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35 pages, 1 figure
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Riemann-Hilbert problem, Toda lattice
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\begin{document}


\title[Stability of the Periodic Toda Lattice]{Stability of the Periodic Toda Lattice under Short Range Perturbations}

\author[S. Kamvissis]{Spyridon Kamvissis}
\address{Department of Applied Mathematics, University of Crete \\
714 09 Knossos, Greece}
\email{\href{mailto:spyros@tem.uoc.gr}{spyros@tem.uoc.gr}}

\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{\href{mailto:Gerald.Teschl@univie.ac.at}{Gerald.Teschl@univie.ac.at}}
\urladdr{\href{http://www.mat.univie.ac.at/~gerald/}{http://www.mat.univie.ac.at/\~{}gerald/}}

\thanks{Research supported in part by the ESF programme MISGAM, and
the Austrian Science Fund (FWF) under Grant No.\ P17762 and Y330.}


\keywords{Riemann--Hilbert problem, Toda lattice}
\subjclass[2000]{Primary 37K40, 37K45; Secondary 35Q15, 37K10}

\begin{abstract}
We consider the stability of the periodic Toda lattice 
(and slightly more generally of the algebro-geometric finite-gap lattice)
under a short range perturbation. We prove that the perturbed lattice asymptotically
approaches a modulated lattice.

More precisely, let $g$ be the genus of the hyperelliptic curve associated with
the unperturbed solution. We show that, apart from the phenomenon of the
solitons travelling on the quasi-periodic background, the $n/t$-pane 
contains  $g+2$ areas where
the perturbed solution is close to a finite-gap 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 lattice ($g=0$) the
isospectral torus consists of just one point and we recover the known result.

Both the solutions in the isospectral torus and the phase transition are explicitly
characterized in terms of Abelian integrals on the underlying hyperelliptic curve.

Our method relies on the equivalence of the inverse spectral problem to
a matrix Riemann--Hilbert problem defined on the hyperelliptic curve
and generalizes the so-called nonlinear 
stationary phase/steepest descent method for
Riemann--Hilbert problem deformations to Riemann surfaces.
\end{abstract}

\maketitle



\section{Introduction}



A 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. 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 periodic one. 

Of course periodic solutions, as well as
solitons travelling on a periodic background, are well understood and one
might expect the natural generalization in this situation. However,
as pointed out in \cite{kt} this naive expectation is wrong:
\begin{figure}
\includegraphics[width=8cm]{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 Figure~\ref{fig1} the two observed lines express the variables $a(n,t)$ at a frozen time $t$.
In areas where the lines seem to be continuous this is due to the fact that we have plotted
a huge number of particles 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 rigorous and complete mathematical explanation of
this picture. We provide the details in the case of the Toda lattice though it is
clear that our methods apply to other soliton equations as well.

Consider the doubly infinite Toda lattice in Flaschka's
variables (see e.g.\ \cite{tjac}, \cite{taet}, or \cite{ta})
\be \label{TLpert}
\aligned
\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)),
\endaligned
\ee
$(n,t) \in \Z \times \R$,
where the dot denotes differentiation with respect to time. We
will consider a quasi-periodic algebro-geometric background
solution $(a_q,b_q)$, to be described in the next section, plus
a short range perturbation $(a,b)$ satisfying
\be \label{decay}
\sum_n |n| (|a(n,t) - a_q(n,t)| + |b(n,t)- b_q(n,t)|) < \infty
\ee
for one (and hence for all, see \cite{emt2}) $t\in\R$. 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}), the more
general case we want here was solved only recently in \cite{emt2}
(see also \cite{mt}). We will also assume that the reflection coefficient
is continuous. This is for example the case if we replace $|n|$ by
$|n|^2$ in (\ref{decay}).

Assume for simplicity that the Jacobi operator $H_q$ 
corresponding to the perturbed problem (\ref{TLpert}) has no eigenvalues.
In this paper we prove that for long times the perturbed Toda lattice
is  asymptotically close to the following limiting  lattice 
defined by
\be \label{limlat}
\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)} 
\log (1-|R|^2) \omega_{\infty_+ \infty_-}\right),\\
\delta_\ell(n,t)= & \frac{1}{2\pi\I} \int_{C(n/t)} \log (1-|R|^2) \zeta_\ell,
\endaligned
\ee
where $R$ is the associated reflection coefficient,
$\omega_{\infty_+ \infty_-}$ is an Abelian differential of the third kind defined in
(\ref{ominfpm}), $C(n/t) = \pi^{-1}(\sig(H_q)\cap (-\infty,z_j(n/t)))$, and
$z_j(n/t)$ is a special stationary phase point for the phase defined in
the beginning of Section~\ref{secSPP}.
From the products above, one easily recovers $a_l(n,t)$.
More precisely, we have the following.

\begin{theorem}\label{thmMain}
Let $C$ be any  (large) positive number and $\delta$ be any (small)
positive number. Let $E_s \in S$ be the 'resonance points' defined by
$S= \{ E_s : |R(E_s)| =1\}.$ (There are at most $2g+2$ such points, since they are always
endpoints $E_j$ of the bands that constitute the spectrum of the Jacobi operator.) 
Consider the  region 
$D= \{(n,t): |\frac{n}{t}| < C \} \cap \{(n,t): |z_j(\frac{n}{t})-E_s|>\delta$\}, 
where  $z_j(\frac{n}{t})$ is the special stationary phase point for the phase defined in
the beginning of Section~\ref{secSPP}. Then one has
\be
\prod_{j=n}^{\infty} \frac{a_l(j,t)}{a(j,t)} \to 1
\ee
uniformly in $D$, as $t \to \infty$.
\end{theorem}

The proof of this theorem will be given in Section~\ref{secSIE} of this paper.

\begin{remark}
(i) If eigenvalues are present we can apply appropriate Darboux transformations
to add the effect of such eigenvalues (\cite{emt4}). What we then see asymptotically is
travelling solitons in a periodic background. Note that this will change the
asymptotics on one side. However, our method works for such situations (cf.\ \cite{emt3})
unaltered.\\
(ii) Employing the very same methods of the paper it is 
very easy to show that in any region
$|\frac{n}{t}|> C$, one has
\be
\prod_{j=n}^{\infty} \frac{a_l(j,t)}{a(j,t)} \to 1
\ee
uniformly in $t$, as $n \to \infty$.\\
(iii) The effect of the resonances $E_s$ is only felt locally (and to higher 
order in $1/t$) in some small
(decaying as $t \to \infty$) region, where in fact $ |z_j(\frac{n}{t})-E_s| \to 0$
as $t \to \infty$. We expect a 'collisionless shock' phenomenon
to appear (\cite{dvz}, \cite{km2}). We will study this in a future paper.
\end{remark}

By dividing in (\ref{limlat}) one recovers the $a(n,t)$.
It follows from the main Theorem and the last remark above that
\be
|a(n,t) - a_l(n,t)| \to 0
\ee
uniformly in $D$, as $t \to \infty$.
In other words, 
the perturbed  Toda lattice
is  asymptotically close to the  limiting  lattice above.

A similar theorem can be proved for the  velocities $b(n,t)$.

\begin{theorem}\label{thmMain2}
In the  region
$D= \{(n,t): |\frac{n}{t}| < C \} \cap \{(n,t): |z_j(\frac{n}{t})-E_s|>\delta$\},
of Theorem ~\ref{thmMain} we also have
\be
\sum_{j=n}^\infty \big( b_l(j,t) - b_q(j,t) \big) \to 0
\ee
uniformly in $D$, as $t \to \infty$, where $b_l$ is given by
\be
\aligned
\sum_{j=n}^\infty \big( b_l(j,t) - b_q(j,t) \big) = &
\frac{1}{2\pi\I} \int_{C(n/t)} \log(1-|R|^2) \Omega_0\\
& {} + \frac{1}{2}\frac{d}{ds}
\log\left( \frac{\theta(\ulz(n,s) + \ul{\delta}(n,t) )}{\theta(\ulz(n,s))} \right) \Big|_{s=t},
\endaligned
\ee
where $\Omega_0$ is an Abelian differential of the second kind defined in (\ref{Om0}).
\end{theorem}

The proof of this theorem will also be given in Section~\ref{secSIE} of this paper.


\section{Algebro-geometric quasi-periodic finite-gap solutions}



As a preparation we need some facts on our background
solution $(a_q,b_q)$ which we want to choose from the class of
algebro-geometric quasi-periodic finite-gap solutions, that is the
class of stationary solutions of the Toda hierarchy, \cite{bght}.
In particular, this class contains all periodic solutions. We will
use the same notation as in \cite{tjac}, where we also refer to for proofs.
As a reference for Riemann surfaces in this context we recommend \cite{fk}.

To set the stage let $\M$ be the Riemann surface associated with the following function
\begin{equation}
\Rg{z}, \qquad R_{2g+2}(z) = \prod_{j=0}^{2g+1} (z-E_j), \qquad
E_0 < E_1 < \cdots < E_{2g+1},
\end{equation}
$g\in \N$. $\M$ is a compact, hyperelliptic Riemann surface of genus $g$.
We will choose $\Rg{z}$ as the fixed branch
\begin{equation}
\Rg{z} = -\prod_{j=0}^{2g+1} \sqrt{z-E_j},
\end{equation}
where $\sqrt{.}$ is the standard root with branch cut along $(-\infty,0)$.

A point on $\M$ is denoted by 
$p = (z, \pm \Rg{z}) = (z, \pm)$, $z \in \C$, or $p = (\infty,\pm) = \infty_\pm$, and
the projection onto $\C \cup \{\infty\}$ by $\pi(p) = z$. 
The points $\{(E_{j}, 0), 0 \leq j \leq 2 g+1\} \subseteq \M$ are 
called branch points and the sets 
\begin{equation}
\Pi_{\pm} = \{ (z, \pm \Rg{z}) \mid z \in \C\setminus
\bigcup_{j=0}^g[E_{2j}, E_{2j+1}]\} \subset \M
\end{equation}
are called upper, lower sheet, respectively.

Let $\{a_j, b_j\}_{j=1}^g$ be loops on the surface $\M$ representing the
canonical generators of the fundamental group $\pi_1(\M)$. We require
$a_j$ to surround the points $E_{2j-1}$, $E_{2j}$ (thereby changing sheets
twice) and $b_j$ to surround $E_0$, $E_{2j-1}$ counter-clockwise on the
upper sheet, with pairwise intersection indices given by
\begin{equation}
a_i \circ a_j= b_i \circ b_j = 0, \qquad a_i \circ b_j = \delta_{i,j},
\qquad 1 \leq i, j \leq g.
\end{equation}
The corresponding canonical basis $\{\zeta_j\}_{j=1}^g$ for the space of
holomorphic differentials can be constructed by
\begin{equation}
\underline{\zeta} = \sum_{j=1}^g \underline{c}(j)  
\frac{\pi^{j-1}d\pi}{R_{2g+2}^{1/2}},
\end{equation}
where the constants $\underline{c}(.)$ are given by
\[
c_j(k) = C_{jk}^{-1}, \qquad 
C_{jk} = \int_{a_k} \frac{\pi^{j-1}d\pi}{R_{2g+2}^{1/2}} =
2 \int_{E_{2k-1}}^{E_{2k}} \frac{z^{j-1}dz}{\Rg{z}} \in
\R.
\]
The differentials fulfill
\begin{equation} \label{deftau}
\int_{a_j} \zeta_k = \delta_{j,k}, \qquad \int_{b_j} \zeta_k = \tau_{j,k}, 
\qquad \tau_{j,k} = \tau_{k, j}, \qquad 1 \leq j, k \leq g.
\end{equation}

Now pick $g$ numbers (the Dirichlet eigenvalues)
\be
(\hat{\mu}_j)_{j=1}^g = (\mu_j, \sigma_j)_{j=1}^g
\ee
whose projections lie in the spectral gaps, that is, $\mu_j\in[E_{2j-1},E_{2j}]$.
Associated with these numbers is the divisor $\dimuz$ which
is one at the points $\hat{\mu}_j$  and zero else. Using this divisor we
introduce
\begin{align} \nn
\ulz(p,n,t) &= \hAmap(p) - \hamap(\dimuz) - n\ul{\hat A}_{\infty_-}(\infty_+)
+ t\ul{U}_0 - \hvrc \in \C^g, \\
\ulz(n,t) &= \ulz(\infty_+,n,t),
\end{align}
where $\vrc$ is the vector of Riemann constants
\begin{equation}
\hat{\Xi}_{p_0,j} = \frac{1- \sum_{k=1}^g \tau_{j,k}}{2},
\qquad p_0=(E_0,0),
\end{equation}
$\ul{U}_0$ are the $b$-periods of the Abelian differential $\Omega_0$ defined below,
and $\Amap$ ($\amap$) is Abel's map (for divisors). The hat indicates that we
regard it as a (single-valued) map from $\hat{\M}$ (the fundamental polygon
associated with $\M$ by cutting along the $a$ and $b$ cycles) to $\C^g$.
We recall that the function $\theta(\ulz(p,n))$ has precisely $g$ zeros
$\hmu_j(n)$ (with $\hmu_j(0)=\hmu_j$), where $\theta(\ul{z})$ is the
Riemann theta function of $\M$.

Then our background solution is given by
\begin{align} \nn
a_q(n,t)^2 &= \ti{a}^2 \frac{\theta(\ulz(n+1,t)) \theta(\ulz(n-1,t))}{\theta(
\ulz(n,t))^2},\\ \label{imfab}
b_q(n,t) &= \tilde{b} + \frac{1}{2}
\frac{d}{dt} \log\Big(\frac{\theta(\ulz(n,t)) }{\theta(\ulz(n-1,t))}\Big).
\end{align}
The constants $\ti{a}$, $\tilde{b}$ depend only on the Riemann surface
(see \cite[Section~9.2]{tjac}).

Introduce the time dependent Baker-Akhiezer function
\begin{align}
\psi_q(p,n,t) &= C(n,0,t) \frac{\theta (\ulz(p,n,t))}{\theta(\ulz (p,0,0))}
\exp \Big( n \int_{E_0}^p \om_{\infty_+,\infty_-} + t\int_{E_0}^p \Omega_0
\Big),
\end{align}
where $C(n,0,t)$ is real-valued,
\begin{equation}
C(n,0,t)^2 = \frac{ \theta(\ulz(0,0)) \theta(\ulz(-1,0))}
{\theta (\ulz (n,t))\theta (\ulz (n-1,t))},
\end{equation}
and the sign has to be chosen in accordance with $a_q(n,t)$.
Here
\be
\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) ,\qquad \ulz \in \C^g,
\ee
is the Riemann theta function associated with $\M$,
\be\label{ominfpm}
\omega_{\infty_+ \infty_-}= \frac{\prod_{j=1}^g (\pi -\lambda_j) }{R_{2g+2}^{1/2}}d\pi
\ee
is the Abelian differential of the third kind with poles at $\infty_+$ and $\infty_-$ and
\be\label{Om0}
\Omega_0 = \frac{\prod_{j=0}^g (\pi - \ti\lambda_j) }{R_{2g+2}^{1/2}}d\pi,
\qquad \sum_{j=0}^g \ti\lambda_j = \frac{1}{2} \sum_{j=0}^{2g+1} E_j,
\ee
is the Abelian differential of the second kind with second order poles at
$\infty_+$ respectively $\infty_-$ (see \cite[Sects.~13.1, 13.2]{tjac}).
All Abelian differentials are normalized to have vanishing $a_j$ periods.


The Baker-Akhiezer function is a meromorphic function on $\M\setminus\{\infty_\pm\}$
with an essential singularity at $\infty_\pm$. The two branches are denoted by
\begin{equation}
\psi_{q,\pm}(z,n,t) = \psi_q(p,n,t), \qquad p=(z,\pm)
\end{equation}
and it satisfies
\bea  \nonumber
H_q(t) \psi_q(p,n,t) &=& \pi(p) \psi_q(p,n,t), \\ \nonumber
\frac{d}{dt} \psi_q(p,n,t) &=& P_{q,2}(t) \psi_q(p,n,t),
\eea
where $H_q$, $P_{q,2}$ are the operators from the Lax pair for the
Toda lattice.

It is well known that the spectrum of $H_q(t)$ is time independent and
consists of $g+1$ bands
\begin{equation}
\sig(H_q) = \bigcup_{j=0}^g [E_{2j},E_{2j+1}].
\end{equation}
For further information and proofs we refer to \cite[Chap.~9 and Sect.~13.2]{tjac}.



\section{The Inverse scattering transform and the Riemann--Hilbert problem}
\label{secISTRH}



In this section our notation and results are  taken from \cite{emt} and \cite{emt2}.
Let $\psi_{q,\pm}(z,n,t)$ be the branches of the  Baker-Akhiezer function defined
in the previous section. Let $\psi_\pm(z,n,t)$ be the Jost functions for the perturbed
problem defined by
\be
\lim_{n \to \pm \infty} 
w(z)^{\mp  n} ( \psi_{\pm}(z,n,t) - \psi_{q, \pm}(z,n,t))
=0,
\ee
where $w(z)$ is the quasimomentum map
\be
w(z)= \exp (\int^p_{E_0} \omega_{\infty_+, \infty_-}), \quad p=(z,+).
\ee
The asymptotics of the  two projections of the Jost function are
\begin{align} \nn
\psi_\pm(z,n,t) =&  \frac{z^{\mp n} \Big(\prod_{j=0}^{n-1} a_q(j,t)\Big)^{\pm 1}}{A_\pm(n,t)}  \times\\
& \times
\Big(1 + \Big(B_\pm(n,t) \pm \sum_{j=1}^{n} b_q(j- {\scriptstyle{0 \atop 1}},t) \Big)\frac{1}{z}
+ O(\frac{1}{z^2}) \Big),
\end{align}
as $z \to \infty$, where
\be \label{defABpm}
\aligned
A_+(n,t) &= \prod_{j=n}^{\infty} \frac{a(j,t)}{a_q(j,t)}, \quad
B_+(n,t)= \sum_{j=n+1}^\infty (b_q(j,t)-b(j,t)), \\
A_-(n,t) &= \!\!\prod_{j=- \infty}^{n-1}\! \frac{a(j,t)}{a_q(j,t)}, \quad
B_-(n,t) = \sum_{j=-\infty}^{n-1} (b_q(j,t) - b(j,t)).
\endaligned
\ee

One has the scattering relations
\be \label{relscat}
T(z) \psi_\mp(z,n,t) =  \ol{\psi_\pm(z,n,t)} +
R_\pm(z) \psi_\pm(z,n,t),  \qquad z \in\sigma(H_q).
\ee
Here $\psi_\pm(z,n,t)$ is defined such that 
$\psi_\pm(z,n,t)= \lim_{\eps\downarrow 0}\psi_\pm(z + \I\eps,n,t)$,
$z\in\sigma(H_q)$. If we take the limit from the other side we
have $\ol{\psi_\pm(z,n,t)}= \lim_{\eps\downarrow 0}\psi_\pm(z - \I\eps,n,t)$.

The transmission $T(z)$ and reflection $R_\pm(z)$ coefficients satisfy
\be \label{reltrpm} 
T(z) \ol{R_+(z)} + \ol{T(z)} R_-(z)=0, \qquad |T(z)|^2 + |R_\pm(z)|^2=1.
\ee
In particular one reflection coefficient, say $R(z)=R_+(z)$, suffices.

We will define a Riemann--Hilbert problem on the Riemann
surface $\M$ as follows:
\be
m(p,n,t)= \left\{\begin{array}{c@{\quad}l}
\begin{pmatrix} T(z) \psi_-(z,n,t)  & \psi_+(z,n,t) \end{pmatrix},
& p=(z,+)\\
\begin{pmatrix} \psi_+(z,n,t) & T(z) \psi_-(z,n,t) \end{pmatrix}, 
& p=(z,-)
\end{array}\right..
\ee
Note that $m(p,n,t)$ inherits the poles at $\hat{\mu}_j(n,t)$ and the
essential singularity at $\infty_\pm$ from the Baker--Akhiezer function.

We are interested in the jump condition of $m(p,n,t)$ on $\Sigma$,
the boundary of $\Pi_\pm$ (oriented counterclockwise when viewed from top sheet $\Pi_+$).
It consists of two copies $\Sigma_\pm$ of $\sigma(H_q)$ which correspond to
non-tangential limits from $p=(z,+)$ with $\pm\im(z)>0$, respectively to non-tangential
limits from $p=(z,-)$ with $\mp\im(z)>0$.

To formulate our jump condition we use the following convention:
When representing functions on $\Sigma$, the lower subscript denotes
the non-tangential limit from $\Pi_+$ or $\Pi_-$, respectively,
\be
m_\pm(p_0) = \lim_{ \Pi_\pm \ni p\to p_0} m(p), \qquad p_0\in\Sigma.
\ee
Using the notation above implicitly assumes that these limits exist in the sense that
$m(p)$ extends to a continuous function on the boundary away from
the band edges.

Moreover, we will also use symmetries with respect to the
the sheet exchange map
\be
p^*= \begin{cases}
(z,\mp) & \text{ for } p=(z,\pm),\\
\infty_\mp & \text{ for } p=\infty_\pm,
\end{cases}
\ee
and complex conjugation
\be
\ol{p} = \begin{cases}
(\ol{z},\pm) & \text{ for } p=(z,\pm)\not\in \Sigma,\\
(z,\mp) & \text{ for } p=(z,\pm)\in \Sigma,\\
\infty_\pm & \text{ for } p=\infty_\pm.
\end{cases}
\ee
In particular, we have $\ol{p}=p^*$ for $p\in\Sigma$.

Note that we have $\ti{m}_\pm(p)=m_\mp(p^*)$ for $\ti{m}(p)= m(p^*)$
(since $*$ reverses the orientation of $\Sigma$) and $\ti{m}_\pm(p)= \ol{m_\pm(p^*)}$ for
$\ti{m}(p)=\ol{m(\ol{p})}$.


With this notation, using (\ref{relscat}) and (\ref{reltrpm}), we obtain
\be
m_+(p,n,t) = m_-(p,n,t)
\begin{pmatrix} |T(p)|^2 & -\ol{R(p)} \\ R(p) & 1 \end{pmatrix},
\ee
where we have extend our definition of $T$ to $\Sigma$ such that
it is equal to $T(z)$ on $\Sigma_+$ and equal to $\ol{T(z)}$ on $\Sigma_-$.
Similarly for $R(z)$. In particular, the condition on $\Sigma_+$ is just the
complex conjugate of the one on $\Sigma_-$ since we have $R(p^*)= \ol{R(p)}$
and $m_\pm(p^*,n,t)= \ol{m_\pm(p,n,t)}$ for $p\in\Sigma$.

To remove the essential singularity at $\infty_\pm$ and to get a meromorphic
Riemann--Hilbert problem we set
\be \label{defm2}
m^2(p,n,t)=  m(p,n,t)
\begin{pmatrix} \psi_q(p^*,n,t)^{-1} & 0 \\ 0 & \psi_q(p,n,t)^{-1} \end{pmatrix}.
\ee
Its divisor satisfies
\be
(m^2_1) \ge -\dimus{n,t}, \qquad (m^2_2) \ge -\dimu{n,t},
\ee
and the jump conditions become
\begin{align} \nn
m^2_+(p,n,t) &= m^2_-(p,n,t) J^2(p,n,t)\\ \label{rhpm2.1}
J^2(p,n,t) &=
\begin{pmatrix}
1 -|R(p)|^2  &- \ol{R(p) \Theta(p,n,t)}  \E^{-t \phi(p)}\\
R(p)  \Theta(p,n,t) \E^{t \phi(p)} & 1
\end{pmatrix},
\end{align}
where
$$
\Theta(p,n,t) = \frac{\theta(\ulz(p,n,t))}{\theta(\ulz(p,0,0))}
\frac{\theta(\ulz(p^*,0,0))}{\theta(\ulz(p^*,n,t))}
$$
and
\be\label{defsp}
\phi(p,\frac{n}{t}) =
2 \int_{E_0}^p \Omega_0 + 2 \frac{n}{t} \int_{E_0}^p \omega_{\infty_+,\infty_-}
\in \I \R
\ee
for $p \in\Sigma$. Furthermore,
\be \label{m2infp}
m^2(\infty_+,n,t) = \begin{pmatrix} A_+(n,t)  & \frac{1}{A_+(n,t)} \end{pmatrix}.
\ee
and
\be  \label{m2infm}
m^2(\infty_-,n,t) = \begin{pmatrix} \frac{1}{A_+(n,t)} & A_+(n,t)  \end{pmatrix}.
\ee
Here we have used
\be
T(\infty) = A_-(n,t) A_+(n,t) \big( 1 - \frac{B_+(n,t) + B_-(n,t)}{z} + O(\frac{1}{z^2}\big)
\ee
where $A_\pm(n,t)$ and $B_\pm(n,t)$ are defined in (\ref{defABpm}).

Next we show how to normalize the problem at infinity.

\begin{theorem} \label{thmm3}
Let $\ell $ be a fixed positive integer $\leq 2g+1$.
Consider the following Riemann--Hilbert problem. 
Find a $2\times 2$ matrix $m^3(p)$ that is meromorphic
off $\Sigma$, 
\be
(m^3_{j1}) \ge -\dimus{n,t}, \quad (m^3_{j2}) \ge -\dimu{n,t}, \qquad j=1,2;
\ee
with the following behavior at the band edges $E_k$, $1\le k \le \ell$:
\be\label{m3OTp}
(m^3_{j1})(p) =O (T(z)),\quad  (m^3_{j2})(p) = O (T(z)^{-1}), \qquad j=1,2,
\ee
for $p=(z,+)$ near $E_k$ on the top sheet and
\be\label{m3OTm}
(m^3_{j1})(p) =O (T(z)^{-1}),\quad  (m^3_{j2})(p) = O (T(z)), \qquad j=1,2,
\ee
for $p=(z,-)$ near $E_k$ on the bottom sheet;
that  is bounded near $\Sigma \setminus (\cup_{j>l}E_j 
\cup \cup_j 
[\hat \mu_j(n,t) \cup \hat \mu_j(n,t)^* ])$ and that  has
same jumps as  $m^2(p)$, that is
\begin{align}\nn
m^3_+(p) &= m^3_-(p) J^3(p),\\ \label{rhpm3.1}
J^3(p) &=
\begin{pmatrix}
1 -|R(p)|^2  &- \ol{R(p) \Theta(p,n,t)}  \E^{-t \phi(p)}\\
R(p)  \Theta(p,n,t) \E^{t \phi(p)} & 1
\end{pmatrix},
\end{align}
$z\in\Sigma \setminus \cup_j \{ E_j\}$, but such that 
\be  \label{rhpm3.2}
m^3 (\infty_+) = \id.
\ee
A unique such $m^3(p)$ exists
and furthermore
the Jacobi operator can be recovered from $m^3(\infty_-)$ via the formulae
\be \label{Apm3}
A_+(n,t)  = \sqrt{{{1 + (m^3_{12}(\infty_-,n,t) )} \over {(m^3_{11}(\infty_-,n,t) ) }}}.
\ee
\end{theorem}

The integer $\ell$ will be chosen later according to the location of the stationary
phase points.

\begin{remark}\label{remuniq}
It is not a priori obvious that a solution of  the above Riemann--Hilbert problem exists.
However, existence and uniqueness follow
from a general theorem in Section~\ref{secETFP}. On the other hand existence 
will  also be justified a posteriori, since existence of the solution $Q$ of problem
(\ref{rhpm6}) guaranteed by Theorem~\ref{thmexm6}  will imply  existence
of the solution of the problem centered on the small crosses of Section~\ref{secSPP},
which will then imply existence of the solution $m^5$ of the  
Riemann--Hilbert problem (\ref{rhpm5}), which is equivalent to existence of
the problem above, using definitions
(\ref{defm5}) and (\ref{defd}). 

Uniqueness is actually not necessary for our
purposes. All we need is uniqueness at $\infty_-$ and that follows easily,
without the use of the vanishing theorem of Section~\ref{secETFP}.
Indeed,  note that the jump matrix $J^3$ has determinant one and hence $\det(m^3)$
has no jump. Thus it extends to a meromorphic function by the Schwarz reflection
principle. The singularities at the band edges are removable by (\ref{m3OTp})--(\ref{m3OTm})
(unless one of the  $\hat{\mu}_j$ is sitting there). So its divisor satisfies
$(\det(m^3)) \ge -\dimuz - \dimuzs$ and the Riemann--Roch
theorem implies
\be
\det(m^3(p)) = \prod_{j=1}^g\frac{z - \nu_j}{z - \mu_j}, \qquad z=\pi(p),
\ee
for some complex numbers $\nu_j$. Hence, if $\ti{m}^3$ is a second solution, then
$\ti{m}^3 (m^3)^{-1}$ has no jump and by a similar argument
\be
\ti{m}^3(p) = \frac{h(z)}{\prod_{j=1}^g (z-\nu_j)} m^3(p),
\ee
where $h(z)$ is a polynomial matrix with $h_{11}, h_{22}$ monic polynomials of degree $g$
and $h_{12}, h_{21}$ polynomials of degree $g-1$.
Hence $m^3$ is uniquely defined at $\infty_-$.

We also note that the  behavior at the band edges $E_j$ is imposed by the zeros and poles 
introduced (later in the text) by  the sequence of transformations from $m^6$ to $m^3$.
\end{remark}


\begin{proof}
As noted above, existence and uniqueness follows 
from a general theorem in Section~\ref{secETFP}. Next,
note the symmetry
\be \label{symJ3}
J^3(p) = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}
J^3(p^*)^{-1} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}
\ee
of the jump matrix. It follows that
\be \label{m3sym}
m^3(p) = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}
(m^3(\infty_-))^{-1}
m^3(p^*) \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.
\ee
Indeed, it is simple to check that both sides satisfy the same 
Riemann--Hilbert problem (\ref{rhpm3.1})--(\ref{rhpm3.2}). Hence, by uniqueness, they are equal.

Now suppose 
$$
m^3(\infty_-) = \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix}.
$$
Noting that $\det (m^3(\infty_-))=1$ and setting $p=\infty_-$ in (\ref{m3sym}) we get
$\beta =-\gamma$. So
\be \label{m3infm}
m^3(\infty_-) = \begin{pmatrix} \alpha &\beta \\
-\beta & \frac{1 - \beta^2}{ \alpha } \end{pmatrix}.
\ee
Next note that both the matrix $m^3(p)$ and the row vector 
$m^2(p)$ satisfy the same  jump conditions (\ref{rhpm3.1}).
It follows that the row vector $m^2(p) (m^3(p))^{-1} $ has no jumps
and is bounded near the band edges. Arguing as in the remark above we see
\be \label{m2km3}
m^2(p) = \frac{k(z)}{\prod_{j=1}^g (z-\nu_j)} m^3(p),
\ee
where, by (\ref{m2infp}), $k(z)$ is of the form
\be
k(z) = \begin{pmatrix}
A_+ \prod_{j=1}^g (z-\nu_{1j}) & \frac{1}{A_+} \prod_{j=1}^g (z-\nu_{2j})
\end{pmatrix}.
\ee
Setting  $p = \infty_-$ in (\ref{m2km3}) we get, in view of (\ref{m2infm}) and (\ref{m3infm}),
$$
\begin{pmatrix} \frac{1}{A_+} & A_+ \end{pmatrix} =
\begin{pmatrix} A_+  & \frac{1}{A_+} \end{pmatrix} 
\begin{pmatrix}
\alpha &\beta \\
-\beta & {{1 - \beta^2} \over { \alpha }}
\end{pmatrix}.
$$
We thus arrive at
$$
A_+ = \sqrt{\frac{1 + \beta}{\alpha}}
$$
and the theorem follows by recalling (\ref{m3infm}).
\end{proof}

Note that setting $R(z)\equiv 0$ we recover the purely periodic solution, as we should.
Moreover, by $J(p)=\ol{J(\ol{p})}$, $p\in\Sigma$, we also have $m^3(p) = \ol{m^3(\ol{p})}$
(since both sides satisfy the same Riemann--Hilbert problem).

\begin{remark}
It is important to note here that the Riemann--Hilbert problem 
arising from the analysis of the perturbed periodic problem (for any 
discrete in space integrable system)
has to be set on a Riemann surface. There is no way one can use a symmetry 
like (\ref{symJ3}) to normalize the Riemann--Hilbert problem
at infinity (i.e. get rid of the inappropriate condition (\ref{m2infp}) at $\infty_+$)
if one stays in the complex plane.
\end{remark}

\begin{remark}
In the proof of Theorem~\ref{thmMain2} we will show that in (\ref{m2km3}) we
actually have
\be
m^2(p) = \begin{pmatrix} A_+  & \frac{1}{A_+} \end{pmatrix} m^3(p).
\ee
\end{remark}




\section{The stationary phase points}
\label{secSPP}


The phase in the factorization problem (\ref{rhpm2.1}) is $t\, \phi$ where
$\phi$ was defined in (\ref{defsp}). Invoking (\ref{ominfpm}) and
(\ref{Om0}), we see that the stationary phase points are given by 
\be
\prod_{j=0}^g (z -\tilde\lambda_j) + \frac{n}{t} \prod_{j=1}^g (z -\lambda_j) =0.
\ee
Due to the normalization of our Abelian differentials, the numbers
$\lam_j$, $0\le j \le g$, are real and different with precisely one lying in each
spectral gap, say $\lam_j$ in the $j$'th gap.
Similarly, $\ti\lam_j$, $0\le j \le g$, are real and different and
$\ti\lam_j$, $1\le j \le g$, sits in the $j$'th gap. However $\ti\lam_0$ can be
anywhere (see \cite[Sect.~13.5]{tjac}).

As a first step let us clarify the dependence of the stationary phase points
on $\frac{n}{t}$.

\begin{lemma}
Denote by $z_j(\eta)$, $0\le j \le g$, the stationary phase points, where
$\eta=\frac{n}{t}$. Set $\lam_0=-\infty$ and $\lam_{g+1}= \infty$, then
\be
\lam_j < z_j(\eta) < \lam_{j+1}
\ee
and there is always at least one stationary phase point in the $j$'th spectral gap.
Moreover, $z_j(\eta)$ is monotone decreasing with
\be
\lim_{\eta\to-\infty} z_j(\eta) = \lam_{j+1} \quad\text{and}\quad
\lim_{\eta\to\infty} z_j(\eta) = \lam_j.
\ee
\end{lemma}

\begin{proof}
Due to the normalization of the Abelian differential $\Omega_0 + \eta \omega_{\infty_+,\infty_-}$ there is at least one stationary phase point in each gap and they are
all different. Furthermore,
$$
z_j' = - \frac{q(z_j)}{\ti{q}'(z_j)+ \eta q'(z_j)},
$$ 
where
$$
\ti{q}(z) =\prod_{j=0}^g (z -\tilde\lambda_j),
q(z)= \prod_{j=1}^g (z -\lambda_j),
$$
shows that $z_j(\eta)$ is monotone decreasing (note that the denominator cannot vanish
since the $z_j$'s are always different) and must stay between $\lam_j$ and
$\lam_{j+1}$.
\end{proof}

In summary, the lemma tells us that we have the following picture:
As $\frac{n}{t}$ 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, $z_{g-1}(\eta)$ 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
$-\infty$.

So, depending on $n/t$ there is at most 
one single stationary phase point belonging to the union of the bands
$\sigma(H_q)$, say $z_j(n/t)$. 
On the Riemann surface, there are two such points $z_j$ and its flipping
image $z^*_j$ which may (depending on $n/t$) lie in $\Sigma$.

There are three possible cases.

\begin{enumerate}
\item
One stationary phase point, say  $z_j$,  belongs to the interior of a band
$[E_{2j}, E_{2j+1}]$  and all other stationary phase points
lie in open gaps.
\item
$z_j =z_j^* = E_j$ for some $j$ and all other stationary phase points
lie in open gaps.
\item
No stationary phase point belongs to $\sigma(H_q)$.
\end{enumerate}

\bigskip

Case (i).
Let us fix $\ell=2j$ in Theorem~\ref{thmm3} and introduce the following "lens" contour
near the band  $[E_{2j}, E_{2j+1}]$ as shown in Figure~\ref{fig2}.
\begin{figure}
\begin{picture}(10,6)

\put(1.2,4.8){$D_{j1}$}
\put(0,5.1){$C_{j1}$}
\put(8.2,4.8){$D_{j2}$}
\put(9.4,5.1){$C_{j2}$}
\put(5,4.5){\circle*{0.06}}
\put(4.9,4.8){$z_j$}
\put(2,4.5){\circle*{0.06}}
\put(2,4.1){$E_{2j}$}
\put(8,4.5){\circle*{0.06}}
\put(7.5,4.1){$E_{2j+1}$}

\put(3,4.6){$+$}
\put(3,4.2){$+$}
\put(6.5,4.6){$+$}
\put(6.5,4.2){$+$}

\put(1.7,5.6){$+$}
\put(1.7,5.2){$-$}
\put(8.1,3.6){$+$}
\put(8.1,3.2){$-$}

\put(2,4.5){\line(1,0){6}}
\put(5,4.5){\curve(0, 0, -2.645, -0.951, -4.28, -0.588, -4.28, 0.588, -2.645, 0.951, 0, 0)}
\curvedashes{0.05,0.05}
\put(5,4.5){\curve(0, 0, 2.645, -0.951, 4.28, -0.588, 4.28, 0.588, 2.645, 0.951, 0, 0)}
\curvedashes{}

%%%%%

\put(1.2,1.8){$D_{j1}^*$}
\put(0,2.1){$C_{j1}^*$}
\put(8.2,1.8){$D_{j2}^*$}
\put(9.4,2.1){$C_{j2}^*$}
\put(5,1.5){\circle*{0.06}}
\put(4.9,1.8){$z_j^*$}
\put(2,1.5){\circle*{0.06}}
\put(2,1.1){$E_{2j}$}
\put(8,1.5){\circle*{0.06}}
\put(7.5,1.1){$E_{2j+1}$}

\put(3,1.6){$-$}
\put(3,1.2){$-$}
\put(6.5,1.6){$-$}
\put(6.5,1.2){$-$}

\put(1.7,2.6){$-$}
\put(1.7,2.2){$+$}
\put(8.1,0.6){$-$}
\put(8.1,0.2){$+$}

\put(5,1.5){\curve(0, 0, 2.645, -0.951, 4.28, -0.588, 4.28, 0.588, 2.645, 0.951, 0, 0)}
\curvedashes{0.05,0.05}
\put(2,1.5){\curve(0,0,6,0)}
\put(5,1.5){\curve(0, 0, -2.645, -0.951, -4.28, -0.588, -4.28, 0.588, -2.645, 0.951, 0, 0)}

\end{picture}
\caption{The lens contour near a band containing a stationary phase point
$z_j$ and its flipping image containing $z_j^*$. Views from the top 
and bottom sheet. Dotted curves lie in the bottom sheet.} \label{fig2}
\end{figure}
The oriented paths $C_j = C_{j1} \cup C_{j2}$,  $C_j^* = C_{j1}^* \cup C_{j2}^*$
are meant to be close to the band $[E_{2j}, E_{2j+1}]$.

We have
$$
\re(\phi) >0, \quad \text{in } D_{j1},\qquad
\re(\phi) <0, \quad \text{in } D_{j2}.
$$
Indeed
\be
\im(\phi') <0, \quad \text{in } [E_{2j}, z_j], \qquad
\im(\phi') >0, \quad \text{in } [z_{j}, E_{2j+1}]
\ee
noting that $\phi$ is imaginary in $[E_{2j}, E_{2j+1}]$ and writing 
$\phi' = d\phi/dz$. Using the Cauchy-Riemann equations we 
find that the above inequalities are true, as long as 
$C_{j1}, C_{j2}$ are  close enough
to the band $[E_{2j}, E_{2j+1}]$. A similar picture appears in the lower
sheet.

Concerning the  other bands, one simply constructs a "lens" contour
near each  of the other bands $[E_{2k}, E_{2k+1}]$  and $[E_{2k}^*, E_{2k+1}^*]$ 
as shown in Figure~\ref{fig3}.
\begin{figure}
\begin{picture}(10,6)

\put(3.4,4.8){$D_k$}
\put(0.6,5.15){$C_k$}
\put(2,4.5){\circle*{0.06}}
\put(2,4.1){$E_{2k}$}
\put(8,4.5){\circle*{0.06}}
\put(7.5,4.1){$E_{2k+1}$}

\put(4.8,4.6){$+$}
\put(4.8,4.2){$+$}

\put(5.5,5.6){$+$}
\put(5.5,5.2){$-$}

\put(2,4.5){\line(1,0){6}}
\put(5,4.5){\curve(0, 1, 2.645, 0.809, 4.28, 0.309, 4.28, -0.309, 2.645, -0.809, 0, -1, %
-2.645, -0.809, -4.28, -0.309, -4.28, 0.309, -2.645, 0.809, 0, 1)}

%%%%%

\put(3.4,1.8){$D_k^*$}
\put(0.6,2.2){$C_k^*$}
\put(2,1.5){\circle*{0.06}}
\put(2,1.1){$E_{2k}$}
\put(8,1.5){\circle*{0.06}}
\put(7.5,1.1){$E_{2k+1}$}

\put(4.8,1.6){$-$}
\put(4.8,1.2){$-$}

\put(5.5,2.6){$-$}
\put(5.5,2.2){$+$}

\curvedashes{0.05,0.05}
\put(2,1.5){\curve(0,0,6,0)}
\put(5,1.5){\curve(0, 1, 2.645, 0.809, 4.28, 0.309, 4.28, -0.309, 2.645, -0.809, 0, -1, %
-2.645, -0.809, -4.28, -0.309, -4.28, 0.309, -2.645, 0.809, 0, 1)}
\end{picture}
\caption{The lens contour near a band not including any stationary phase point.
Views from the top and bottom sheet.} \label{fig3}
\end{figure}
The oriented paths $C_k, C_k^*$ are meant to be  close
to the band $[E_{2k}, E_{2k+1}]$. The appropriate transformation is now obvious.
Arguing as before, for all bands $[E_{2k}, E_{2k+1}]$ we will have
$$
\aligned
\re(\phi) < (>) 0, \quad \text{in } D_k, \quad k > (<) j.
\endaligned
$$

Now observe that our jump condition (\ref{rhpm3.1}) has the  following important
factorization
\be
J^3=(b_-)^{-1} b_+,
\ee
where 
$$
\aligned
b_-
=\begin{pmatrix}
1 & \ol{ R  \Theta}  \E^{-t\, \phi} \\ 0 &1
\end{pmatrix}, \qquad
b_+= \begin{pmatrix}
1 & 0 \\ R  \Theta \E^{t\, \phi} & 1
\end{pmatrix}.
\endaligned
$$
This is the right factorization for $z>z_j(n/t)$. Similarly, we have
\be\label{facB}
J^3=(B_-)^{-1} \begin{pmatrix} 1-|R|^2 & 0 \\ 0 & \frac{1}{1-|R|^2}\end{pmatrix} B_+,
\ee
where 
$$
\aligned
B_-
=\begin{pmatrix}
1 & 0 \\ - \frac{R  \Theta  \E^{t\, \phi}}{1-|R|^2} &1
\end{pmatrix}, \qquad
B_+= \begin{pmatrix}
1 & -\frac{\ol{ R  \Theta}  \E^{-t\, \phi}}{1-|R|^2} \\ 0 & 1
\end{pmatrix}.
\endaligned
$$
This is the right factorization for $z<z_j(n/t)$. To get rid of the diagonal part
we need to solve the corresponding scalar Riemann--Hilbert problem.
Again we have to search for a meromorphic solution. This means
that the poles of the scalar Riemann--Hilbert problem will be added
to the resulting Riemann--Hilbert problem. On the other hand, a
pole structure similar to the one of $m^3$ is crucial for uniqueness.
We will address this problem by choosing the poles of the scalar problem in such
a way that its zeros cancel the poles of $m^3$. The right choice
will turn out to be $\dinuz$ (that is, the Dirichlet divisor corresponding
to the limiting lattice defined in (\ref{limlat})).

\begin{lemma}
Define a divisor $\dinu{n,t}$ of degree $g$ via
\be
\amap(\dinu{n,t}) = \amap(\dimu{n,t}) + \ul{\delta}(n,t),
\ee
where
\be \label{defdel}
\delta_\ell(n,t) = \frac{1}{2\pi\I} \int_{C(n/t)} \log(1-|R|^2) \zeta_\ell.
\ee
Then $\dinu{n,t}$ is nonspecial and $\pi(\hat{\nu}_j(n,t))=\nu_j(n,t)\in\R$ with precisely one
in each spectral gap.
\end{lemma}

\begin{proof}
Using (\ref{ominfpm}) one checks that $\delta_\ell$ is real. Hence it follows from \cite[Lem.~9.1]{tjac}
that the $\nu_j$ are real and that there is one in each gap. In particular, the divisor $\dinuz$
is nonspecial by \cite[Lem.~A.20]{tjac}.
\end{proof}

Now we can formulate the scalar Riemann--Hilbert problem required
to eliminate the diagonal part in the factorization (\ref{facB}):
\be \label{rhpd}
\aligned
&d_+(p,n,t) = d_-(p,n,t) (1-|R(p)|^2), \quad p \in C(n/t),\\
&(d)\ge -\dinu{n,t},\\
&d(\infty_+,n,t) = 1,
\endaligned
\ee
where $C(n/t) = \Sigma \cap \pi^{-1}((-\infty,z_j(n/t))$. 
Since the index of the (regularized) jump is zero (see remark below), 
there will be no solution in general unless we admit $g$ additional poles
(see e.g. \cite[Thm.~5.2]{ro}).


\begin{theorem}\label{thmplf}
The unique solution of (\ref{rhpd}) is given by
\be \label{defd}
\aligned
d(p,n,t) = & \frac{\theta(\ulz(n,t)+\ul{\delta}(n,t))}{\theta(\ulz(n,t))}
\frac{\theta(\ulz(p,n,t))}{\theta(\ulz(p,n,t)+ \ul{\delta}(n,t))} \times\\
& \times \exp\left( \frac{1}{2\pi\I} \int_{C(n/t)} 
\log (1-|R|^2) \omega_{p\,\infty_+}\right),
\endaligned
\ee
where $\ul{\delta}(n,t)$ is defined in (\ref{defdel}) and $\omega_{p\,q}$
is the Abelian differential of the third kind with poles at $p$ and $q$.

The function $d(p)$ is meromorphic in $\M\setminus\Sigma$ with first order poles at
$\hat{\nu}_j(n,t)$ and first oder zeros at $\hat{\mu}_j(n,t)$.
Also $d(p)$ is uniformly bounded in $n,t$ away from the poles.

In addition, we have $d(p)=\ol{d(\ol{p})}$.
\end{theorem}

Note that this formula is different (in fact much simpler) from the explicit solution
formula from Rodin \cite[Sec.~1.8]{ro}. It is the core of our explicit formula
(\ref{limlat}) for the limiting lattice.


\begin{proof}
On the Riemann sphere, a scalar Riemann-Hilbert problem is solved by the Plemelj--Sokhotsky formula.
On our Riemann surface we need to replace the Cauchy kernel $\frac{d\lam}{\lam-z}$
by the Abelian differential of the third kind $\om_{p\,\infty_+}$. But now it is
important to observe that this differential
is not single-valued with respect to $p$. In fact, if we move $p$ across the $a_\ell$ cycle,
the normalization $\int_{a_\ell} \om_{p\,\infty_+}=0$ enforces a jump by $2\pi\I\zeta_\ell$.
One way of compensating for these jumps is by adding to $\om_{p\,\infty_+}$
suitable integrals of Abelian differentials of the
second kind (cf. \cite[Sec~1.4]{ro} respectively Section~\ref{secSIE}). Since this will produce
essential singularities after taking exponentials we prefer to rather
leave $\om_{p\,\infty_+}$ as it is and
compensate for the jumps (after taking exponentials) by proper use of Riemann theta functions.

To this end recall that the Riemann theta function satisfies
\be\label{thetajump}
\theta( \ulz + \ul{m} + \ul{\tau} \, \ul{n}) = \exp [2 \pi \I \left( -
\spr{\ul{n}}{\ulz} - \frac{\spr{\ul{n}}{\ul{\tau} \, \ul{n}} }{2}\right)]
\theta(\ulz), \quad \ul{n}, \ul{m} \in \Z^g,
\ee
where $\ul{\tau}$ is the matrix of $b$-periods defined in (\ref{deftau}) and $\spr{.}{..}$
denotes the scalar product in $\R^g$ (cf., e.g.\ \cite{fk} or \cite[App.~A]{tjac}).
By definition both the theta functions (as functions on $\M$) and the exponential
term are only defined on the "fundamental polygon" $\hat{\M}$ of
$\M$ and do not extend to
single-valued functions on $\M$ in general. However, multi-valuedness apart, $d$
is a (locally) holomorphic solution of our Riemann--Hilbert problem
which is one at $\infty_+$ by our choice of the second pole of the Cauchy kernel
$\om_{p\,\infty_+}$. The ratio of theta functions is, again apart from multi-valuedness,
meromorphic with simple zeros at $\hat{\mu}_j$ and simple poles at $\hat{\nu}_j$
by Riemann's vanishing theorem. Moreover, the normalization is chosen again such
that the ratio of theta functions is one at $\infty_+$. Hence it remains to verify that (\ref{defd})
gives rise to a single-valued function on $\M$.

Let us start by looking at the values from the left/right on the cycle $b_\ell$. Since our
path of integration in $\ul{z}(p)$ is forced to stay in $\hat{\M}$, the difference
between the limits from the right and left is the value of the integral along $a_\ell$.
So by (\ref{thetajump}) the limits of the theta functions match. Similarly, since
$\om_{p\,\infty_+}$ is normalized along $a_\ell$ cycles, the limits from the left/right
of $\om_{p\,\infty_+}$ coincide. So the limits of the exponential
terms from different sides of $b_\ell$ match as well.

Next, let us compare the values from the left/right on the cycle $a_\ell$. Since our
path of integration in $\ul{z}(p)$ is forced to stay in $\hat{\M}$, the difference
between the limits from the right and left is the value of the integral along $b_\ell$.
So by (\ref{thetajump}) the limits of the theta functions will differ by a multiplicative
factor $\exp(2\pi\I \delta_\ell)$. On the other hand, since $\om_{p\,\infty_+}$ is normalized
along $a_\ell$ cycles, the values from the right and left will differ by $-2\pi\I \zeta_\ell$.
By our definition of $\ul{\delta}$ in (\ref{defdel}), the jumps of the ration of theta
functions and the exponential term compensate each other which shows that
(\ref{defd}) is single-valued.

To prove uniqueness let $\ti{d}$ be a second solution and consider $\ti{d}/d$. Then
$\ti{d}/d$ has no jump and the Schwarz reflection principle implies that it extends
to a meromorphic function on $\M$. Since the poles of $d$ cancel the poles of $\ti{d}$,
its divisor satisfies $(\ti{d}/d) \ge -\dimuz$. But  $\dimuz$ is nonspecial and thus
$\ti{d}/d$ must be constant by the Riemann--Roch theorem. Setting $p=\infty_+$
we see that this constant is one, that is, $\ti{d}=d$ as claimed.

Finally, $d(p)=\ol{d(\ol{p})}$ follows from uniqueness since both functions solve (\ref{rhpd}).
\end{proof}

\begin{remark}\label{remd}
Once the last stationary phase point has left the spectrum, that is,
once $C(n/t)=\Sigma$ we have $d(p) = A^{-1} T(z)^{\pm 1}$, $p=(z,\pm)$ (compare \cite{tag}).
Here $A=A_+(n,t) A_-(n,t) = T(\infty)$.
\end{remark}

In particular,
\be \label{defdinfm}
\aligned
d(\infty_-,n,t) = & \frac{\theta(\ulz(n-1,t))}{\theta(\ulz(n,t))}
\frac{\theta(\ulz(n,t)+ \ul{\delta}(n,t))}{\theta(\ulz(n-1,t)+\ul{\delta}(n,t))} \times\\
& \times \exp\left( \frac{1}{2\pi\I} \int_{C(n/t)} 
\log (1-|R|^2) \omega_{\infty_-\infty_+}\right),
\endaligned
\ee
since $\ulz(\infty_-,n,t)=\ulz(\infty_+,n-1,t)=\ulz(n-1,t)$. Note that $\ol{d(\infty_-,n,t)}=
d(\ol{\infty_-},n,t) = d(\infty_-,n,t)$ shows that $d(\infty_-,n,t)$ is real-valued. Using
(\ref{ominfpm}) one can even show that it is positive.

The next lemma characterizes the singularities of $d(p)$ near the stationary phase points
and the band edges.


\begin{lemma} \label{lemd}
For $p$ near a stationary phase point $z_j$ or $z_j^*$ (not equal to a band edge) we have
\be
d(p)= \left(\frac{z-z_j}{z-E_{2j}}\right)^{\pm\I\nu} e_\pm(z),
\quad p=(z,\pm),
\ee
where $e_\pm(z)$ is holomorphic near $z_j$ and
\be
\nu = - \frac{1}{2\pi} \log (1-|R(z_j)|^2)>0.
\ee
For $p$ near a band edge $E_k\in C(n/t)$ we have
\be
d(p) = T^{\pm 1}(z) \ti{e}_\pm(z), \quad p=(z,\pm),
\ee
where $\ti{e}_\pm(z)$ is holomorphic near $E_k$ if none of the $\nu_j$ is equal
to $E_k$ and $\ti{e}_\pm(z)$ has a first order pole at $E_k=\nu_j$ else.
\end{lemma}

\begin{proof}
The first claim is a local property of the Cauchy integral and hence follows as in the case of
Cauchy integrals in the plane (cf.\ \cite{mu}) by factorizing the jump according to
$1-|R(p)|^2 = (1-|R(z_j)|^2) \frac{1-|R(p)|^2}{1-|R(z_j)|^2}$.

For the second claim note that (compare the Remark~\ref{remd})
$$
t(p) = \frac{1}{T(\infty)} \begin{cases} T(z), & p =(z,+)\in\Pi_+,\\ T(z)^{-1}, & p=(z,-)\in\Pi_-,\end{cases}
$$
satisfies the (holomorphic) Riemann-Hilbert problem
$$
\aligned
&t_+(p) = t_-(p) (1-|R(p)|^2), \quad p \in \Sigma,\\
&t(\infty_+) = 1.
\endaligned
$$
Hence $d(p)/t(p)$ has no jump along $C(n,t)$ and is thus holomorphic near $C(n/t)$
away from band edges $E_k=\nu_j$ (where there is a simple pole) by the Schwarz
reflection principle.
\end{proof}

Having solved the scalar problem above for $d$ we can introduce
the new Riemann--Hilbert problem
\be \label{defm4}
m^4(p) = \begin{pmatrix} \frac{1}{d(\infty_-)} & 0 \\
0 &1 \end{pmatrix} m^3(p) D(p), \quad
D(p) = \begin{pmatrix} d(p^*) & 0 \\
0 &d(p) \end{pmatrix},
\ee
where $d^*(p)= d(p^*)$ is the unique solution of
\begin{align*}
& d^*_+(p) = d^*_-(p) (1-|R(p)|^2)^{-1}, \quad p \in C(n/t),\\
& (d^*)\ge -\dinus{n,t},\\
& d^*(\infty_-) = 1.
\end{align*}
Note that
$$
\det(D(p)) =  d(p)d(p^*) = d(\infty_-)
\prod_{j=1}^g \frac{z-\mu_j}{z-\nu_j}.
$$

Then a straightforward calculation shows that $m^4$ satisfies
\begin{align}\nn
& m^4_+(p) = m^4_-(p) J^4(p), \quad p\in\Sigma,\\ \label{rhpm4}
& (m^4_{j1}) \ge -\dinus{n,t}, \quad (m^3_{j2}) \ge -\dinu{n,t}, \quad j=1,2,\\ \nn
& m^4(\infty_+) = \id,
\end{align}
where the jump is given by
\be
J^4(p) = D_-(p)^{-1} J^3(p) D_+(p), \quad p\in\Sigma.
\ee
In particular, $m^4$ has its poles shifted from $\hat\mu_j(n,t)$ to $\hat\nu_j(n,t)$
and the behavior (\ref{m3OTp}), (\ref{m3OTm}) at the band edges is canceled
by $d(p)$ (cf.\ Lemma~\ref{lemd}).

Furthermore, $J^4$ can be factorized as
\be
J^4 = \begin{pmatrix}
1-|R|^2 & - \frac{d}{d^*} \ol{R \Theta} \E^{-t\, \phi}\\
\frac{d^*}{d} \Theta R \E^{t\, \phi} & 1
\end{pmatrix}
= (\ti{b}_-)^{-1} \ti{b}_+, \quad p\in\Sigma\setminus C(n/t),
\ee
where
$$
\aligned
\ti{b}_- &= D^{-1} b_- D =
\begin{pmatrix}
1 & \frac{d}{d^*}\ol{R \Theta} \E^{-t\, \phi} \\
0 & 1 \end{pmatrix},\\
\ti{b}_+ &= D^{-1} b_+ D =
\begin{pmatrix}
1 & 0 \\
\frac{d^*}{d} \Theta R \E^{t\, \phi} & 1 \end{pmatrix},
\endaligned
$$
for $\pi(p)< z_j(n/t)$ and
\be
J^4 = \begin{pmatrix}
1 & - \frac{d_+}{d^*_-} \ol{R \Theta} \E^{-t\, \phi}\\
\frac{d^*_-}{d_+} \Theta R \E^{t\, \phi} & 1-|R|^2
\end{pmatrix}
= (\ti{B}_-)^{-1} \ti{B}_+, \quad p\in C(n/t),
\ee
where
$$
\aligned
\ti{B}_- &= D_-^{-1} B_- D_- =
\begin{pmatrix} 1 & 0 \\
-\frac{d^*_-}{d_-}  \frac{R  \Theta}{1-|R|^2} \E^{t\, \phi} & 1
\end{pmatrix},\\
\ti{B}_+ &= D_+^{-1} B_+ D_+ =
\begin{pmatrix}
1 & - \frac{d_+}{d^*_+} \frac{\ol{R \Theta}}{1-|R|^2} \E^{-t\, \phi} \\
0 & 1 \end{pmatrix},
\endaligned
$$
for $\pi(p)> z_j(n/t)$.

Note that by $\ol{d(p)}=d(\ol{p})$ we have
\be
\frac{d^*_-(p)}{d_+(p)} = \frac{d^*_-(p)}{d_-(p)} \frac{1}{1-|R(p)|^2} =
\frac{\ol{d_+(p)}}{d_+(p)}, \qquad p\in C(n/t),
\ee
respectively
\be
\frac{d_+(p)}{d^*_-(p)} = \frac{d_+(p)}{d^*_+(p)} \frac{1}{1-|R(p)|^2} =
\frac{\ol{d^*_-(p)}}{d^*_-(p)} , \qquad p\in C(n/t).
\ee

We finally define $m^5$ by
\be \label{defm5}
\aligned
m^5 &= m^4  \ti{B}_+^{-1}, \quad p \in D_k, \: k < j,\\
m^5 &= m^4  \ti{B}_-^{-1}, \quad p \in D_k^*, \: k < j,\\
m^5 &= m^4  \ti{B}_+^{-1}, \quad p \in D_{j1},\\
m^5 &= m^4  \ti{B}_-^{-1}, \quad p \in D_{j1}^*,\\
m^5 &= m^4  \ti{b}_+^{-1}, \quad p \in D_{j2},\\
m^5 &= m^4  \ti{b}_-^{-1}, \quad p \in D_{j2}^*,\\
m^5 &= m^4  \ti{b}_+^{-1}, \quad p \in D_k, \: k > j,\\
m^5 &= m^4  \ti{b}_-^{-1}, \quad p \in D_k^*, \: k > j,\\
m^5 &= m^4, \quad \text{otherwise},
\endaligned
\ee
where we assume that the deformed contour is sufficiently close to
the original one (in particular so close such that we don't cross any
poles). The new jump matrix is given by
\be \label{rhpm5}
\aligned
m^5_+(p,n,t) &= m^5_-(p,n,t) J^5(p,n,t),\\ 
J^5 &= \ti{B}_+, \quad p \in C_k , ~~k < j,\\
J^5 &= \ti{B}_-^{-1}, \quad p \in C_k^*,~~k < j,\\
J^5 &= \ti{B}_+, \quad p \in C_{j1}, \\ 
J^5 &= \ti{B}_-^{-1}, \quad p \in C_{j1}^{*}, \\
J^5 &= \ti{b}_+, \quad p \in C_{j2}, \\
J^5 &=  \ti{b}_-^{-1}, \quad p \in C_{j2}^*,\\
J^5 &= \ti{b}_+, \quad p \in C_k , ~~k > j,\\
J^5 &= \ti{b}_-^{-1}, \quad p \in C_k^*,~~k > j.
\endaligned
\ee
Here we have assumed that the function $R(p)$ admits an analytic extension in the
corresponding regions. Of course this is not true in general, but we can always evade this
obstacle by approximating $R(p)$ by analytic functions. We refer to the discussion
of \cite{dz} for the details.

Eventually we will see that, asymptotically, $m^5(p) \to \id$
uniformly, as $p \to \infty_\pm$, hence
\be\label{asymm3}
\aligned
m^3(p) &= \begin{pmatrix} d(\infty_-) & 0 \\
0 &1 \end{pmatrix} m^4(p) \begin{pmatrix} d(p^*)^{-1} & 0 \\
0 &d(p)^{-1} \end{pmatrix}\\
&= \begin{pmatrix} d(\infty_-) & 0 \\
0 &1 \end{pmatrix}m^5(p) \begin{pmatrix} d(p^*)^{-1} & 0 \\
0 &d(p)^{-1} \end{pmatrix}\\
&\sim \begin{pmatrix} d(\infty_-) d(p^*)^{-1} & 0 \\
0 &d(p)^{-1} \end{pmatrix}
\endaligned
\ee
for $p$ near $\infty_\pm$.
Finally, by Theorem~\ref{thmm3} we will recover
\be
A_+(n,t)^2 \sim d(\infty_-,n,t)^{-1}.
\ee


\begin{remark}
$|R|$ is generically $1$ at the endpoints $E_j$
and hence ${R  \over { 1-|R|^2}}$ is generically singular there, so if $m^5$ is to be
regular (which is still to be proved) then 
$m^3$ must be considered  (generically) as  singular at the endpoints.
This observation is the origin of the imposed behavior of $m^3$ at $E_j$ in the definition 
of $m^3$ in Theorem~\ref{thmm3}. If we are in the non-generic case where $|R| <1$  at 
some of the band edge  $E_j$ the imposed conditions on $m^3$ are of course void.
\end{remark}

The crucial observation now is that the jumps 
$J^5$ on the 
oriented paths
$C_{j1}, C_{j2}, C_{j1}^*, C_{j2}^*$ are
of the form $\id+ exponentially~small$
asymptotically as $t \to \infty$,
at least away from
the stationary phase points $z_j, z^*_j$.
We thus hope we can simply replace these jumps by the identity matrix
(asymptotically as $t \to \infty$). Then, we can  delete
the contours $C_{j1}, C_{j2}, C_{j1}^*, C_{j2}^*$ from the picture.
In any case, the  jumps across
$C_k, C_k^*$, $k \neq j$, are uniformly exponentially small.
So we hope we can also  delete
the contours $C_k, C_k^*$, $k \neq j$.


Consider then the following "model" Riemann--Hilbert problem
which is meant to "approximate" (\ref{rhpm5}). Let $m^6$ be the function 
meromorphic of index $0$ in $X \setminus L$ where $L$ is the set of two crosses centered
at $z_j$, $z_j^*$ each branch of which has length $=t^{-1/2+\eps}$,
for some small $\eps >0$ which we do not need to specify.
(We also do not specify the exact angles between the branches of the crosses
as long as the situation is as shown in the two figures.)
Assume the jump along the cross is given by
\be \label{rhpm6}
\aligned
m^6_+(p,n,t)= m^6_-(p,n,t) J^6(p,n,t), \quad p \neq z_j, z_j^*,
\endaligned
\ee
where $J^6(p,n,t) = J^5(p,n,t)$, $p\in L$.

Note that the Riemann--Hilbert problems for $m^3$, $m^4$, $m^5$ are exactly,
not approximately, equivalent. The step from $m^5$ to $m^6$ is the first where
an approximation is involved.

Once we show that a solution of (\ref{rhpm6}) exists, it is easy to see by simple 
rescaling that the solution is decaying as  $t \to \infty$.
Thus the remaining cross can also be neglected, up to
leading order. To be more precise, once we have chosen the crosses so that their
length is of order $t^{-1/2+\eps}$, $\eps >0$, 
then the complement of the crosses
will be deleted since the exponential terms are uniformly small there,
while the small crosses will eventually be ignored simply because
their length is small. All this will be done rigorously in Sections~\ref{secSIE} and
\ref{secETFP}.

Existence of  solutions $m^3, m^4, m^5$ and the approximate problem (\ref{rhpm6}) 
will follow directly as a corollary of Theorem~\ref{thmrhpQ}.

We expect (\ref{rhpm6}) to be explicitly solvable since it is easily reducible to
a problem with constant jump matrix, even though the underlying space
is a Riemann surface. In any case, finding an exact solution is not necessary
so long as we are only interested in leading asymptotics for the perturbed lattice.

Case (ii).
In the  special case where the two stationary phase points coincide
(so $z_j = z_j^*= E_k$ for some $k$) the 
Riemann--Hilbert problem arising above is of a different nature, even in the simpler
non-generic case $|R(E_k)|<1$.
In analogy to the case of the free lattice one might expect different local
asymptotics expressed in terms of (generalized) Painlev\'e functions.
In the case $|R(E_k)| <1$ the two crosses coalesce and the discussion of Section~\ref{secETFP}
goes through virtually unaltered.
If $|R(E_k)| =1$ the problem is 
singular in an essential way 
and we expect a "collisionless shock" phenomenon in the region where
$z_j(n/t) \sim E_k$, similar to the one studied
in  \cite{as}, \cite{dvz}, \cite{km2}. The main difficulty
arises from the singularity of  ${R  \over { 1-|R|^2}}$.
We expect that an appropriate  "local" Riemann--Hilbert problem however is still 
explicitly solvable.
It is definitely  interesting to understand the
actual contribution of the band edges and the way it is different
from the free case due to the fact that the underlying Riemann surface has
nonzero genus. We plan to investigate this issue in a future publication.
But in the present work, we will assume
that the stationary phase points stay away from the $E_k$ (see Theorems~
\ref{thmMain}, \ref{thmexm6}).
 
Case (iii). 
In the case where no stationary phase points lie in the spectrum the situation is
similar to the case (i). There is a gap (the $j$-th gap, say) in which two stationary
phase points exist. We construct  "lens-type" contours $C_k$ around every single
band lying to the left of the $j$-th gap and make use of the factorization
$J^3=(\ti{b}_-)^{-1} \ti{b}_+$. We also construct  "lens-type" contours $C_k$ around every single
band lying to the right of the $j$-th gap and make use of the factorization
$J^3=(\ti{B}_-)^{-1} \ti{B}_+$. Indeed, in place of (\ref{defm5})
we set
\be 
\aligned
m^5 &= m^4  \ti{B}_+^{-1}, \quad p \in D_k, \: k < j,\\
m^5 &= m^4  \ti{B}_-^{-1}, \quad p \in D_k^*, \: k < j,\\
m^5 &= m^4  \ti{b}_+^{-1}, \quad p \in D_k, \: k > j,\\
m^5 &= m^4  \ti{b}_-^{-1}, \quad p \in D_k^*, \: k > j,\\
m^5 &= m^4, \quad \text{otherwise}.
\endaligned
\ee

It is now easy to check that in both cases (i) and (iii) 
formula  (\ref{defdinfm}) is still true.

\section{A singular integral equation}
\label{secSIE}


In the complex plane, the solution of a Riemann--Hilbert problem
can be reduced to the solution of a singular integral equation (see \cite{bc}).
In our case the underlying space is a Riemann surface. Hence we have
to replace the classical Cauchy kernel by  a "generalized" Cauchy
kernel  appropriate to our
Riemann surface. In order to get a single valued kernel we need again
to admit $g$ poles. We follow the construction from \cite[Sec.\ 4]{ro}.
Allowing poles at the nonspecial divisor $\dimuz$ the corresponding Cauchy
kernel is given by
\be
\Omega_p^\uhmuz = \om_{p \infty_+} + \sum_{j=1}^g I_j^\uhmuz(p) \zeta_j,
\ee
where
\be
I_j^\uhmuz(p) = \sum_{\ell=1}^g c_{j\ell} \int_{\infty_+}^p \om_{\hmu_\ell,0}.
\ee
Note that $I_j^\uhmuz(p)$ has first order poles at the points $\uhmuz$.

The constants $c_{j\ell}$ are chosen such that $\Omega_p^\uhmuz$ is single
valued (cf.\ the discussion in the proof of Theorem~\ref{thmplf}). That is,
$$
\int_{b_k} dI_j^\uhmuz = \sum_{\ell=1}^g c_{j\ell} \int_{b_k} \om_{\hmu_\ell,0}
= \sum_{\ell=1}^g c_{j\ell} \eta_k(\hmu_\ell) = \delta_{j k},
$$
where $\zeta_k = \eta_k(z) dz$ is the chart expression in a local chart near
$\hmu_\ell$ (here the $b_k$ periods are evaluated using the usual bilinear relations,
see \cite[Sect.~III.3]{fk} or \cite[Sect.~A.2]{tjac}). That the matrix
$\eta_k(\hmu_\ell)$ is indeed invertible can be seen as follows:
If $\sum_{k=1}^g \eta_k(\hmu_\ell) c_k =0$ for $1\le \ell \le g$, then the divisor of
$\zeta= \sum_{k=1}^g c_k \zeta_k$ satisfies $(\zeta)\ge \dimuz$. But since we
assumed the divisor $\dimuz$ to be nonspecial, $i(\dimuz)=0$, we have
$\zeta=0$ implying $c_k=0$.



We will always assume that none of the poles lie on our contour $\Sigma$.
This can be done without loss of generality since we can move the contour
a little without changing the value at $\infty_-$ (which is the only value
we are eventually interested in). Furthermore, we will, for simplicity, also assume that
$\Sigma$ is sufficiently smooth, compact, and does not contain $\infty_\pm$. We will
abbreviate $L^p(\Sigma)=L^p(\Sigma,\C^{2\times 2})$.

\begin{theorem} \label{thmCrs}
Set
\be
\ul{\Omega}_p^\uhmuz = \begin{pmatrix}
\Omega_p^{\uhmuz^*} & 0 \\ 0 & \Omega_p^\uhmuz
\end{pmatrix}
\ee
and define the matrix operators as follows. Given a $2\times2$ matrix $f$ defined 
on $\Sigma$ with H\"older continuous entries, let
\be
(C f)(p) = \frac{1}{2 \pi \I} \int_{\Sigma} f\, \ul{\Omega}_p^\uhmuz,
\quad\text{for}\quad p \not \in \Sigma,
\ee
and
\be \label{defCpm}
(C_\pm f)(q) = \lim_{p \to q \in \Sigma} (C f)(p)
\ee
from the left and right of $\Sigma$ respectively (with respect to its orientation).
Then 
\begin{enumerate}
\item
The operators $C_\pm$ are given by the Plemelj formulas
$$
\aligned
(C_+ f)(q) - (C_- f)(q) &= f(q),\\
(C_+ f)(q) + (C_- f)(q) &= \frac{1}{\pi \I}\; \dashint_\Sigma f\, \ul{\Omega}_q^\uhmuz,
\endaligned
$$
and extend to bounded operators on $L^2(\Sigma)$. Here $\dashint$ denotes the
principal value integral, as usual.
\item
$C f$ is a meromorphic function off $\Sigma$, with divisor given
by $((C f)_{j1}) \ge -\dimuzs$ and $((C f)_{j2}) \ge -\dimuz$.
\item
$(C f) (\infty_+) =0$.
\end{enumerate}
\end{theorem}

\begin{proof}
In a chart $z=z(p)$ near $q_0\in\Sigma$, the differential $\Omega_q^\uhmuz = 
(\frac{1}{z-z(q)} + O(1)) dz$ and hence the first part follows as in the Cauchy case
on the complex plane (cf.\ \cite{mu}) using a partition of unity. To see (ii) note that
the integral over $\om_{p \infty_+}$ is a (multivalued) holomorphic function, while
the integral over the rest is a linear combination of the (multivalued) meromorphic
functions $I_j^\uhmuz$ respectively $I_j^{\uhmuz^*}$. By construction, $I_j^\uhmuz$
has at most simple poles at the points $\uhmuz$ and thus (ii) follows.
Finally, to see (iii) observe that $\om_{p \infty_+}$ restricted to $\Sigma$ converges
uniformly to zero as $p\to\infty_+$. Moreover, $I_j^{\uhmuz^*}(\infty_+)=0$ and
hence (iii) holds.
\end{proof}

Now, given any $b_-, b_+ \in L^\infty(\Sigma)$ with
determinant equal to $1$,
let the operator $C_w: L^2(\Sigma) \to L^2(\Sigma) $ be defined by
\be\label{defcw}
C_w f = C_+ (fw_-) + C_- (fw_+)
\ee
for a $2\times2$ matrix valued $f$, where
$$
w_+ = b_+ - \id  \quad\text{ and }\quad   w_- = \id - b_-.
$$

\begin{theorem}\label{thmQ}
Assume that  $\mu$ solves the singular integral equation
\be \label{musie}
\mu = \id + C_w \mu  \quad\text{ in }\quad L^2(\Sigma).
\ee
Let $Q$ be defined by the integral formulae
\be \label{fQ}
Q = \id + C (\mu w) \quad\text{ on } \M \setminus \Sigma ,
\ee
where $w=w_+ + w_-$. Then $Q$ is a solution of the following
meromorphic Riemann--Hilbert problem.
\be \label{rhpQb}
\aligned
Q_+(p) = Q_-(p) b^{-1}_-(p) b_+ (p), \quad p \in \Sigma,\\
Q(\infty_+) = \id,\\
(Q_{j1}) \ge -\dimuzs, \quad (Q_{j2}) \ge -\dimuz.
\endaligned
\ee
\end{theorem}

\begin{proof}
To show that $Q$ defined above solves (\ref{rhpQb}) note that
$$
Q_\pm = \id + C_\pm(\mu w).
$$
Thus, using $C_+-C_- = \id$ and the definition of $C_w$ we obtain
$$
\aligned
Q_+ &= (\id + C_+ (\mu w)) = (\id+C_+(\mu w_+) + C_+(\mu w_-))\\
& = (\id+\mu w_+ + C_- (\mu w_+) + C_+ (\mu w_-) )
= (\id+ \mu w_+ + C_w \mu )\\
& = \mu (\id + w_+) 
\endaligned
$$
and similarly $Q_- = \mu (\id - w_-)$.
Hence $ Q_+ b_+^{-1} = \mu = Q_- b_-^{-1} $ and thus
thus $Q_+ = Q_- (b_-)^{-1} b_+.$ This proves the jump condition.
That $Q$ has the right devisor and the correct normalization at $\infty_+$
follows from  Theorem~\ref{thmCrs} (ii) and (iii), respectively.
\end{proof}

\begin{remark}\label{remQ}
A few remarks are in order:

(i). The theorem stated above does not address uniqueness. However, we do have uniqueness
of $Q(\infty_-)$. For the proof of this, just follow Remark~\ref{remuniq}.

(ii). If we only assume $\mu = \id + C_w \mu$ in $L^2(\Sigma)$,
we are only guaranteed an $L^2$-solution of the Riemann--Hilbert problem, that is,
the limits are taken in the $L^2$ sense and
the values of $Q_+$, $Q_-$ on $\Sigma$ are $L^2$. This is all we need here.
On the other hand, we could have alternatively used the H\"older theory
described in \cite{kmm}.  Of course, in our current concrete situation the analyticity
of the jumps for $m^4$, $m^5$  implies the continuity of the limiting values $m^4_{\pm}$,
$m^5_{\pm}$ (except at the cross center) and thus also of $m^3_{\pm}$ away from
the band edges $E_j$.

(iii). The notation $b_+, b_-$ is meant to make one think of the example
$J^3 =(b_-)^{-1} b_+ $ in Section~\ref{secSPP}, but the theorem above is fairly general.
In particular it also applies to the 
trivial factorizations $J^3 = \id J^3 = J^3 \id$.
\end{remark}

We  are interested in the  formula (\ref{fQ}) evaluated at $\infty_-$.
We write it as
\be \label{ifQ}
\aligned
Q(\infty_-) &=  (\id + C (\mu w)) (\infty_-)\\ & = 
\id + \int_{\Sigma} (\id-C_w)^{-1}(\id)\, w \, \ul{\Omega}_{\infty_-}^\uhmuz
\endaligned
\ee and we perturb it
with respect to $w$ while keeping the contour $\Sigma$ fixed. 

\begin{theorem}\label{thmexm6}
A unique solution $Q$ of the Riemann--Hilbert problems (\ref{rhpm5}) and
(\ref{rhpm6})
exists such that $Q$ is uniformly bounded in $(n, t)$ away from the poles.
\end{theorem}

\begin{proof}
Existence and uniqueness follows from Theorem~\ref{thmrhpQ}. Moreover, note
that the jump for both $m^5$ and $m^6$
is of the form $\id+ \mbox{\it off-diagonal}$ where the off-diagonal part
is bounded in norm by $|R|$.

Furthermore the operator $(\id-C_{w})^{-1}$ corresponding to both problems
(with trivial factorization of the jump) is clearly uniformly bounded in norm. 
This in turn follows from the
Neumann series
\be
(\id-C_w)^{-1} =  \id + C_w + C_w^2 +C_w^3 + ....
\ee
and the fact that $\|C_{w^6}\| $ is bounded by a constant that is strictly
less than $1$ and independent of $n,t$. The bound for 
$\|C_{w^6}\| $ follows from the following facts:

The singular part of the Cauchy operators $C_\pm$ as defined by (\ref{defCpm})
has norm equal to one (see e.g. \cite{stein}, p.58). Hence the singular part of
$C_{w^6}$ has norm bounded by the supremum of $|R|$,
which is less than one in the domain we are interested.

The nonsingular part of the Cauchy operators is 
small (for large times) because
the contour where $J^6$ lies is of
length $=t^{-1/2+\eps}$.

So, after all, 
$(\id-C_{w^6})^{-1}$ is uniformly bounded in norm, and by, say, the second
resolvent identity 
$(\id-C_{w^5})^{-1}$ is uniformly bounded in norm, hence $m^5, m^6$ 
are uniformly bounded in $(n, t)$ away from the poles.
\end{proof}

The next  theorem will enable us to
\begin{enumerate}
\item first remove the bulk of the contour in
problem (\ref{rhpm5}),
away from the stationary phase points,
\item also remove the rest of the contour.
\end{enumerate}

\begin{theorem}\label{thmQubd}
Suppose we have a Riemann--Hilbert problem as
in (\ref{rhpQb}) above with $w$ depending on
parameters $n,t$ such that the solution given by (\ref{fQ}),
or equivalently $ (\id-C_w)^{-1}(\id) (s)$,
is uniformly bounded in $n,t$ on $\Sigma$.
Suppose first  that
the  jump matrix is such that $\|w_\pm\|_{\infty} < \eps$,
where $\eps$ is a given small number (independent of $n,t$).

Then the solution of the Riemann--Hilbert problem given by
(\ref{fQ}) satisfies $\|Q(\infty_-) - \id \| < C \eps$,
where $C$ is some constant independent of $n,t$.

Suppose then that the length of the contour
is $o(1)$ as $t \to \infty$. Then the solution of the Riemann--Hilbert
problem given by (\ref{fQ}) satisfies $\|Q(\infty_-) - \id \| \to 0$,
as $t \to \infty$ uniformly in $n,t$.
\end{theorem}

\begin{proof}
Both assertions
follow from the integral formula (\ref{ifQ}) above.
\end{proof}

Theorem~\ref{thmMain}  follows  in view of Theorem~\ref{thmm3}.

\begin{proof}[Proof of Theorem~\ref{thmMain}]
In order to compare the solution $m^6$ of (\ref{rhpm6}) and the solution $m^5$ of (\ref{rhpm5})
observe that the difference of the jumps lives in the complement of the two small crosses $L$
with respect to the jump for $m^5$, that is $\cup_j (C_j \cup C^*_j) \setminus L$.
Hence the difference of the jumps is a uniformly exponentially small quantity, so 
both solutions are uniformly close at least near $\infty_-$ (which is all we care here)
by the second resolvent identity.

From Theorem~\ref{thmQubd} above we see that
$m^6 (\infty_-)$ is uniformly  close to the identity.
Then $m^5 (\infty_-)$ is uniformly  close to the identity and 
by tracing back to
$m^4$ and $m^3$ (according to the discussion in Section~\ref{secSPP})
and using (\ref{Apm3}) we have our result.
\end{proof}


\begin{proof}[Proof of Theorem~\ref{thmMain2}]
We begin by showing
$$
m^2(p) = \begin{pmatrix} A_+  & \frac{1}{A_+} \end{pmatrix} m^3(p).
$$
In order to avoid the possible singularities of $m^3$ at the band edges,
we will show the equivalent statement
$$
\ti{m}^2(p) = \begin{pmatrix} d(\infty_-) A_+  & \frac{1}{A_+} \end{pmatrix} m^4(p),
$$
where
$$
\ti{m}^2(p) = m^2(p) D(p)
$$
(compare the definition (\ref{defm4}) of $m^4$). By the factor $T(z)$ in the definition
(\ref{defm2}) of $m^2$, the new quantity $\ti{m}^2$ is bounded near the band edges and
thus both sides solve the vector Riemann-Hilbert problem
\begin{align}\nn
& \ti{m}^2_+(p) = \ti{m}^2_-(p) J^4(p), \quad p\in\Sigma,\\
& (\ti{m}^2_1) \ge -\dinuzs, \quad (\ti{m}^2_2) \ge -\dinuz,\\ \nn
& \ti{m}^2(\infty_+) = \begin{pmatrix} d(\infty_-) A_+  & \frac{1}{A_+} \end{pmatrix},
\end{align}
whose solution is unique by Theorem~\ref{thmrhpQ} (cf.\ Remark~\ref{remrhpQ}).

Combining this with (\ref{asymm3}) we obtain
\be
m^2(p,n,t) \sim \begin{pmatrix} \frac{A_+(n,t) d(\infty_-,n,t)}{d(p^*,n,t)} &
\frac{1}{A_+(n,t) d(p,n,t)} \end{pmatrix}
\ee
for $p$ near $\infty_\pm$. Moreover, for $p=(z,+)$ we have
$$
m^2(p) = \begin{pmatrix} A_+  & \frac{1}{A_+} \end{pmatrix} +
\begin{pmatrix} -A_+B_+  & \frac{B_+}{A_+} \end{pmatrix} \frac{1}{z} + O(\frac{1}{z^2})
$$
and thus
$$
B_+(n,t) \sim - d_1(n,t),
$$
where $d_1$ is given by
$$
d(p) = 1 + \frac{d_1}{z} + O(\frac{1}{z^2}).
$$
Hence it remains to compute $d_1$. Proceeding as in \cite[Thm.~9.4]{tjac}
respectively \cite[Sec.~4]{tag} one obtains
\begin{align*}
d_1 =& - \frac{1}{2\pi\I} \int_{C(n/t)} \log(1-|R|^2) \Omega_0\\
& {} - \frac{1}{2}\frac{d}{ds} \log\left( \frac{\theta(\ulz(n,s) + \ul{\delta}(n,t) )}{\theta(\ulz(n,s))} \right) \Big|_{s=t}.
\end{align*}
where $\Omega_0$ is the Abelian differential of the second kind defined in (\ref{ominfpm}).
\end{proof}



\section{Conclusion}


We have considered here the stability problem for the periodic Toda lattice under a
short range perturbation. We have discovered that a nonlinear stationary phase
method (cf.\ \cite {dz}, \cite{km}) is applicable and as a result we have shown that
the long time behavior of the perturbed lattice is described by a modulated lattice
which undergoes a continuous phase transition (in the Jacobian variety).

Although the most celebrated applications of the deformation method
initiated by \cite{dz} for  the asymptotic evaluation of solutions of
Riemann-Hilbert factorization problems have been in statistical mechanics
(via orthogonal polynomials, random matrices and combinatorial probability),
most mathematical innovations have appeared in the study of nonlinear dispersive
PDEs or systems of ODEs (cf. \cite{dz}, \cite{dvz}, \cite{kmm}).
It is thus interesting that another mathematical extension of the theory
(setting the factorization problem on a Riemann surface rather than the complex plane)
arises in the study of an innocent looking stability problem for the periodic
Toda lattice.

\appendix


\section{An existence theorem for factorization problems on a Riemann surface}
 \label{secETFP}
 
 
 
In the case where the underlying spectral curve is the complex plane
it is often useful to have a theorem guaranteeing 
existence of a solution of a Riemann--Hilbert problem
under some symmetry conditions. One such is, for example,
the Schwarz reflection theorem  provided in \cite{zh}.
In this section we state and prove an analogous theorem where the underlying
spectral curve is our hyperelliptic curve with real branch cuts.

We state and prove the following theorem assuming local analyticity of
the jump and thus proving continuity of the limiting values. But
(cf.\ Remark~\ref{remQ} (ii)) we could alternatively drop the analyticity condition and then
only guarantee an $L^2$-solution of the Riemann--Hilbert problem, which is
all we need here.
On the other hand, we could have alternatively used the H\"older theory
described in \cite{kmm}.  

For any matrix $M$ we denote its adjoint (transpose of complex conjugate)
as $M^*$. Then we have

\begin{theorem} \label{thmrhpQ}
The following matrix Riemann--Hilbert problem on $\M$ has always a 
unique solution and the limits $Q_+$, $Q_-$ on $\Sigma$ are continuous.
\be \label{rhpQ}
\aligned
Q_+(p) = Q_-(p) J(p), \quad p \in \Sigma,\\
Q(\infty_+) = \id, \quad
 (Q_{j1}) \ge -\dimuzs, \quad (Q_{j2}) \ge -\dimuz
\endaligned
\ee
where $\Sigma$ is an oriented  contour, consisting of a union of
finitely many smooth arcs, which is symmetric under sheet exchange plus
conjugation ($\Sigma= \ol{\Sigma}^*$) such that  

(i) $J(p^*)=  J(\ol{p})^*$, for $p \in \Sigma \setminus \pi^{-1}(\sig(H_q))$,

(ii) $\re(J(p)) = \frac{1}{2}(J(p)+ J(p)^*)$ is positive definite for $p \in \pi^{-1}(\sig(H_q))$,

(iii) $J$ is analytic in a neighborhood of $\Sigma$.

\noindent
The divisor $\dimuz$ is such that $\pi(\mu_j)\in[E_{2j-1},E_{2j}]$.
\end{theorem}

Note here that the $+$-side of the contour is mapped to the $-$-side under
sheet exchange. In particular, the theorem holds if $J=\id$, 
that is there is no jump, on $\pi^{-1}(\sig(H_q))$.

\begin{proof}
Continuity of the limits $Q_+$, $Q_-$ follows immediately from the analyticity of $J$.

Let $J(z) = b_-^{-1} (z) b_+ (z)= \id J(z)$ be the trivial  factorization
$b_- =\id$, $b_+=J$.
The operator $\id-C_w$ is defined as in Section~\ref{secSIE} with
$w_+= J - \id$ and $w_-=0$.
By Theorem~\ref{thmQ}, we only need to show that the operator
$\id-C_w$ is invertible. Note first that $\id-C_w$ is
a Fredholm operator with index,  $\ind(\id-C_w)$,  zero. Indeed
one can easily check that
\be
(\id-C_w) (\id-C_{\tilde w}) = \id+ T_w,
\ee
where $ T_w(f) = C_-[C_-(f \tilde w_+)w_+]$ and
\be
\tilde w_+ = -b_+ +\id = -J+\id, \qquad  \tilde w_- = \id - \id = 0.
\ee
But   $ T_w(f)$ is a compact operator.
Indeed, suppose $f_n \in L^2(\Sigma)$ converges weakly to zero.
We will show that $\|T_w f_n\|_{L^2} \to 0$.

Using the analyticity of $w=w_-$ in a neighborhood of $\Sigma$
and the definition of $C_-$, we can slightly deform
the contour $\Sigma$ to some contour 
$\Sigma'$ close to $\Sigma$, on the right, and
have, by Cauchy's theorem,
\be
T_w f_n(p) = \frac{1}{2 \pi \I}
\int_{\Sigma'} (C_-(f_n w) \tilde w) \ul{\Omega}_p^\uhmuz.
\ee
Now clearly $(C_-(f_n w) \tilde w)(p) \to 0$ as $n \to \infty$.
Also $|(C_-(f_n w) \tilde w)(p)| < const\, \|f_n\|_{L^2} \|\tilde w \|_{L^2}
< const$. By the dominated convergence theorem,
$\|T_w f_n\|_{L^2} < const\, \|(C_-(f_n w) \tilde w)(p)\| \to 0$.

Similarly, one can show that there exists a compact operator $T_v$ such that
\be
(\id-C_{\tilde w}) (\id-C_w)  = \id + T_v.
\ee
Hence by \cite[Thm.~1.4.3]{proe} $\id-C_w$ is Fredholm.

Now considering $\ind(\id- \eps C_w)$ for $0 \leq \eps \leq 1$ and recall
that $C_w$ is bounded. 
It is a fact \cite[Thm.~1.3.8]{proe} that $\ind(\id- \eps C_w)$
is continuous with $\eps$.  Since it is an integer, it has to be constant.
So in particular $\ind(\id-C_w)= \ind(\id) =0.$

By the Fredholm alternative, it follows that to show the bounded invertibility of $\id-C_w$
we only need to show that $\ker (\id-C_w) =0$.

So suppose there is a $\nu$ such that $(\id-C_w) \nu =0$. We will prove that $\nu =0$.

Define
\be
M = C(\nu w)=C(\nu w_+).
\ee
Then, as in the proof of Theorem~\ref{thmQ} one verifies $M_- = \nu$ and $M_+= \nu J$.
Hence $M$ solves (\ref{rhpQ}) except for the normalization at $\infty_+$, which
now reads  $M(\infty_+) =0$. In other words $M$ solves the associated
vanishing Riemann--Hilbert problem. Also, as above, $M_+$, $M_-$ are continuous.

Next we want to apply Cauchy's integral theorem to $M(p) M^*(\ol{p}^*)$.
To this end we will multiply it by a meromorphic differential $d\Omega$
which has zeros at $\ul{\mu}$ and $\ul{\mu}^*$ and simple poles
at $\infty_\pm$ such that the differential $M(p) M^*(\ol{p}^*) d\Omega(p)$
is a holomorphic away from the contour.

Indeed let 
\be
d\Omega= -\I \frac{\prod_{j=1}^g (\pi -\mu_j)}{R_{2g+2}^{1/2}} d\pi
\ee
and note that ${\prod_j(z-\mu_j) \over R_{2g+2}^{1/2}(z)}$ is a Herglotz--Nevanlinna
function. That is, it has positive imaginary part in the upper half-plane (and it is
purely imaginary on $\sig(H_q)$). Hence $ M (p) \ol{M^T(p)}d\Omega(p)$ will be positive
on $\pi^{-1}(\sig(H_q))$.

Consider then the integral
\be
\int_D M(p)  M^*(\ol{p}^*) d\Omega(p),
\ee
where $D$ is a $\ol{*}$-invariant contour consisting of one small loop in every
connected component of $\M\setminus\Sigma$. Clearly the above integral
is zero by Cauchy's residue theorem. We will deform $D$ to a $\ol{*}$-invariant
contour consisting of two parts, one, say $D_+$, wrapping around the part of $\Sigma$
lying on $\Pi_+$ and the $+$ side of $\pi^{-1}(\sig(H_q))$ and the other being
$D_- = \ol{D_+}^*$.

For each component $\Sigma_{\nu} $ of $\Sigma \setminus \pi^{-1}(\sig(H_q))$
there are two contributions to the integral on the deformed contour: 
$$
\aligned
\int_{ \Sigma_\nu} M_+(p)  M^*_-(\ol{p}^*) d\Omega & =
\int_{ \Sigma_\nu} M_-(p) J(p) M^*_-(\ol{p}^*) d\Omega \quad\text{and}\\
\int_{-\Sigma_\nu} M_-(p) M^*_+(\ol{p}^*) d\Omega &=
\int_{-\Sigma_\nu} M_-(p) J^*(\ol{p}^*) M^*_-(\ol{p}^*) d\Omega. 
\endaligned
$$
Because of condition (i) the two integrals cancel each other.

In view of the above and using Cauchy's  theorem, one gets
\begin{align} \nn
0 &= \int_D M(p) M^*(\ol{p}^*) d\Omega\\ \nn
&= \int_{\pi^{-1}(\sig(H_q))} [M_+(p) M_-^*(\ol{p}^*) + M_-(p) M_+^*(\ol{p}^*)] d\Omega\\ \nn
&= \int_{\pi^{-1}(\sig(H_q))}  M_-(p) (J(p) +J^*(\ol{p}^*)) M_-^*(\ol{p}^*) d\Omega.
\end{align}
By condition (ii) it now follows that $M_- = \nu = 0$  and hence $M = 0$.

To prove uniqueness, suppose there were two solutions. Their difference would then satisfy the
same jump  and it would  be zero at $\infty_+$. Using the above argument for $M$ we
see that the difference would have to vanish everywhere.
\end{proof}

\begin{remark} \label{remrhpQ}
The proof of the previous theorem also shows uniqueness for the following
vector Riemann--Hilbert problem on $\M$
\be
\aligned
Q_+(p) = Q_-(p) J(p), \quad p \in \Sigma,\\
Q(\infty_+) = \begin{pmatrix} \alpha & \beta \end{pmatrix}, \quad
 (Q_1) \ge -\dimuzs, \quad (Q_2) \ge -\dimuz
\endaligned
\ee
where $J(z)$, $\Sigma$, and $\dimuz$ satisfy the same assumptions as in the
previous theorem and $\begin{pmatrix} \alpha & \beta \end{pmatrix}$ is some
constant vector.
\end{remark}

The above Theorem does not apply directly
to the Riemann--Hilbert problem for $m^3$ of Theorem~\ref{thmm3},
because of the (possible) singularities at the
band edges (see (\ref{m3OTp}), (\ref{m3OTm})). However, since
condition (ii) holds for the Riemann--Hilbert problem (\ref{rhpm4}) for $m^4$
(while condition (i) is irrelevant), we get existence and uniqueness for $m^4$ and
thus also for $m^3$ by (\ref{defm4}).

\begin{theorem} \label{thmuniq}
The  Riemann--Hilbert problem of Theorem~\ref{thmm3}
admits a unique solution.
\end{theorem}

Again,  the analyticity condition (iii) is not always
satisfied for the jump matrix of  Theorem~\ref{thmm3}. But, as noted by
\cite{dz} (see also the discussion after (4.22)), 
one can always approximate the reflection coefficient by a
holomorphic function near the jump contour. Since the nearby holomorphic
Riemann--Hilbert problem has a solution, so does the possibly non-holomorphic
one (see \cite{dz} for the details).

Alternatively, we can use the $L^2$ version of Theorem 6.1, without the analyticity 
condition. 



\bigskip

\noindent {\bf Acknowledgments.}

Spyridon Kamvissis gratefully acknowledges the support of the European Science
Foundation (MISGAM program) during two visits to the University of Vienna in 2005 and
2006. He also acknowledges the support of the Austrian Science Fund (FWF)
during a second visit in 2006.




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