%% @texfile{ %% filename="KdVStepPer2.tex", %% version="1.1", %% date="Sep-2011", %% cdate="20090430", %% filetype="LaTeX2e", %% journal="J. d'Analyse Math. 115:1, 71-101 (2011)", %% doi="10.1007/s11854-011-0024-9", %% copyright="Copyright (C) I. Egorova and G. Teschl". %% } \documentclass{amsart} \usepackage{hyperref} \usepackage{enumerate} %\usepackage{showkeys} %%%%%%%%% \newcommand{\arxiv}[1]{\href{http://arxiv.org/#1}{arXiv:#1}} \newcommand*{\mailto}[1]{\href{mailto:#1}{\nolinkurl{#1}}} %%%%%%%%%%%%%%%%%%%%%% Declaration section %%%%%%%%%%%%%%%%%%% \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \renewcommand{\theenumi}{\roman{enumi}} \numberwithin{equation}{section} \unitlength1cm %%%%%%%%%%%%%%%%%%%%%%% Command section %%%%%%%%%%%%%%%%%% \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\E}{\mathrm{e}} \newcommand{\I}{\mathrm{i}} \newcommand{\siul}{\sigma^{\mathrm{u,l}}} \newcommand{\siu}{\sigma^{\mathrm{u}}} \newcommand{\sil}{\sigma^{\mathrm{l}}} \newcommand{\sipmu}{\sigma_\pm^{\mathrm{u}}} \newcommand{\sipml}{\sigma_\pm^{\mathrm{l}}} \newcommand{\sipmul}{\sigma_\pm^{\mathrm{u,l}}} \newcommand{\simpul}{\sigma_\mp^{\mathrm{u,l}}} \newcommand{\lau}{\la^{\mathrm{u}}} \newcommand{\lal}{\la^{\mathrm{l}}} \newcommand{\laul}{\la^{\mathrm{u,l}}} \newcommand{\spr}[2]{\langle #1 , #2 \rangle} \newcommand{\floor}[1]{\lfloor#1 \rfloor} \newcommand{\diag}{\mathop{\mathrm{diag}}} \newcommand{\dom}{\mathfrak{D}} \newcommand{\Res}{\mathop{\mathrm{Res}}} \newcommand{\sign}{\mathop{\mathrm{sign}}} \newcommand{\dist}{\mathop{\mathrm{dist}}} \renewcommand{\Im}{\mathop{\mathrm{Im}}} \renewcommand{\Re}{\mathop{\mathrm{Re}}} \newcommand{\supp}{\mathop{\mathrm{supp}}} \newcommand{\clos}{\mathop{\mathrm{clos}}} \newcommand{\inte}{\mathop{\mathrm{int}}} \newcommand{\beq}{\begin{equation}} \newcommand{\eeq}{\end{equation}} \newcommand{\bal}{\begin{align}} \newcommand{\eal}{\end{align}} \newcommand{\nn}{\nonumber} \newcommand{\al}{\alpha} \newcommand{\be}{\beta} \newcommand{\ga}{\gamma} \newcommand{\de}{\delta} \newcommand{\si}{\sigma} \newcommand{\pa}{\partial} \newcommand{\ep}{\varepsilon} \newcommand{\la}{\lambda} \newcommand{\ov}{\overline} \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\\ University of Vienna\\ 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{J. d'Analyse Math. {\bf 115:1}, 71--101 (2011)} \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 quasi-periodic, finite-gap initial conditions under the assumption that the perturbations have a given number of derivatives with finite moments. \end{abstract} \maketitle \section{Introduction} The purpose of this paper 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 real-valued steplike initial conditions $q(x)$. More precisely, we will assume that $q(x)$ is asymptotically close to (in general) different quasi-periodic, 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 $m_0, n_0$. Here by quasi-periodic, finite-gap potentials we mean algebro-geometric, quasi-periodic, finite-gap potentials which arise naturally as the stationary solutions of the KdV hierarchy as discussed in \cite{GH} (further details will be given in Section~\ref{secist}). If \eqref{2.111} holds for all $m_0, n_0$ we call $q$ a Schwartz-type perturbation. Ever since the seminal work of Gardner et al.\ \cite{GGKM} in 1967 the inverse scattering transform has become one of the main tools used for solving this Cauchy problem. Numerous articles have been devoted to this subject since the publication of the GGKM paper. In particular, the case that the initial condition is asymptotically close to $p_\pm(x)=0$ is well understood. 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}. There are two natural cases which have been considered in the past when extending this classical situation. The first case is that of equal quasi-periodic, finite-gap potentials $p_-(x)=p_+(x)$ and the second is the case of steplike constant asymptotics $p_\pm(x) = c_\pm$ (with $c_- \neq c_+$). The aim of our present paper is to combine both cases and to solve some open problems in these special cases (to be discussed in detail below) along the way. The underlying scattering theory in the case of asymptotically periodic solutions was first investigated by Firsova \cite{F1}--\cite{F3}. The first ones to consider the Cauchy problem with a periodic background seem to be Kuznetsov and Mikha\u\i lov \cite{kumi}, who informally treated the Korteweg--de Vries equation with the Weierstra{\ss} elliptic function as background solution. It turns out, due to the poles of the Baker--Akhiezer functions, which reflect the fact that the underlying hyperelliptic Riemann surface is no longer simply connected, that the periodic case is much more complicated. The only known results concerning the existence of the solution seem to be by Ermakova \cite{Er}, \cite{Er1} and Firsova \cite{F4} (where the evolution of the scattering data for periodic background was given). However, both works are incomplete from the point of view of a rigorous application of the inverse scattering method. Surprisingly, much more is know about the asymptotical behavior (assuming existence) of such solutions; see for example \cite{Ba}, \cite{Bik}--\cite{Bik2}, \cite{iu}, \cite{Kh}--\cite{KhS}, \cite{N}. A complete and rigorous treatment of the inverse scattering transform for the KdV equation in the case of initial conditions which are Schwartz-type perturbations of finite-gap solutions was given only recently by Grunert and the present authors \cite{EGT}. Let us now turn to the case of steplike constant potentials, $p_\pm(x)=c_\pm$. 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 Khruslov \cite{Kh}, who derives the time evolution of the scattering data and analyzes the long-time asymptotics. Later, Cohen \cite{C} solved the case that $q(x)$ is the Heaviside step function. Kappeler \cite{Kap}, based on some advances in scattering theory of Cohen and Kappeler \cite{CK1}, showed how to handle general initial conditions with only a fixed number of moments finite. The time evolution of the scattering data for the entire KdV hierarchy was computed recently by Khasanov and Urazboev \cite{KU}. However, while Kappeler's 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 only very mild information is given concerning the decay properties at the other side. In particular, even if one starts with a Schwartz-type initial condition, the results in \cite{Kap} do 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. The case of power-like asymptotic behavior (including some unbounded initial conditions) was investigated by Bondareva and Shubin \cite{Bo}, \cite{BS};see also \cite{KPST} for the case of the mKdV equation. Finally, we mention that in the discrete steplike finite-gap case (Toda lattice), the same problem was completely solved in \cite{EMT1}. For analysis of the corresponding long-time asymptotic behavior, see \cite{EBM}, \cite{dkkz}, \cite{km2}, \cite{kt}, \cite{kt2}, \cite{krt2} and \cite{vdo}. To state our main result we 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 are 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_10 \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}, that satisfy 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 quasimomenta 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 quasimomenta 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{for}\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 quasimomenta 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 in $\la$ of $\deg(H_\pm)\leq r_\pm-1$ with real-valued coefficients which are 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} that satisfies $\breve\psi_\pm(\la,\cdot,t)\in L^2(\R_\mp)$ 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 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 throughout this discussion, we omit the dependence on $t$. Let $n_1\geq 0$ and $m_1\geq 2$ be given natural numbers and let $q(x)\in C^{n_1}(\mathbb R)$ 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 $q(x)$ that satisfies \eqref{S.2}. The spectrum of $L$ consists of a purely absolutely continuous part $\sigma:=\sigma_+\cup\sigma_-$, plus a finite number of discrete 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 the set $\sigma_+^{(1)}\cup\sigma_-^{(1)}$ with $\sigma_\pm^{(1)}= \clos(\sigma_\pm\setminus\sigma_\mp)$ is the spectrum of 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 $q(x)$ satisfies \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 $\la\in\C\backslash\si_\pm$ and have the same poles as $\psi_\pm(\la, .)$. The functions $\hat\delta(\la)\phi_\pm(\la, \cdot)$ are continuous up to the boundary $\siu_\pm\cup\sil_\pm$. Moreover, $\hat\delta(\la)\phi_\pm(\la,\cdot)$ 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 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 from existence of solutions of these integral equations. Existence is proved 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 also need to know the asymptotic behavior of the Jost solutions as $\la\to\infty$. To determine it, we recall the well-know expansion (cf.\ the proof of Lemma~1.19 in \cite{GH}) \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 $q(x)$ satisfies \eqref{S.2}. Then the Jost solutions have the asymptotic expansions \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 quasi-periodic 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 $x$, 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 from $n$ partial integrations and the Riemann-Lebesgue Lemma (cf.\ also \cite[Theorem~3.2]{MT}). As in the previous lemma, the claims for the derivatives follow by considering the corresponding integral equations. \end{proof} \begin{corollary}\label{corasymphi} Assume $q(x)$ satisfies \eqref{S.2}. Then the Weyl $m$-functions $m_{q,\pm}(\la,x)= \frac{\phi_\pm'(\la,x)}{\phi_\pm(\la,x)}$ have the 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}, we have the estimate \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 result. \begin{lemma}\label{lemMhat} Let $E\in\pa\si_\pm$ and let $\varepsilon>0$ 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 \beq\label{phige} \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) \eeq for $\la [E-\varepsilon, E+\varepsilon]$. The functions $\phi_{\pm,E}(\la,x)$ admit 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 $c_{\pm,E}(\cdot,x), s_{\pm,E}(\cdot,x)\in C^{m_0-1}([E-\varepsilon, E+\varepsilon])$ and $c_{\pm,E}(\cdot,x), s_{\pm,E}(\cdot,x)\in \mathbb R$. Analogous representations hold for $\frac{\pa}{\pa x} \phi_{\pm,E}(\la,x)$. Moreover, \beq\label{realval} \phi_{\pm,E}(\la,x)\in\mathbb{R},\qquad \la\in [E-\varepsilon, E+\varepsilon]\setminus\si_\pm, \eeq and \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 $\mu_j^\pm\neq E$. The functions $\phi_\pm(\la,x)$ admit the same representation \eqref{struct3} on the set $[E-\varepsilon/2, E+\varepsilon/2]$. \end{lemma} Next, recalling \eqref{S2.6}, set \beq\label{S2.12} \tilde\phi_\pm(\la,x)=\delta_\pm(\la) \phi_\pm(\la,x) \eeq so that the functions $\tilde\phi_\pm(\la,x)$ have no poles in the interior of the gaps of the spectrum $\si$. For each 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, by \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 $q(x)\in C^{n_1}(\mathbb R)$ satisfies \eqref{S.2}. 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} has 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 to 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 also $m_1$ 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[Lemma~4.1]{EGT}. To prove the missing items, we need to prove that $\wronsk(\phi_\pm,\ov{\phi_\mp})=o(\la^{-n_1/2})$ and all its 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} As in \cite[Lemma~4.2]{EGT}, we have the following result: \begin{lemma} Under the same assumptions as in Lemma~\ref{lem2.3}, the functions $F_\pm(x,y)$ satisfy \begin{enumerate}[\bf I.] \addtocounter{enumi}{3} \item $F_\pm(x,y)\in C^{(n_1+1)}(\R^2)$. There exist real-valued continuous functions $\tilde q_{\pm}(x)$, with $x^{m_1}\tilde q_{\pm}\in L^1(\mathbb R_\pm)$, and monotone positive continuous functions $C_\pm(x)$, $Q_\pm(x)$, which decay as $x\to\pm\infty$, with $x^{m_1-1}Q_\pm(x)\in L^1(\mathbb R_\pm)$, such that for $\pm x>\pm a$, $\pm y >\pm a$ and $0\leq n+l\leq n_1+1$ the inequalities hold \beq \label{4.4} \left|\frac{\pa^{n+l}}{\pa x^n\pa y^l} F_\pm (x,y)\right|\leq C_\pm(a) \left( Q_\pm(x+y) +\tilde q_\pm(x+y)(1-\delta_{n+l,0})\right). \eeq Here $\delta_{n,m}$ is the Kronecker delta and $a\in\mathbb R$ is an arbitrary fixed number. 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} \end{lemma} \begin{proof} The GLM equation \eqref{ME} and \eqref{4.2} are derived in \cite{BET}. Estimate \eqref{4.4} follows directly from \eqref{ME} and \eqref{S2.3}. Equation \eqref{4.5} is then immediate from \eqref{4.4}. \end{proof} As demonstrated in \cite{BET} and \cite{EGT}, 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 $q(x)$ satisfy \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 shown in \cite{BET}, under condition \textbf{IV} the GLM equations \eqref{ME} have unique smooth real-valued solutions $K_\pm(x,y)$, that satisfy 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 \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 We obtain the following result. \begin{theorem}[\cite{BET}]\label{theor4} Let the set of data ${\mathcal S}$, defined as in \eqref{S4.6}, satisfy 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 $q(x)$ satisfying \eqref{S.2}. \end{theorem} Our next step is to describe a formal scheme for using the inverse scattering method to solve the initial value problem for the KdV equation with initial conditions $q(x)$ satisfying \eqref{2.111} with with some quasi-periodic, finite-gap potentials $p_\pm(x)$ and fixed $m_0\geq 8$ and $n_0\geq m_0+5$. 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. Also set \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 (\cite[Lemma~5.3]{EGT}) by: \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 satisfy \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 \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 are able to prove that $F_\pm$ 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 $Q_\pm(x)= (1+|x|^{m_1+2})^{-1}$, $\tilde q_\pm(x)=0$, and $C_\pm(a)$, that exists due to continuity of functions $F_\pm(x,y,t)$ together with their derivatives. 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 yields 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 $q$ 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 following the arguments in Section~6 of \cite{EGT} verbatim (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{1.37} and we have set \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 $M_\pm(0)$ and square-root singularities $\hat M_\pm(0)$. 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. 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) together with all its derivatives decays exponentially as $x+y\to\pm\infty$. Therefore, it satisfies \eqref{4.311} for all natural $m_1$ and $n_1$. In the second summand, $\hat\psi_\pm(\la,x,t)\hat \psi_\pm(\la,y,t)$ together with all derivatives decays exponentially 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. In both integrals of \eqref{Fhat} we make a change of variables from $\la$ to the quasimomentum variables $\theta_\pm$ 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 $\rho_\pm$ together with their derivatives are smooth and uniformly bounded with respect to $x,y\in\R$. 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 boundary terms either cancel or vanish. To investigate the validity of integration by parts for the first summand in \eqref{Fhat} we use \eqref{1.23}--\eqref{1.25} to represent the first summand 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 have 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. The boundary terms arising from integration by parts (except for the last one, corresponding to $+\infty$) become \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 possible 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 which takes care of 1), is an immediate consequence of the proof of \cite[Lemma~6.2]{EGT}. \begin{lemma} \label{lemestim3} Let $E\in\pa\si_\pm\cap\pa\si$. Then the following limits exists and assume either real or purely 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 no such points pose any problems; 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 write it locally 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 Since this singularity is non-integrable for $s\ge 2$, integration by parts is not an option near such points. Hence we ill split off the leading behavior near such a point. The 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{for } |\xi|\geq 1. \end{cases} \eeq We choose $\varepsilon>0$ so small that the supports of the functions $B_i(\theta)$ neither intersect nor 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*} Because of \eqref{funk2} the first term can be integrated by parts $m_0$ times and thus is covered by Lemma~\ref{lemestim3}. For the second term, we 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 \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 that \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 the interval $(1,1)$ to the interval$(-\infty,\infty)$, 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 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}). This 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 obtained from integration by parts and can hence be replaced with 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 $s=0$, so we need to take the second summand in \eqref{Fhat} into account. For notational convenience we consider only the $+$ case and assume, without loss of generality, that $E=E_{2j}^+$ . In this case, $\si^{(2)}$ is located to the right of $E$ and $\si_-^{(1)}$ to the left. Moreover, without loss of generality, we assume that the other boundary terms are already covered by the previous considerations so that we do not have to worry about them. Choose $\varepsilon>0$ so small that \[ [\la(\theta_+(E) + \I\varepsilon), E]\subset \left((\xi_j^+, E]\cap \si_-^{(1)}\right),\quad (E,\la(\theta_+(E +\varepsilon))]\subset \inte \si^{(2)}. \] Introduce in these two small intervals the two new (positive) variables \beq\label{ash} h:=\frac{\theta_+ - \theta_+(E)}{\I},\quad k:=\theta_+ - \theta_+(E). \eeq We 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 $\wronsk_0(\cdot,\cdot)=\wronsk(\cdot,\cdot)|_{t=0}$. Equation \eqref{defP} was obtained by using \eqref{S2.18} together with the fact that $\overline{g_-(\la,0)} = -g_-(\la,0)$ if $\la\in\si_-$. Integrating \eqref{Rint} and \eqref{T} by parts with respect to $k$ and $h$, respectively, 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 \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 limit is taken from the side of the spectrum of multiplicity two and the right limit is taken from the side of the spectrum of multiplicity one. 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 suffices to prove the following lemma. \begin{lemma}\label{Lemma 3.3} Let $h, k, P(h), R(k)$ be defined by \eqref{ash}, \eqref{defR}, and \eqref{defP}. If $E\in\pa\si_\pm\cap\inte(\si_\mp)$, then \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 $\phi_-(\cdot,x)$, $\ov{\phi_-(\cdot,x)}\in C^{m_0}(E-\varepsilon, E+\varepsilon)$ (and similarly for the $x$ derivative) 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.\ \eqref{phige}) 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 $\sqrt{\la - E}$. For $\la>E$ introduce the function \[ \tilde{P}(k):= \frac{-\I}{2g_+(\la,0)g_-(\la,0)\wronsk_0(\phi_-,\phi_+) \wronsk_0(\overline{\phi_-},\phi_+)}. \] Then \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 From $g_\pm(\la,0)^{-1}=\pm\wronsk_0(\phi_\pm,\overline{\phi_\pm})$ we see that \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 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 numerator of \eqref{def P1} gives \[ \tilde{P}(k)=\frac{\I}{2}\left(-\frac{\wronsk_0(\phi_-,\ov{\phi_+})} {\wronsk_0(\phi_-,\phi_+)}+ \frac{\wronsk_0(\ov{\phi_-},\ov{\phi_+})}{\wronsk_0(\ov{\phi_-},\phi_+)} \right). \] 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 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 the 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 gives \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 that $V(k)= W_{m_0-1}(-k)+o(k^{m_0-1})$, 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 that $R^{-1}(k)=R_{m_0-1}(-k) + o(k^{m_0-1})$, 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} Lemma \ref{Lemma 3.3} settles case 2). Case 3) will follow from the next lemma. \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 $\phi_\pm(\la,x,0)$ by $\phi_{\pm,E}(\la,x,0)$ and $g_\pm(\la)$ by $g_{\pm,E}(\la)$ (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}) \beq\label{defin5} \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} \eeq 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 \beq\label{defin6} \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} \eeq and note that $\ov{\phi_-(\la,x,0)} = \breve\phi_-(\la,x,0) +o(h^{m_0-1})$ for $E-\varepsilon<\la \le E$. Similarly, we write \beq\label{defin7} \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} \eeq and define \beq\label{defin8} \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} \eeq implying $\ov{\phi_+(\la,x,0)} = \breve\phi_+(\la,x,0)+o(k^{m_0-1})$ 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, \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 While the above Wronskian depends on $x$ ($\breve\phi_\pm$ do not solve \eqref{S.4} in general), the leading order is independent of $x$. Here and in all following Wronskians below, we set $x=0$ (and of course $t=0$). Now consider (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 implying \[ P(\la) = \tilde{P}(\la)+o\big((\la-E)^{(m_0-2)/2} \big), \quad E-\varepsilon<\la\leq E \] In particular,\footnote{Note that if $\wronsk(\phi_+,\phi_-)(E)=0$, we loose one derivative, 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 $\tilde{P}(\la)$ 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 \eqref{defin5}--\eqref{defin8}, one can verify that \[ \frac{\breve W(k)}{\breve V(k)} = \ov{\left(\frac{V(-k)}{W(-k)}\right)} + o(k^{m_0-1}). \] Now, since $V(k)= \wronsk_0(\phi_-,\ov{\phi_+}) + o(k^{m_0-1})$, 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 integrating the last (unbounded) integrand of \eqref{sumFc} by parts. 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). To begin, we recall the well-known asymptotic expressions \[ m_+(\la)=\I\sqrt{\la}(1+o(1)),\quad \theta(\la)=\sqrt{\la}(1+o(1)) \] as $\la\to\infty$. Moreover, (recall $\alpha_+(\la,t)=4\I(\sqrt\la)^3t (1+o(1))$) we also have \[ \frac{\pa^s}{\pa\theta^s} u_+(\la,x,t)=O(1), \qquad \frac{\pa^s}{\pa\theta^s} \E^{\alpha_+(\la,t)} =O(t \theta^{2s}). \] As in the previous cases, the only interesting part is 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 that \[ \frac{\pa^s}{\pa\theta_+^s} \rho_+(\la(\theta),x,y,t) =O(\theta^{2s-n_0-1}), \quad \mbox{as } \la\to\infty, \] 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 it is determined by the points $\inte \si_+\cap\pa\si_-$, and is given by $\floor{\frac{m_0}{2}}$. So, excluding case 3), we have an almost complete picture. However, up to this point, we have only looked at $F_{R,+}(x,y,t)$ and have not considered derivatives with respect to $x,y,t$. Fortunately, since $R_+(\la,0)$ is evidently independent of these variables and all other terms of the integrand \eqref{sumFc} (except for the last summand in \eqref{sumFc}) can be differentiated as often as we please, these derivatives do not affect our analysis. Moreover, for this last summand, one has only 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 $q(x)$ satisfy \eqref{2.111}. 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} Lemma \ref{lemfinal} shows that for the integral in \eqref{est10} and its derivatives with respect to $x,y,t$ to converge, it is sufficient that $l+s+3k +2\floor{\frac{m_0}{2}}-n_0<0$. A comparison to \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 at least $l+s= 4, k=0$ and $l+s= 1, k=1$ to hold; that is, we need $m_0$ and $n_0$ to satisfy $4+2\floor{\frac{m_0}{2}}-n_0<0$. 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. 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. 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