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Riemann-Hilbert problem, solitons, periodic, Toda lattice
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\begin{document}

%\preprint{nlin.SI/0607053}

\title[Stability of Periodic Soliton Equations]{Stability of Periodic Soliton Equations under Short Range Perturbations}

\author{Spyridon Kamvissis}
 \email{spyros@tem.uoc.gr}
 \affiliation{Department of Applied Mathematics, University of Crete \\
714 09 Iraklion, Greece}
\author{Gerald Teschl}%
 \email{Gerald.Teschl@univie.ac.at}
 \homepage{http://www.mat.univie.ac.at/~gerald/}%
 \affiliation{Faculty of Mathematics, University of Vienna,
Nordbergstrasse 15, 1090 Vienna, Austria\\ and\\ International Erwin Schr\"odinger
Institute for Mathematical Physics, Boltzmanngasse 9, 1090 Vienna, Austria}


\date{\today}% It is always \today, today,
             %  but any date may be explicitly specified
\thanks{Research supported in part by the ESF programme MISGAM, and
the Austrian Science Fund (FWF) under Grant No.\ P17762.}

\begin{abstract}
We consider the stability of (quasi-)periodic solutions of soliton equations under
short range perturbations
and give a complete description of the long time asymptotics in this situation.
We show that, apart from the phenomenon of the solitons travelling on the quasi-periodic
background, the perturbed solution asymptotically approaches a modulated
solution.
We use the Toda lattice as a model but the same methods and ideas are applicable to
all soliton equations in one space dimension.

More precisely, let $g$ be the genus of the hyperelliptic Riemann surface associated with
the unperturbed solution. We show that the $n/t$-pane contains $g+2$ areas where
the perturbed solution is close to a quasi-periodic solution in the same isospectral
torus. In between there are $g+1$ regions where the perturbed solution is asymptotically
close to a modulated lattice which undergoes a
continuous phase transition (in the Jacobian variety) and which interpolates
between these isospectral solutions. In the special case of  the free solution ($g=0$) the
isospectral torus consists of just one point and we recover the classical result.

Both the solutions in the isospectral torus and the phase transition are explicitly
characterized in terms of Abelian integrals on the underlying hyperelliptic Riemann surface.
\end{abstract}

\pacs{02.30.Ik, 02.70.Hm, 05.45.Yv}
\keywords{Riemann-Hilbert problem, solitons, periodic, Toda lattice}%Use showkeys class option if keyword
                              %display desired
%\subjclass[2000]{Primary 37K40, 37K45; Secondary 35Q15, 37K10}

\maketitle

\section{Introduction}

One of the most important defining properties of solitons  is their
stability under perturbations. The classical result going back to
Zabusky and Kruskal \cite{zakr} states that a short range perturbation
of the constant solution of a soliton equation
eventually splits into a number of  stable solitons. So the solitons constitute
the stable part of arbitrary short range initial conditions. This is the  motivation
for the result presented here. Our aim is 
to investigate the case where the constant background
solution is replaced by a quasi-periodic one. 

Solitons on a (quasi-)periodic background have a long tradition
and are used to model localized excitements on a phonon, lattice, or magnetic
field background. Of course periodic solutions, as well as solitons travelling
on a periodic background, are well understood (see e.g.\ \cite{km2}
or \cite{emt3}) and one might expect that,
in analogous fashion, a
small initial perturbation of the periodic lattice splits into a number of solitons 
on a periodic background.
The reason why this naive expectation has to be wrong  is
related to the fact that solitons produce a phase shift; something that  was probably first
observed in \cite{kumi} who investigated solitons of the Korteweg-de Vries
equation on the background  of the two-gap Weierstrass
solution. Since the phase shifts of the solitons do not necessarily add up to zero,
there must be an additional feature making up for this phase shift. This feature
has  not been understood so far and will be  our main topic here.

To illustrate these facts, let us consider the doubly infinite Toda lattice in Flaschka's
variables (see e.g.\ \cite{tjac} or \cite{ta})
\begin{align*}
\dot b(n,t) &= 2(a(n,t)^2 -a(n-1,t)^2),\\
\dot a(n,t) &= a(n,t) (b(n+1,t) -b(n,t)),
\end{align*}
$(n,t) \in \Z \times \R$,
where the dot denotes differentiation with respect to time. We
will consider a quasi-periodic algebro-geometric background
solution $(a_q,b_q)$ (e.g., any periodic solution) plus
a short range (in the sense of \cite{emt2})
perturbation $(a,b)$. The
perturbed solution can be computed via the inverse
scattering transform. The case where $(a_q,b_q)$ is constant
is classical (see again \cite{tjac} or \cite{ta}), but the more
general case applicable  here has  only recently been
analyzed in \cite{emt2}
(see also \cite{emt}).

In Figure~\ref{fig1} the numerically computed solution corresponding to
the initial condition $a(n,0)=1$, $b(n,0)=(-1)^n + 2 \delta_{0,n}$ is shown.
\begin{figure}
\includegraphics[width=7cm]{TodaStab1}
\caption{Numerically computed solution of the Toda lattice, with initial
condition a period two solution perturbed at one point in the middle.} \label{fig1}
\end{figure}
In this picture the two observed lines express the variables $a(n,t)$ at a frozen time $t=90$.
In areas where the lines seem to be continuous this is due to the fact that we have plotted
a huge number of particles (around $1000$) and also due to the $2$-periodicity in space.
So one can think of the two lines as the even- and odd-numbered particles of the lattice.
We first note the single soliton which separates two regions of
apparent periodicity on the left. Also, after the soliton, we observe three different 
areas with apparently periodic
solutions of  period two.
Finally there are some transitional regions in between
which interpolate between the different period two regions. It is the
purpose of this paper to give a complete explanation of
this picture. We provide the details in case of the Toda lattice though it is
clear that our arguments apply to other soliton equations as well.


\section{Quasiperiodic solutions in terms of Riemann theta functions}


We begin by recalling that quasi-periodic solutions are most conveniently written
in terms of Riemann theta functions (\cite{dub}).
In case of the Toda lattice the underlying Riemann surface is associated with the
square root
\[
\Rg{z} = -\prod_{j=0}^{2g+1} \sqrt{z-E_j}.
\]
We can picture this surface as  the end result of cutting and pasting; we consider two 
"sheets" (copies of the complex plane), we cut them along the segments
$[E_0,E_1]$, $[E_2, E_3]$, \dots{} and paste the top side of the upper sheet
to the bottom side of the lower sheet and the bottom side of the upper
sheet to the top side of the lower sheet across every such segment.
The numbers $E_j\in\R$ are the band edges of the spectrum $\Sigma =
\bigcup_{j=0}^g [E_{2j},E_{2j+1}]$ of the corresponding
Jacobi operator in the Lax pair. On this Riemann surface we have a standard
basis of normalized holomorphic differentials
\[
\zeta_j = \frac{\sum_{k} c_j(k) z^k}{\Rg{z}}dz, \quad1\le j \le g
\]
(the constants $c_j(k)$ have to be determined from the usual normalization
with respect to a canoncial homology basis).

Introduce the vector
\begin{align*}
z_j(n,t) &= \int_{E_0}^{\infty_+} \zeta_j - \sum_{k=1}^g \int_{E_0}^{\mu_k} \zeta_j\\
& - n \int_{\infty_-}^{\infty_+} \zeta_j + t 2 c_j(g) - \vrc \in \C^g.
\end{align*}
Here $\infty_\pm$ are the points above $\infty$ on the upper/lower sheet and
$\vrc$ is the vector of Riemann constants. The numbers $\mu_j$ are some arbitrary points whose images in the complex plane lie in the $j$-th interior spectral gap.
All possible choices form the isospectral class of quasi-periodic Jacobi operators
with the given spectral bands.  This isospectral class is just a $g$ dimensional torus,
since the preimage of each gap with respect to the map that maps the surface to 
the complex plane (seen as one of the two sheets) consists of two parts,
one on the upper and one the lower sheet, which form a circle.

Then the well-known formulas for the solutions read
\begin{align*}
a(n,t)^2 &= \ti{a}^2 \frac{\theta(\ulz(n+1,t))
\theta(\ulz(n-1,t))}{\theta(\ulz(n,t))^2},\\
b(n,t) &= \ti{b} + \frac{1}{2}\frac{d}{dt}
\ln\left( \frac{\theta(\ulz(n,t))}{\theta(\ulz(n-1,t))} \right),
\end{align*}
where $\ti{a}$, $\ti{b}$ are the averages and
\[
\theta(\ulz) = \sum_{\ul{m} \in \Z^g} \exp 2 \pi \I \left( \spr{\ul{m}}{\ulz} +
\frac{\spr{\ul{m}}{\ul{\tau} \, \ul{m}}}{2}\right) ,\quad \ulz \in \C^g,
\]
is the Riemann theta function of our surface. The matrix $\tau$ is the matrix of
$b$-periods of the normalized basis of holomorphic differentials $\zeta_j$.


\section{The main result}

We will assume for simplicity that no solitons are present.
In other words, we assume that the Jacobi operator
above has no eigenvalues. The assumption is not crucial, since the solitons
can be easily incorporated using a Darboux transform.

To obtain the long time asymptotics one reformulates the problem
as a Riemann-Hilbert problem (RHP) on the underlying Riemann surface.
This shows that the solution can be read off from the two by two matrix valued
function $m$ that is meromorphic off the preimage of the spectrum $\Sigma$,
with divisor satisfying
\[
(m_{j1}) \ge -\sum_k \di_{\mu_k^*(n,t)}, \quad
(m_{j2}) \ge -\sum_k \di_{\mu_k(n,t)},
\]
$ j=1,2$, jump matrix given by
\[
J(p,n,t)= \begin{pmatrix}
1 - |R(p,n,t)|^2 & - \overline{R(p,n,t)} \\
R(p,n,t) & 1
\end{pmatrix},
\]
$p\in\Sigma$, and normalization
\[
m(p,n,t) \to \id, \quad \text{as } p\to \infty_+.
\]
This is  similar to the RHP applicable to  the constant background case, 
with the main difference being that we allow poles at the points
$\mu_j(n,t)$, and their flip images $\mu_j^*(n,t)$ on the other sheet.
The $g$ points $\mu_j(n,t)$ are uniquely defined by the Jacobi inversion problem
\[
\sum_{k=1}^g \int_{E_0}^{\mu_k(n,t)} \zeta_j= 
\sum_{k=1}^g \int_{E_0}^{\mu_k} \zeta_j\\
+ n \int_{\infty_-}^{\infty_+} \zeta_j - t 2 c_j(g).
\]
This is a crucial point and related to the fact that our RHP is no
longer formulated in the complex plane. While a holomorphic RHP with
jump of index zero has a solution in the complex plane, this
is no longer true on a surface of genus $g$ unless we admit
at least $g$ poles (\cite{ro}). In particular, the above RHP
has no holomorphic solutions except in special cases
(e.g.\ if there is no jump). In fact, a main issue in the mathematical
analysis of the RHP (\cite{katept}) is to define an appropriate space of solutions
which makes the problem well-posed.

The matrix elements of the jump are of the form
\[
R(p,n,t) = R(p) \Theta(p,n,t) \exp( t \phi(p) ),
\]
where $R(p)$ is the reflection coefficient at $t=0$, $\Theta$ is a ratio of four
theta functions
\[
\Theta(p,n,t) = \frac{\theta(\ulz(p,n,t))}{\theta(\ulz(p,0,0))}
\frac{\theta(\ulz(p^*,0,0))}{\theta(\ulz(p^*,n,t))}
\]
(of modulus one), and the phase $\phi$ is
given by
\[
\phi(p) = 2 \int_{E_0}^p \Omega_0 + 2 \frac{n}{t} \int_{E_0}^p \omega_{\infty_+\infty_-}.
\]
Here
\[
\omega_{\infty_+ \infty_-}= \frac{\prod_{j=1}^g (z -\lambda_j) }{\Rg{z}}dz
\]
is the normalized Abelian differential of the third kind with poles
at $\infty_+$ respectively $\infty_-$ and
\[
\Omega_0 = \frac{\prod_{j=0}^g (z - \ti\lambda_j) }{\Rg{z}}dz,
\quad \sum_{j=0}^g \ti\lambda_j = \frac{1}{2} \sum_{j=0}^{2g+1} E_j,
\]
is the normalized Abelian differential of the second kind with second order poles at
$\infty_+$ respectively $\infty_-$. The constants $\lambda_j$ respectively
$\ti\lambda_j$ again have to be determined from the normalization.

There are $g+1$ stationary phase points $z_j(n/t)$ which behave as follows:
As $\eta=\frac{n}{t}$ runs from $-\infty$ to $+\infty$ we start with $z_g(\eta)$
moving from $\infty$ towards $E_{2g+1}$ while the others stay in their
spectral gaps until $z_g(\eta)$ has passed the first spectral band.
After this has happened, $z_{g-1}$ can leave its gap, while $z_g(\eta)$
remains there, traverses the next spectral band and so on. Until
finally $z_0(\eta)$ traverses the last spectral band and escapes to
$-\infty$.

Factorizing the jump matrix and using the asymptotic  analysis of oscillatory
Riemann-Hilbert problems introduced in (\cite{dz}), but generalized accordingly
in \cite{katept}, one obtains that for long times the perturbed
Toda lattice is asymptotically close to the 
following limiting lattice defined by
\[
\aligned
\prod_{j=n}^{\infty} (\frac{a_l(j,t)}{a_q(j,t)} )^2 = &
\frac{\theta(\ulz(n,t))}{\theta(\ulz(n+1,t))}
\frac{\theta(\ulz(n+1,t)+ \ul{\delta}(n,t))}{\theta(\ulz(n,t)+\ul{\delta}(n,t))} \times\\
& \times \exp\left( \frac{1}{2\pi\I} \int_{C(n/t)} 
\hspace*{-7mm} \log (1-|R|^2) \omega_{\infty_- \infty_+}\right),\\
\delta_\ell(n,t)= & \frac{1}{2\pi\I} \int_{C(n/t)}
\hspace*{-7mm} \log (1-|R|^2) \zeta_\ell,
\endaligned
\]
where $R$ is the associated reflection coefficient,
$C(n/t) = \Sigma \cap (-\infty,z_j(n/t))$, and
$z_j(n/t)$ is the single stationary phase
point lying in the spectrum, if there is such a point, or
otherwise, one of the two stationary phase points 
lying in the same spectral gap.

In summary, for any short range perturbation $a(n,t)$ of a quasi-periodic solution of the
Toda lattice one has that
\begin{equation} \label{eqlim}
\prod_{j=n}^{\infty} \frac{a(j,t)}{a_l(j,t)} \to 1
\end{equation}
uniformly in $n$, as $t \to \infty$.

From \eqref{eqlim} one recovers the $a(n,t)$ and by differentiating
one recovers the $b(n,t)$. It thus follows that
\[
|a(n,t) - a_l(n,t)| + |b(n,t)- b_l(n,t)| \to 0
\]
uniformly in $n$, as $t \to \infty$.

A complete mathematical proof will be given in \cite{katept}.

If solitons are present we can apply appropriate Darboux transformations
to add the effect of such solitons. What we then see asymptotically is
additional travelling solitons on a periodic background (\cite{emt3} or \cite{km2}).

We end this note by pointing out that the limiting lattice can be easily computed
numerically. For the initial data defined in  the introduction the result is shown in
Figure~\ref{fig2}.
\begin{figure}
\includegraphics[width=7cm]{TodaStab2}
\caption{Solution of the Toda lattice (black) plus corresponding limiting lattice (gray).
No gray points visible implies that black points overlap gray points.} \label{fig2}
\end{figure}
The soliton was added using a Darboux transformation.







\begin{acknowledgments}
Spyridon Kamvissis gratefully acknowledges the support of the European Science
Foundation (ESF) and the kind hospitality of the Faculty of Mathematics
during two visits to the University of Vienna from April to July 2005
and in April 2006.
\end{acknowledgments}

\begin{thebibliography}{XXXX}

\bibitem{bght} W. Bulla, F. Gesztesy, H. Holden, and G. Teschl,
%{\em Algebro-geometric quasi-periodic finite-gap solutions of the Toda and Kac-van Moerbeke hierarchies},
Mem. Amer. Math. Soc. {\bf 135}, no. 641, 1--79 (1998).

\bibitem{dz} P. Deift, X. Zhou,
%{\em A Steepest Descent Method for Oscillatory Riemann-Hilbert Problems},
Ann. of Math. {\bf 137}, 295--368 (1993).

\bibitem{dub} B.A. Dubrovin,
%{\em Theta functions and non-linear equations},
Russian Math.\ Surv.\ {\bf 36:2}, 11--92 (1981).

\bibitem{emt} I. Egorova, J. Michor, and G. Teschl,
%{\em Scattering Theory for Jacobi Operators with Quasi-Periodic Background}, 
Comm. Math. Phys. {\bf 264}, no. 3, 811--842 (2006).

\bibitem{emt2} I. Egorova, J. Michor, and G. Teschl,
{\em Inverse scattering transform for the Toda hierarchy with quasi-periodic background}, 
Proc. Amer. Math. Soc. (to appear).

\bibitem{emt3} I. Egorova, J. Michor, and G. Teschl,
{\em Soliton solutions of the Toda hierarchy on quasi-periodic background revisited},
(in preparation).

\bibitem{km} S. Kamvissis,
%{\em On the Long Time Behavior of the Doubly Infinite Toda  Lattice Under Initial Data Decaying at Infinity}, 
Comm. Math. Phys., {\bf 153}, no. 3, 479--519 (1993).

\bibitem{km2} S. Kamvissis,
%{\em On the  Toda Shock Problem},
Physica D, {\bf 65}, no. 3, 242--266 (1993).

\bibitem{katept} S. Kamvissis and G. Teschl,
{\em  Stability of the periodic Toda lattice under short range perturbations},
in preparation.

\bibitem{kumi} E.A. Kuznetsov and A.V. Mikha\u\i lov,
%{\em Stability of stationary waves in nonlinear weakly dispersive media},
Soviet Phys.~JETP {\bf 40}, no. 5, 855--859 (1975).

\bibitem{ro} Yu. Rodin,
{\em The Riemann Boundary Problem on Riemann Surfaces},
Mathematics and its Applications (Soviet Series), {\bf 16},
D. Reidel Publishing Co., Dordrecht, 1988.

\bibitem{tjac} G. Teschl,
{\em Jacobi Operators and Completely Integrable Nonlinear Lattices},
Math. Surv. and Mon. {\bf 72}, Amer. Math. Soc., Rhode Island, 2000. 

\bibitem{ta} M. Toda,
{\em Theory of Nonlinear Lattices}, 2nd enl. ed.,
Springer, Berlin, 1989.

\bibitem{zakr} N. J. Zabusky and M. D. Kruskal,
%{\em Interaction of solitons in a collisionless plasma and the recurrence of initial states},
Phys. Rev. Lett. {\bf 15}, 240--243 (1963).

\bibitem{zh} X. Zhou,
%{\em The Riemann-Hilbert Problem and Inverse Scattering},
SIAM J. Math. Anal.  {\bf 20}, no. 4, 966--986 (1989).



\end{thebibliography}

\end{document}
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