Thus we obtain the following result:

\begin{lemma}\label{lemfinal}
Let \eqref{2.111} be fulfilled. Then
\beq\label{est10}
F_+(x,y,t)=\frac{1}{(x+y)^{\floor{\frac{m_0}{2}}}}
\left(H(x,y,t) +\int_{\tilde E}^{+\infty} \E^{\I\sqrt\la(x+y)
+4\I(\sqrt\la)^3 t}\,H_1(\la,x,y,t)d\la\right),
\eeq
where $\tilde E> \max\{E_{2r_+}^+, E_{2r_-}^-, 1\}$.

The function $H(x,y,t)$ is smooth on the set  $\mathcal D:=
[0,+\infty)\times[0,+\infty)\times[0,T]$. All
partial derivatives with respect to $x,y,t$ of function $H$ are
bounded on $\mathcal D$.

The function $H_1(\la,x,y,t)$ is bounded on $\la$ and smooth with
respect to $x,y,t\in\mathcal D$.  Moreover,
\beq\label{main5}
\frac{\pa^{l+s+k}}{\pa x^l\pa y^s\pa
t^k}\,H_1(\la,x,y,t)=o\left(\left(\sqrt\la\right)^{l+s+3k-n_0-2
+2 \floor{\frac{m_0}{2}}}\right)\quad \mbox{as } \la\to\infty,
\eeq
uniformly on $\mathcal D$.
\end{lemma}

This lemma shows, that for the convergence of the integral and its
derivatives with respect to $x,y,t$  in \eqref{est10} it is
sufficient that $l+s+3k +2\floor{\frac{m_0}{2}}-n_0<0$.
Comparing with \eqref{4.311} shows that to guarantee a
classical solution (three derivatives with respect to $x$ and
one with respect to $t$) of the KdV equation we need to be able to admit at least
$l+s= 4, k=0$ and  $l+s= 1, k=1$, that is, we need at least $4+2\floor{\frac{m_0}{2}}-n_0<0$.
Since we also need $\floor{\frac{m_0}{2}}-2\geq 2$, this yields the conditions
$m_0\geq 8$ and $n_0 \geq 2\floor{\frac{m_0}{2}} +5$.

In particular, if \eqref{2.111} holds for all $m_0, n_0\in \N$ (Schwartz-type
perturbations), then the same is true for the solution and thus this provides
a generalization of the main result from \cite{EGT} without any restriction
on the background spectra.



\bigskip
\noindent{\bf Acknowledgments.} We are very grateful to F. Gesztesy, E.Ya. Khruslov,
and V.A. Marchenko for helpful discussions. I.E. gratefully acknowledges the
extraordinary hospitality  of the Faculty of Mathematics at the University of
Vienna during extended stays 2008--2009, where parts of this paper were
written. G.T. gratefully acknowledges the stimulating atmosphere at the
Centre for Advanced Study at the Norwegian Academy of Science and Letters in Oslo
during June 2009 where parts of this paper were written as part of  the international
research program on Nonlinear Partial Differential Equations.





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