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10 pages
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Unique continuation, Toda lattice, Kac-van Moerbeke lattice, Ablowitz-Ladik equations,
discrete nonlinear Schroedinger equation, Schur flow
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\begin{document}
\title{Unique continuation for discrete nonlinear wave equations}
\author[H. Kr\"uger]{Helge Kr\"uger}
\address{Department of Mathematics\\ Rice University\\ Houston\\ TX 77005\\ USA}
\email{\mailto{helge.krueger@rice.edu}}
\urladdr{\url{http://math.rice.edu/~hk7/}}
\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 and
the National Science Foundation (NSF) under Grant No.\ DMS--0800100.}
\keywords{Unique continuation, Toda lattice, Kac--van Moerbeke lattice, Ablowitz--Ladik equations,
discrete nonlinear Schr\"odinger equation, Schur flow}
\subjclass[2000]{Primary 35L05; 37K60 Secondary 37K15, 37K10}
\begin{abstract}
We establish unique continuation for various discrete nonlinear wave equations.
For example, we show that if two solutions of the Toda lattice coincide for one lattice point in some
arbitrarily small time interval, then they coincide everywhere. Moreover, we establish analogous results
for the Toda, Kac--van Moerbeke, and Ablowitz--Ladik hierarchies. Although all these equations
are integrable, the proof does not use integrability and can be adapted to other equations as well.
\end{abstract}
\maketitle
\section{Introduction}
Unique continuation results for wave equations have a long tradition and seem to originate in control theory.
One of the first result seems to be the one by Zhang \cite{zh}, where he proves that if a short range solution
of the Korteweg--de Vries (KdV) equation vanishes on an open subset in the $xt$-plane, then it must vanish everywhere.
Since then, this result has been extended in various directions and for different equations (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).
However, all the results so far seem to only deal with wave equations which are continuous in the spatial direction and
this clearly raises the question for such unique continuation results for wave equations which are discrete in the spatial
variable. In particular, to the best of our knowledge there are no results for example for the Toda equation,
one of the most prominent discrete system. While in principle the strategy from Zhang \cite{zh} would be applicable
to the Toda lattice, it is the purpose of this note to advocate a much simpler direct approach in the
discrete case. We will start with the Toda lattice as our prototypical example and then show how the entire Toda
hierarchy as well as the Kac--van Moerbeke and Ablowitz--Ladik hierarchies can be treated. It is important to
stress that our approach does not use integrability of these equations and hence can be adapted to more
general systems. On the other hand, our approach seems to need one dimensionality in the spatial variable and
thus does not apply to the discrete Schr\"odinger equation on $\Z^d$. Due to the connections with
localization for discrete Anderson--Bernoulli models, unique continuation for this
model is an important open problem; see \cite{bu2}, \cite{buk}.
\section{The Toda lattice}
In this section we want to treat the Toda lattice as the prototypical example.
To this end, recall the Toda lattice \cite{ta} (in Flaschka's variables \cite{fl1})
\begin{align} \nn
\dot{a}(n,t) &= a(n,t) \Big(b(n+1,t)-b(n,t)\Big), \\ \label{todeqfl}
\dot{b}(n,t) &= 2 \Big(a(n,t)^2-a(n-1,t)^2\Big), \qquad n\in \Z,
\end{align}
where the dot denotes a derivative with respect to $t$.
It is a well studied physical model and one of the prototypical discrete integrable wave equations.
We refer to the monographs \cite{fad}, \cite{tjac}, \cite{ta} or the review articles \cite{krt}, \cite{taet}
for further information.
\begin{theorem}\label{thmToda}
Assume that $a_0(n,t)\ne 0, b_0(n,t)$ and $a(n,t), b(n,t)$ are complex-valued solutions
of \eqref{todeqfl} such that there is one $n_0 \in \Z$ and two times $t_01$. Here $R(n,m)$ denotes terms which involve only $a(\ell)$ and $b(\ell)$ with $m \le \ell \le n$ and
we set $R(n,m)=0$ if $n0$ and $\rho(n,t)>0$ are solutions
of the Kac--van Moerbeke equation
\be
\dot{\rho}(n,t) = \rho(n,t) \big(\rho(n+1,t) - \rho(n-1,t) \big)
\ee
such that there is one $n_0 \in \Z$ and two times $t_00$ and set $p=p_++p_--1$.
Assume that $\alpha_0(n,t), \beta_0(n,t)$, with $\rho_0(n,t) \ne 0$, and $\alpha(n,t), \beta(n,t)$
are solutions of some equation in the Toda hierarchy $\AL_{\ul p}$ such that there is one
$n_0 \in \Z$ and two times $t_0