Content-Type: multipart/mixed; boundary="-------------0909030936776"

This is a multi-part message in MIME format.

---------------0909030936776
Content-Type: text/plain; name="09-152.comments"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="09-152.comments"

9 pages
---------------0909030936776
Content-Type: text/plain; name="09-152.keywords"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="09-152.keywords"

Inverse scattering, finite-gap background, KdV, nonlinear Paley--Wiener Theorem
---------------0909030936776
Content-Type: application/x-tex; name="PerScatPW.tex"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="PerScatPW.tex"

%% @texfile{
%%     filename="PerScatPW.tex",
%%     version="1.0",
%%     date="Jun-2009",
%%     cdate="20090629",
%%     filetype="LaTeX2e",
%%     journal="Preprint",
%%     copyright="Copyright (C) I. Egorova and G. Teschl".
%%     }

\documentclass{amsart}
%\newcommand{\href}[2]{ #2 }
\usepackage{hyperref}
%\usepackage{showkeys}
\usepackage{amsmath,amsthm,amscd,amssymb}
\usepackage{enumerate}

%%%%%%%%%
\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}}
%\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{\lau}{\lambda^{\mathrm{u}}}
\newcommand{\lal}{\lambda^{\mathrm{l}}}
\newcommand{\laul}{\lambda^{\mathrm{u,l}}}

\newcommand{\spr}[2]{\langle #1 , #2 \rangle}
\newcommand{\diag}{\mathop{\rm diag}}
\newcommand{\dom}{\mathfrak{D}}
\newcommand{\Res}{\mathop{\rm Res}}
\newcommand{\sign}{\mathop{\rm sign}}
\newcommand{\tr}{\mathop{\rm tr}}
\renewcommand{\Im}{\mathop{\rm Im}}
\renewcommand{\Re}{\mathop{\rm Re}}
\newcommand{\supp}{\mathop{\rm supp}}
\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{\om}{\omega}
\newcommand{\si}{\sigma}
\newcommand{\pa}{\partial}
\newcommand{\Om}{\Omega}
\newcommand{\Ga}{\Gamma}
\newcommand{\ep}{\varepsilon}
\newcommand{\la}{\lambda}
\newcommand{\ov}{\overline}




\numberwithin{equation}{section}

\begin{document}

\title[{A Paley--Wiener Theorem for Periodic Scattering}]{A Paley--Wiener Theorem
for Periodic Scattering with Applications to the Korteweg--de Vries Equation}



\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{Inverse scattering, finite-gap background, KdV, nonlinear Paley--Wiener Theorem}
\subjclass[2000]{Primary 34L25, 35Q53; Secondary 35B60, 37K20}



\begin{abstract}
Consider a one-dimensional Schr\"odinger operator which is a short range perturbation
of a finite-gap operator. We give necessary and sufficient conditions on the left, right reflection
coefficient such that the difference of the potentials has finite support to the left, right, respectively.
Moreover, we apply these results to show a unique continuation type result for solutions
of the  Korteweg--de Vries equation in this context. By virtue of the Miura transform an
analogous result for the modified Korteweg--de Vries equation is also obtained.
\end{abstract}

\maketitle


\section{Introduction}




Since the seminal work of Gardner et al.\ \cite{GGKM} in 1967 the inverse scattering
transform is one of the main tools for solving the Korteweg--de Vries (KdV) equation
\beq\label{KdV}
q_t = -q_{xxx} + 6 q q_x.
\eeq
Since it very much resemblances the use of the classical Fourier transform method
to solve linear partial differential equations, the inverse scattering transform is
also known as the nonlinear Fourier transform. Moreover, the linear and nonlinear Fourier transform
share many other properties one of which, namely the Paley--Wiener theorem, will
be the main subject of this paper.

Let $L_q= -\frac{d^2}{dx^2}+q(x,t)$ be the one-dimensional
Schr\"odinger operator associated with a solution of the KdV
equation via the Lax pair formalism. Let us assume that $q(x,t)$ is
decaying sufficiently fast such that one can associate left/right
reflection coefficients $R_\pm(\la,t)$ with $q(.,t)$. In their
seminal paper Deift and Trubowitz \cite{dt} observed that if $L_q$
has no eigenvalues, then $q(.,t)$ has support in $(-\infty,a)$ if
$R(.,t)$ has an analytic extension satisfying the growth condition
$\sqrt\la\,R_-(\la,t) =O(\E^{2a\I\sqrt{\la}})$. Combining this
result with some Hardy space theory enabled Zhang \cite{zh} to prove
unique continuation results for the KdV equation. To be
able to use the result from Deift and Trubowitz, commutation methods (see
\cite{dt}, \cite{gt}) were used to remove all eigenvalues. In order
to avoid this extra step raises the question what is needed in
addition to the growth condition on $R_-(\la,t)$ in the case when
eigenvalues are present. It seems that Aktuson \cite{A} was the
first to realize that there is an extra condition on the residue of
$R_-(\la,t)$ at an eigenvalue. However, it seems he did not notice
that this condition, together with the growth estimate, is also
sufficient. This Paley--Wiener type theorem will be our fist main
result, Theorem~\ref{thm:pw}. In fact, we will establish the more
general case of potentials which are asymptotically close to a
finite-gap potential. We then apply this to solutions of the KdV
equation and prove a unique continuation result
(Theorem~\ref{thm:kdv2}) for the KdV equation in this setting. Again
we extend the results from \cite{zh} to solutions which are not
decaying but rather asymptotically are close to some finite-gap
solution $p(x,t)$.

Finally, let us emphasize that our proof is much simpler than the one from \cite{zh}
and does not require Hardy theory, which would not be available in the more
general setting of periodic background envisaged here. Moreover, while
unique continuation results for wave equations have a long tradition, most of them
only cover decaying solutions (see for example \cite{bu}, the introduction in \cite{ik}
for the case of the nonlinear Schr\"odinger equation, \cite{ekpv}, \cite{kpv}, \cite{kpv2} for the
generalized KdV equation, \cite{le} for the Camassa--Holm equation).
The only exception being Kr\"uger and Teschl \cite{kt} which uses an entirely different
approach only working for discrete models.

By virtue of the Miura transform all our results for KdV also hold for the modified
KdV (mKdV) equation.



\section{Some general facts on finite-gap potentials}
\label{secfgp}



In this section we briefly recall some basic facts on finite gap
potentials needed later one. For further information we refer to,
for example, \cite{GH}, \cite{GRT}, \cite{M}, or \cite{NMPZ}.

Let $L_p$ be  a one-dimensional Schr\"odinger operator with a finite
gap potential $p(x)$ associated with the hyperelliptic Riemann
surface of the square root $Y(\la)^{1/2}$, where
\[
 Y(\la)=-\prod_{j=0}^{2r} (\la-E_j),\quad E_0 < E_1 <
  \dots < E_{2r}.
\]
 The spectrum of $L_p$ consists of $r+1$ bands:
\[
\sigma = \sigma(L_p) = [E_0, E_1]\cup\dots\cup[E_{2j-2},E_{2j-1}]\cup\dots\cup[E_{2r},\infty)
\]
and the potential $p(x)$ is uniquely determined by its associated
Dirichlet divisor
\[
 \left\{(\mu_1,\si_1), \dots,(\mu_r, \si_r)\right\},
\]
where $\mu_j \in [ E_{2j-1}, E_{2j}]$ and $\si_j \in \{+1,-1\}$.

We denote by $ \psi_\pm(\la,x)$ the corresponding Weyl solutions of
$L_p\psi_\pm=\la\psi_\pm$, normalized according to
$\psi_\pm(\la,0)=1$ and satisfying $\psi_\pm(\la,.)\in
L^2((0,\pm\infty))$ for $\la\in\mathbb{C}\setminus\si$. These
functions  are meromorphic for $\la\in\C\backslash\si$ with
continuous limits (away from its singularities described below) on
$\si$ from the upper and lower half plane. Unless otherwise stated
we will always chose the limit from the upper half plane (the one
from the lower half plane producing just the corresponding complex
conjugate number).

When there is the need to distinguish between these limits we will cut the complex plane along
the spectrum $\si$ and denote the upper and lower sides of the cuts by $\siu$ and
$\sil$. The corresponding points on these cuts will be denoted by
$\lau$ and $\lal$, respectively. Moreover, we will write
\[
f(\lau) := \lim_{\varepsilon\downarrow0} f(\la+\I\varepsilon),
\qquad f(\lal) := \lim_{\varepsilon\downarrow0}
f(\la-\I\varepsilon), \qquad \la\in\si.
\]
Let $m_\pm(\la)=\frac{\pa}{\pa x}\psi_\pm(\la,0)$ be the Weyl
functions of operator $L_p$. Due to our normalization, for every
Dirichlet eigenvalue $\mu_j$ the Weyl functions might have poles. If
$\mu_j$ is in the interior of its gap, precisely one Weyl function
$m_+(\la)$ or $m_-(\la)$ will have a simple pole. Otherwise, if
$\mu_j$ sits at an edge, both will have a square root singularity.
Hence we divide the set of poles accordingly:
\begin{align*}
M_+ &=\{ \mu_j\mid\mu_j \in (E_{2j-1},E_{2j}) \text{ and } m_+(\la) \text{ has a simple pole}\},\\
M_- &=\{ \mu_j | 1 \le j \le r\} \backslash M_+.
\end{align*}
In addition, we set \beq\label{S2.6} \delta_\pm(z) :=
\prod_{\mu_j\in M_\pm}(z-\mu_j), \quad \tilde{\psi}_\pm(\la,x) :=
\delta_\pm(\la) \psi_\pm(\la,x)\eeq such that $ \tilde{\psi}_\pm$
are analytic for $\la\in\C\backslash\si$.
 Note
that we have chosen $M_-$ such that
\beq\label{1.881}\delta_-(\la)\delta_+(\la) = \prod_{j=1}^r
(\la-\mu_j).\eeq  Finally, introduce the function
\begin{equation}\label{1.88}
g(\la)= -\frac{\prod_{j=1}^r(\la - \mu_j)}{2 Y^{1/2}(\la)} = \frac{1}{W(\psi_+(\la),\psi_-(\la))},
\eeq
where the branch of the square root is chosen such that
\[
\frac{1}{\I} g(\lau) = \Im(g(\lau))  >0 \quad
\mbox{for}\quad \lambda\in\si,
\]
where $W(f,g)= f(x) g'(x) - f'(x) g(x)$ is the usual Wronski determinant.

Recall also the well-known asymptotics
\beq\label{estg}
g(\la) = \frac{\I}{2\sqrt{\la}} + O(\la^{-1})
\eeq
and
\beq\label{asympsi}
\psi_\pm(\la,x,t)=\E^{\pm\I\sqrt{\la} x} \left(1 +
O\Big(\frac{1}{\sqrt{\la}}\Big) \right),
\eeq
as $\la\to\infty$.



\section{Scattering theory in a nutshell}



In this section we give a brief review of scattering theory with respect to
finite-gap backgrounds. We refer to \cite{BET} for further details and proofs
(see also \cite{F1}, \cite{F2}, \cite{F3}, \cite{MT}).

Let $L_p$ be a Schr\"odinger operators with a real-valued
finite-gap potentials $p(x)$ as in the previous section.
Let $q(x)$ be a real-valued function satisfying\footnote{In \cite{BET} we required the second moment,
i.e., $(1+x^2)$ instead of $(1+|x|)$, but all the results used here hold with this weaker requirement.}
\beq\label{S.2}
\int_\R (1+|x|)|q(x) - p(x)| dx < \infty
\eeq
and let
\[
L_q :=- \frac{d^2}{dx^2} +q(x),\quad x\in \R,
\]
be the ``perturbed" operator. The spectrum of $L_q$ consists of a
purely absolutely continuous part $\sigma$ plus a finite number of
eigenvalues situated in the gaps, \[ \si^d:=
\{\lambda_1,\dots,\lambda_p\}\subset\R\setminus\sigma. \]


%-------------------%
The Jost solutions of the equation
\[
\left(-\frac{d^2}{dx^2}+q(x)\right)\phi(x)= \la \phi(x),\quad \la\in \C,
\]
that are asymptotically close to the Weyl solutions of the background
operators as $x\to\pm\infty$
can be represented with the help of the transformation operators as
\begin{equation}\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,
\end{equation}
where $K_\pm(x,y)$ are real-valued functions satisfying
\begin{equation}\label{A.5}
 K_\pm(x,x)=\pm\frac{1}{2}\int_x^{\pm\infty} (q(y)-p(y))dy.
\end{equation}
\beq\label{A.6} |K_\pm(x,y)|\leq C(x_0)
\int_{\frac{x+y}{2}}^{\pm\infty} |q(s)-p(s)|d s,\quad \pm y>\pm
x>\pm x_0.\eeq Representation \eqref{S2.2} shows, that the Jost
solutions inherit all singularities of the background Weyl solutions
as well as the asymptotics
\beq\label{asymphi}\phi_\pm(\la,x,t)=\E^{\pm \I\sqrt{\la} x} \left(1
+ O\Big(\frac{1}{\sqrt{\la}}\Big) \right), \quad \la\to\infty. \eeq
Hence we set (recall \eqref{S2.6})
\[
\tilde\phi_\pm(\la,x)=\delta_\pm(\la) \phi_\pm(\la,x)
\]
such that the functions $\tilde\phi_\pm(\la,x)$ have no poles in the interior of
the gaps of $\si$. For every eigenvalue we can then
introduce the corresponding norming constants
\[
\left(\gamma_k^\pm\right)^{-1}=\int_\R \tilde\phi_\pm^2(\la_k,x) dx.
\]
Since at every eigenvalue the two Jost solutions must be linearly
dependent, we have \beq \label{ck}\tilde\phi_+(\la_k,x) = c_k
\tilde\phi_-(\la_k,x). \eeq  Furthermore, introduce the scattering
relations 
\beq
T(\lambda) \phi_\pm(\lambda,x)
=\overline{\phi_\mp(\lambda,x)} + R_\mp(\lambda)\phi_\mp(\lambda,x),
\quad\lambda\in\siul,
\eeq
where the transmission and reflection
coefficients are defined as usual,
\beq\label{2.17}
T(\lambda):= \frac{W(\overline{\phi_\pm(\la)}, \phi_\pm(\la))}
{W(\phi_\mp(\la),\phi_\pm(\la))}, \quad
R_\pm(\la):= - \frac{W(\phi_\mp(\la),\overline{\phi_\pm(\la)})}
{W(\phi_\mp(\la), \phi_\pm(\la))}, \quad\la\in \siul.
\eeq
Since
\[
T(\lambda)= \frac{W(\psi_+(\la),\psi_-(\la))}{W(\phi_+(\la),\phi_-(\la))} = \frac{1}{g(\la) W(\phi_+(\la),\phi_-(\la))}
\]
the transmission coefficient has a meromorphic extension to the set
$\C\backslash\si$ with simple poles at the eigenvalues $\la_k$ and
residues given by (cf. \cite{BET})\beq\label{residues}
\Res_{\la=\la_k} T(\la) = 2 Y^{1/2}(\la_k) c_k^{\pm 1}\gamma_k^\pm.
\eeq It is important to emphasize that the reflection coefficients
in general do not have a meromorphic extension.

The sets
\begin{align}\nn
\mathcal{S}_\pm(q) := \Big\{ & R_\pm(\la),\; \la\in\si; \:
\lambda_1,\dots,\lambda_p\in\R\setminus \sigma,\;
\gamma_1^\pm,\dots,\gamma_p^\pm\in\R_+\Big\}
\end{align}
are called the right, left scattering data, respectively. Given $p(x)$, the
potential $q(x)$ can be uniquely recovered from each one of them as follows:

The kernels $K_\pm(x,y)$ of the transformation operators satisfy
the Gelfand-Levitan-Marchenko equations
\begin{equation}\label{ME}
K_\pm(x,y) + F_\pm(x,y) \pm \int_x^{\pm\infty} K_\pm(x,s)
F_\pm(s,y)d s =0, \quad \pm y>\pm x,
\end{equation}
where \footnote{Here we have used the notation
$\oint_{\sigma_\pm}f(\lambda)d\la := \int_{\siu} f(\lambda)d\la -
\int_{\sil} f(\lambda)d\la$.}
\begin{align}\label{4.2}
F_\pm(x,y) &= \frac{1}{2\pi\I}\oint_{\sigma_\pm} R_\pm(\lambda)
\psi_\pm(\lambda,x) \psi_\pm(\lambda,y) g(\lambda)d\la\\ \nn &\quad
+ \sum_{k=1}^p \gamma_k^\pm \tilde\psi_\pm(\lambda_k,x)
\tilde\psi_\pm(\lambda_k,y).
\end{align}
Conversely, given $\mathcal{S}_\pm(q)$ we can compute $F_\pm(x,y)$
and solve \eqref{ME} for $K_\pm(x,y)$. The potential $q(x)$ can then
be recovered from \eqref{A.5}.




\section{Perturbations with finite support on one side and the nonlinear Paley--Wiener theorem}



In this section we want to look at the special case where $q(x)$ will be equal to $p(x)$ for
$x\leq a$ or $x\ge b$. Our main result in this section is the following theorem:


\begin{theorem}[Nonlinear Paley--Wiener]\label{thm:pw}
Suppose $q(x)$ satisfies \eqref{S.2}. Then we have $q(x)=p(x)$ for
$x\le a$ if and only if $R_-(\la)$ has an analytic extension to
$\C\backslash(\si\cup\si^d)$ such that
\begin{align}\label{cond1}
&\Res_{\la=\la_k} \frac{g(\la)}{\delta_-(\la)^2} R_-(\la) = \gamma^-_k,\\\label{cond12}
&\sqrt \la R_-(\la)= O(\E^{2a \I\sqrt{\la}}) \quad \text{as}\  \la\to\infty.
\end{align}
Similarly, we have $q(x)=p(x)$ for $x\ge b$ if and only if $R_+(\la)$ has an analytic extension
to $\C\backslash(\si\cup\si^d)$ such that
\begin{align}\label{cond2}
& \Res_{\la=\la_k} \frac{g(\la)}{\delta_+(\la)^2} R_+(\la)= \gamma^+_k,\\\label{cond22}
&\sqrt\la R_+(\la)= O(\E^{-2\I b
\sqrt{\la}}) \quad \text{as}\ \la\to\infty.
\end{align}
\end{theorem}


\begin{proof}
To see the direct direction suppose $q(x)=p(x)$ for $x\le a$. Then we have
\[
\phi_+(\la,x) =  \alpha(\la) \psi_+(\la,x) + \beta(\la) \psi_-(\la,x), \qquad x \leq a,
\]
and thus
\begin{align*}
\alpha(\la) &= \frac{W(\psi_-(\la),\phi_+(\la))}{W(\psi_-(\la),\psi_+(\la))}
= -g(\la) W(\psi_-(\la),\phi_+(\la)),\\
\beta(\la) &= -\frac{W(\psi_+(\la),\phi_+(\la))}{W(\psi_-(\la),\psi_+(\la))}
= g(\la) W(\psi_+(\la),\phi_+(\la)),
\end{align*}
where the Wronskians can be evaluated at any $x\leq a$. In
particular, $\alpha(\la)$ is analytic in $\C\backslash\si$ and
$\beta(\la)$ is meromorphic in $\C\backslash\si$ with the only
simple poles at $\la\in M_+$. Note also that $\beta(\la)$ has simple
zeros at $\la\in M_-$and thus
\beq\label{betatilde}
\tilde{\beta}(\la)= \frac{\delta_+(\la)}{\delta_-(\la)} \beta(\la)
\eeq
is analytic in $\C\backslash\si$. Hence, since $\alpha(\la)$
vanishes at each eigenvalue $\la_k$, evaluating
\[
\tilde\phi_+(\la,x) =  \alpha(\la) \tilde\psi_+(\la,x) + \tilde\beta(\la) \tilde\psi_-(\la,x),
\qquad x \leq a,
\]
at $\la_k$ shows $\tilde{\beta}(\la_k) = c_k$ and formula
\eqref{cond1} follows from \eqref{1.881}, \eqref{1.88},
\eqref{residues}, \eqref{betatilde} and
\[
R_-(\la) = \frac{\beta(\la)}{\al(\la)},
\]
respectively,
\[
g(\la)R_-(\la)\delta_-^{-2}
(\la)=\tilde\beta(\la)T(\la)(2Y^{1/2}(\la))^{-1}
\]
The asymptotic behavior \eqref{cond12} follows using the well-known asymptotical formula $\al(\la)= 
T(\la)^{-1} = 1+o(1)$, \eqref{estg}, \eqref{asympsi}, and \eqref{asymphi}. This finishes
the first part.

To see the converse, note that the growth estimate implies that we
can evaluate the integral in \eqref{4.2} by the residue theorem by
using a large circular arc of radius $r$ whose contribution will
vanish as $r\to\infty$ by the Jordan's Lemma. Hence the integral in
\eqref{4.2} is just the sum over the residues which are precisely at the
eigenvalues $\la_k$ and by our conditions \eqref{cond1}
on the poles of integrand   it will cancel with the other sum in
\eqref{4.2}. Thus $F(x,y)=0$ for $y<x<a$ which by the GLM equation
implies $K_-(x,y)=0$ for $y<x<a$ which finally implies $p(x)-q(x)=0$
for $y<x<a$.
\end{proof}

As a consequence we obtain that the scattering data $\mathcal{S}_\pm(q)$ are determined
by $R_\pm(\la)$ alone in such a situation.

\begin{corollary}
Suppose $q(x)$ satisfies \eqref{S.2} and $q(x)=p(x)$ for
$x\le a$. Then $\mathcal{S}_-(q)$ is determined by $R_-(\la)$ alone.
Similarly, if $q(x)$ satisfies \eqref{S.2} and $q(x)=p(x)$ for $x\ge
b$, then $\mathcal{S}_+(q)$ is determined by $R_+(\la)$ alone.
\end{corollary}

Consequently, this implies that while infinitely many $q(x)$ satisfying \eqref{S.2} will
have the same reflection coefficient $R_-(\la)$, only one will satisfy $q(x)=p(x)$ for
$x\le a$. In fact, one could add or remove eigenvalues (unless $p(x)$ is constant
one has of course take the known algebraic constraints into account \cite{tistalg}) but
only one choice will satisfy $q(x)=p(x)$ for $x\le a$. In particular, this property
is destroyed if one adds or removes eigenvalues using commutation methods
\cite{gs}, \cite{gt}.



\section{Applications to KdV}



As our next step we show how to use the solution of the inverse scattering
problem found in the previous section to give a formal scheme for solving
the initial-value problem for the KdV equation with initial data from the class \eqref{S.2}.

Let $p(x,t)$ be a finite-gap solution of the KdV equation and suppose $q(x,t)$ is
a (classical) solution of the KdV equation 
\beq\label{KdV}
q_t(x,t) = -q_{xxx}(x,t) + 6 q(x,t) q_x(x,t),
\eeq
(where subscripts denote partial derivatives as usual) satisfying \eqref{S.2} for every $t$.
For the existence of such solutions we refer to \cite{EGT}, \cite{ET} (see also \cite{F4}).
Then all considerations from the previous section apply to the operator $L_q(t)$ if we
consider $t$ as an additional parameter. Moreover, the time evolution of the scattering
data can be computed explicitly and is given in the next lemma:

\begin{lemma}[\cite{EGT}]
Let $q(x,t)$ be a solution of the KdV equation satisfying \eqref{S.2}.
Then $\la_k(t)=\la_k(0)\equiv \la_k;$
\begin{align}\label{refl}
R_\pm(\la,t) &= R_\pm(\la,0)\E^{\alpha_\pm(\la,t)
-\alpha_\mp(\la,t)}, \quad \la\in\si_\pm, \\ \label{trans} T(\la,t)
&= T(\la,0),\quad\la\in\C,\\ \label{norm} \gamma_k^\pm(t) &=
\gamma_k^\pm(0) \,
\frac{\delta_\pm^2(\la_k,0)}{\delta_\pm^2(\la_k,t)}\,
\E^{2\alpha_\pm(\la_k,t)},
\end{align}
where $\delta_\pm(\la,t)$ is defined  as in \eqref{S2.6} and
\beq
\alpha_\pm(\la,t) := \int_0^t \left(2(p(0,s) + 2\la)
m_\pm(\la,s) - \frac{\pa p_\pm(0,s)}{\pa x}\right)ds.
\eeq
\end{lemma}

Our first result reads

\begin{theorem}\label{thm:kdv1}
Let $q(x,t)$ be a solution of the KdV equation satisfying \eqref{S.2}.
Suppose that $q(x,t) = p(x,t)$ for $x<a$ at two times $t_0 \ne t_1$.
Then $q(x,t) = p(x,t)$ for all $(x,t)\in\R^2$.
\end{theorem}

\begin{proof}
Without loss we can choose $t_0=0$. Then $R_-(\la,0) = O(\E^{-2a\sqrt{\la}})$.
If $q(0,.) \neq p(0,.)$ we can choose $a$ maximal and this this estimate cannot be improved!
Thus $\alpha_-(\la,t)= \E^{ t \la^{3/2} + \dots}$ shows that the same estimate
cannot hold for another $t\ne 0$ and we are done.
\end{proof}

With the help of this result we also obtain the following unique continuation result for
our situation:


\begin{theorem}\label{thm:kdv2}
Let $p(x,t)$ be a finite-gap solution of the KdV equation and
$q(x,t)$ a solution of the KdV equation satisfying \eqref{S.2}.
Suppose that $q(x,t) = p(x,t)$ for $(x,t)$ in some open set $U\subset\R^2$.
Then $q(x,t) = p(x,t)$ for $(x,t)\in\R^2$.
\end{theorem}

\begin{proof}
Let $[a,b]\times[t_0,t_1] \subset U$ and define
\[
\tilde{q}(x,t)= \begin{cases} p(x,t), & x\leq a,\\
q(x,t), & x \geq a,
\end{cases}
\]
for $t\in[t_0,t_1]$. Then Theorem~\ref{thm:kdv1} implies $\tilde{q}(x,t) = p(x,t)$ for $(x,t)\in\R\times [t_0,t_1]$
and consequently $q(x,t) = p(x,t)$ for $(x,t)\in[a,\infty)\times [t_0,t_1]$. Hence another application of
Theorem~\ref{thm:kdv1} finishes the proof.
\end{proof}

Let $u(x,t)$ be a finite-gap solution of the KdV equation and suppose $v(x,t)$ is
a (classical) solution of the mKdV equation 
\beq\label{mKdV}
v_t(x,t) = -v_{xxx}(x,t) + 6 v(x,t)^2 v_x(x,t).
\eeq
Then by virtue of the Miura transform (see, e.g., \cite{GH}, \cite{gs}),
\beq\label{miurau}
p(x,t) = u(x,t)^2 + u_x(x,t),
\eeq
is a finite-gap solution of the KdV equation and
\beq\label{miurav}
q(x,t) = v(x,t)^2 + v_x(x,t)
\eeq
is a solution of the KdV equation. We will suppose
again that $q(x,t)$ satisfies \eqref{S.2} for every $t$.
Then, since $v(x,t)=u(x,t)$ implies 

\begin{corollary}
Let $u(x,t)$ be a finite-gap solution of the mKdV equation and
$v(x,t)$ a solution of the mKdV equation such that \eqref{miurau}, \eqref{miurav} satisfy \eqref{S.2}.
Suppose that $v(x,t) = u(x,t)$ for $(x,t)$ in some open set $U\subset\R^2$.
Then $v(x,t) = u(x,t)$ for $(x,t)\in\R^2$.
\end{corollary}

\begin{proof}
Since $v(x,t) = u(x,t)$ for $(x,t)\in U$ implies $q(x,t) = p(x,t)$ for $(x,t)\in U$, Theorem~\ref{thm:kdv2}
shows $q(x,t) = p(x,t)$ for $(x,t)\in\R^2$. Hence $w_x(x,t)+w(x,t)^2 + 2 u(x,t) w(x,t) =0$, where
$w(x,t) = v(x,t)-u(x,t)$ and the standard uniqueness result for ordinary differential equations
yields $w(x,t)=0$ for $(x,t)\in \R \times \{t | (x_0,t)\in U \text{ for some } x_0\}$. Thus uniqueness
of solutions of the mKdV equation finally implies $w(x,t)=0$ for all $(x,t)\in \R^2$.
\end{proof}




\noindent{\bf Acknowledgments.} We are very grateful to F. Gesztesy for
hints with respect to the literature. 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.

\begin{thebibliography}{99}
\bibitem{A} T. Aktosun, {\em Bound states and inverse scattering for the Schr\"odinger
equation in one dimension}, J. Math. Phys. {\bf 35:12}, 6231--6236 (1994).
\bibitem{bu} J. Bourgain, {\em On the compactness of the support of solutions of dispersive equations},
Internat. Math. Res. Notices {\bf 1997}, no. 9, 437--447 (1997).
\bibitem{buk} J. Bourgain and C. E. Kenig, {\em On localization in the continuous Anderson-Bernoulli model in higher dimension},
Invent. Math. {\bf 161:2}, 389--426 (2005).
\bibitem{BET} A. Boutet de Monvel, I. Egorova, and G. Teschl,
{\em Inverse scattering theory for one-dimensional Schr\"odinger
operators with steplike finite-gap potentials}, J. d'Analyse Math.
{\bf 106:1}, 271--316, (2008).
\bibitem{dt} P. Deift and E. Trubowitz, {\em Inverse scattering on the line},
Commun. Pure Appl. Math. {\bf 32}, 121--251 (1979).
\bibitem{ET} I. Egorova and G. Teschl, {\em On the Cauchy
problem for the Korteweg--de Vries equation with steplike finite-gap initial data II.
Perturbations with Finite Moments}, Preprint.
\bibitem{EGT} I. Egorova, K. Grunert, and G. Teschl, {\em On the Cauchy
problem for the Korteweg--de Vries equation with steplike finite-gap initial data I.
Schwartz-type perturbations}, Nonlinearity {\bf 22}, 1431--1457 (2009).
\bibitem{ekpv} L. Escauriaza, C. E.  Kenig, G. Ponce, and L. Vega, {\em On uniqueness properties of solutions of
Schr\"odinger equations}, Comm. Partial Differential Equations {\bf 31:12}, 1811--1823 (2006).
\bibitem{F1} N.E. Firsova, {\it An inverse scattering problem for the perturbed Hill
operator},  Mat. Zametki {\bf 18}, no. 6, 831--843 (1975).
\bibitem{F2} N.E. Firsova, {\it A direct and inverse scattering problem for a one-dimensional
perturbed Hill operator} Matem. Sborn. (N.S.) {\bf 130(172)}, no. 3, 349--385 (1986).
\bibitem{F3} N. Firsova, {\em Resonances of the perturbed Hill operator with exponentially decreasing
extrinsic potential}, Mat. Zametki {\bf 36}, 711--724 (1984).
\bibitem{F4} N.E. Firsova, {\em Solution of the Cauchy problem for the Korteweg-de Vries equation
with initial data that are the sum of a periodic and a rapidly decreasing function},
Math. USSR-Sb. {\bf 63}, no. 1, 257--265 (1989).
\bibitem{GGKM} C. S. Gardner, J. M. Green, M. D. Kruskal, R. M. Miura,
 {\it Method for solving the Korteweg-de Vries equation},
Phys. Rev. Lett., {\bf 19}, 1095--1097 (1967).
\bibitem{GH} F. Gesztesy and H. Holden, {\em Soliton Equations and
their Algebro-Geometric Solutions. Volume {I}: $(1+1)$-Dimensional
Continuous Models}, Cambridge Studies in Advanced Mathematics,
Vol. {\bf 79}, Cambridge University Press, Cambridge, 2003.
\bibitem{gs} F. Gesztesy and R. Svirsky, {\em (m)KdV-Solitons on the background of
quasi-periodic finite-gap solutions}, Memoirs Amer. Math. Soc. {\bf 118},
No. 563 (1995).
\bibitem{gt} F. Gesztesy and G. Teschl, {\em On the double commutation
method}, Proc. Amer. Math. Soc. {\bf 124}, 1831--1840 (1996).
\bibitem{GRT} F. Gesztesy, R. Ratnaseelan, and G. Teschl, {\it The KdV
hierarchy and associated trace formulas}, in ``{\it Proceedings of
the International Conference on  Applications of Operator Theory\/}'',
(eds. I. Gohberg, P. Lancaster, and P. N. Shivakumar), Oper. Theory
Adv. Appl., 87, Birkh\"auser, Basel, 125--163 (1996).
\bibitem{ik} A. Ionescu and C. E. Kenig, {\em $L^p$ Carleman inequalities and uniqueness of solutions of
nonlinear Schr\"odinger equations}, Acta Math. {\bf 193:2}, 193--239 (2004).
\bibitem{kpv} C. E.  Kenig, G. Ponce, and L. Vega, {\em On the support of solutions to the
generalized KdV equation}, Ann. I. H. Poincar\'{e} {\bf 19:2}, 191--208 (2002).
\bibitem{kpv2} C. E. Kenig, G. Ponce, and L. Vega, {\em On unique continuation of solutions to the generalized KdV
equation}, Math. Res. Lett. {\bf 10}, 833--846 (2003).
\bibitem{kt} H. Kr\"uger and G. Teschl, {\em Unique continuation for discrete nonlinear wave equations},
\arxiv{0904.0011}.
\bibitem{le} J. Lenells, {\em Infinite propagation speed of the Camassa--Holm equation}, J. Math. Anal. Appl. {\bf 325:2}, 1468--1478 (2007).
\bibitem{M} V. A. Marchenko, {\em Sturm--Liouville Operators and
Applications}, Birkh\"auser, Basel, 1986.
\bibitem{MT} A. Mikikits-Leitner and G. Teschl, {\em Trace formulas for
Schr\"odinger operators in connection with scattering theory for finite-gap backgrounds},
\arxiv{0902.3917}.
 \bibitem{NMPZ} S. P. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov,
{\it Theory of Solitons. The Inverse Scattering Method}, Springer, Berlin, 1984.
\bibitem{tistalg} G. Teschl, {\em Algebro-geometric constraints on solitons with respect to
quasi-periodic backgrounds}, Bull. London Math. Soc. {\bf 39}, No.4, 677--684 (2007).
\bibitem{zh} B. Zhang, {\em Unique continuation for the Korteweg--de Vries equation}, SIAM J. Math. Anal. {\bf 23},
55--71 (1992).
\end{thebibliography}

\end{document}
---------------0909030936776--
