%% @texfile{ %% filename="TodaStab.tex", %% version="1.2", %% date="June-2011", %% cdate="20060603", %% filetype="LaTeX2e", %% pics="TodaStab1", %% journal="Preprint", %% copyright="Copyright (C) S.Kamvissis and G.Teschl". %% } \documentclass{amsart} \usepackage{hyperref} \usepackage{graphicx} \usepackage{curves} \unitlength1cm %%%%%%%%% \newcommand{\arxiv}[1]{\href{http://arxiv.org/abs/#1}{arXiv:#1}} \newcommand*{\mailto}[1]{\href{mailto:#1}{\nolinkurl{#1}}} %%%%%%%%%THEOREMS%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \newtheorem{hypothesis}[theorem]{Hypothesis {\bf H.}\hspace*{-0.6ex}} %%%%%%%%%%%%%%FONTS%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\R}{{\mathbb R}} \newcommand{\N}{{\mathbb N}} \newcommand{\Z}{{\mathbb Z}} \newcommand{\C}{{\mathbb C}} \newcommand{\M}{{\mathbb M}} \newcommand{\W}{{\mathbb W}} %%%%%%%%%%%%%%%%%%ABBRS%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\nn}{\nonumber} \newcommand{\be}{\begin{equation}} \newcommand{\ee}{\end{equation}} \newcommand{\bea}{\begin{eqnarray}} \newcommand{\eea}{\end{eqnarray}} \newcommand{\ul}{\underline} \newcommand{\ol}{\overline} \newcommand{\ti}{\tilde} \newcommand{\spr}[2]{\langle #1 , #2 \rangle} \newcommand{\id}{\mathbb{I}} \newcommand{\I}{\mathrm{i}} \newcommand{\E}{\mathrm{e}} \newcommand{\ind}{\mathrm{ind}} \newcommand{\re}{\mathrm{Re}} \newcommand{\im}{\mathrm{Im}} \newcommand{\floor}[1]{\lfloor#1 \rfloor} \def\Xint#1{\mathchoice {\XXint\displaystyle\textstyle{#1}}% {\XXint\textstyle\scriptstyle{#1}}% {\XXint\scriptstyle\scriptscriptstyle{#1}}% {\XXint\scriptscriptstyle\scriptscriptstyle{#1}}% \!\int} \def\XXint#1#2#3{{\setbox0=\hbox{$#1{#2#3}{\int}$} \vcenter{\hbox{$#2#3$}}\kern-.5\wd0}} \def\dashint{\Xint-} \newcommand{\sigI}{\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}} \newcommand{\rI}{\begin{pmatrix} 1 & 1 \end{pmatrix}} \newcommand{\ulz}{\underline{z}} \newcommand{\di}{\mathcal{D}} \newcommand{\vrc}{\ul{\Xi}_{p_0}} \newcommand{\hvrc}{\ul{\hat{\Xi}}_{p_0}} \newcommand{\hmu}{\hat{\mu}} \newcommand{\uhmu}{{\underline{\hat{\mu}}(n,t)}} \newcommand{\uhmuz}{{\underline{\hat{\mu}}}} \newcommand{\hnu}{\hat{\nu}} \newcommand{\uhnu}{{\underline{\hat{\nu}}(n,t)}} \newcommand{\uhnuz}{{\underline{\hat{\nu}}}} \newcommand{\dimu}[1]{\di_{\ul{\hat{\mu}}(#1)}} \newcommand{\dimus}[1]{\di_{\ul{\hat{\mu}}(#1)^*}} \newcommand{\dimuz}{\di_{\ul{\hat{\mu}}}} \newcommand{\dimuzs}{\di_{\ul{\hat{\mu}}^*}} \newcommand{\dinu}[1]{\di_{\ul{\hat{\nu}}(#1)}} \newcommand{\dinus}[1]{\di_{\ul{\hat{\nu}}(#1)^*}} \newcommand{\dinuz}{\di_{\ul{\hat{\nu}}}} \newcommand{\dinuzs}{\di_{\ul{\hat{\nu}}^*}} \newcommand{\Amap}{\ul{A}_{p_0}} \newcommand{\amap}{\ul{\alpha}_{p_0}} \newcommand{\hAmap}{\ul{\hat{A}}_{p_0}} \newcommand{\hamap}{\ul{\hat{\alpha}}_{p_0}} \newcommand{\Rg}[1]{R_{2g+2}^{1/2}(#1)} %%%%%%%%%%%%%%%GREEK%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\eps}{\varepsilon} \newcommand{\vphi}{\varphi} \newcommand{\sig}{\sigma} \newcommand{\lam}{\lambda} \newcommand{\gam}{\gamma} \newcommand{\om}{\omega} %%%%%%%%%%%%%%%%%%%%%%%%NUMBERING%%%%%%%%%%%%%%%%%%%%%%%% \renewcommand{\labelenumi}{(\roman{enumi})} \numberwithin{equation}{section} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \title[Long-Time Asymptotics of the Perturbed Periodic Toda Lattice]{Long-Time Asymptotics 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{\mailto{spyros@tem.uoc.gr}} \urladdr{\url{http://www.tem.uoc.gr/~spyros/}} \author[G. Teschl]{Gerald Teschl} \address{Faculty of Mathematics\\ University of Vienna\\ Nordbergstrasse 15\\ 1090 Wien\\ Austria\\ and International Erwin Schr\"odinger Institute for Mathematical Physics\\ Boltzmanngasse 9\\ 1090 Wien\\ Austria} \email{\mailto{Gerald.Teschl@univie.ac.at}} \urladdr{\url{http://www.mat.univie.ac.at/~gerald/}} \thanks{Research supported 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 long-time asymptotics of periodic (and slightly more generally of algebro-geometric finite-gap) solutions of the doubly infinite Toda 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 on 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 vector 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 decaying (fast enough) perturbation of the constant solution of a soliton equation eventually splits into a number of "solitons": localized travelling waves that preserve their shape and velocity after interaction, plus a decaying radiation part. 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. We provide the detailed analysis in the case of the Toda lattice though it is clear that our methods apply to other soliton equations as well. In the case of the Korteweg--de Vries equation the asymptotic result was first shown by \v{S}abat \cite{sh} and by Tanaka \cite{ta2}. Precise asymptotics for the radiation part were first formally derived by Zakharov and Manakov \cite{zama} and by Ablowitz and Segur \cite{as}, \cite{as2} with further extensions by Buslaev and Sukhanov \cite{bs}. A detailed rigorous justification not requiring any a priori information on the asymptotic form of the solution was first given by Deift and Zhou \cite{dz} for the case of the mKdV equation, inspired by earlier work of Manakov \cite{ma} and Its \cite{its1} (see also \cite{its2}, \cite{its3}, \cite{ip}). For further information on the history of this problem we refer to the survey by Deift, Its, and Zhou \cite{diz}. A naive guess would be that the perturbed periodic lattice approaches the unperturbed one in the uniform norm. However, as pointed out in \cite{kt} this 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 $a(n,t)$ of the Toda lattice (see \eqref{TLpert} below) 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. This will be done by formulating the inverse spectral problem as a vector Riemann--Hilbert problem on the underlying hyperelliptic curve and extending the nonlinear steepest descent method to this new setting. While Riemann--Hilbert problem on Riemann surfaces have been considered in detail before, see for example the monograph by Rodin \cite{ro}, we extend this theory as well (see e.g.\ our novel solution formula for scalar Riemann--Hilbert problems in Theorem~\ref{thmplf}). Consider the doubly infinite Toda lattice in Flaschka's variables (see e.g.\ \cite{ft}, \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. In case of a constant background the long-time asymptotics were first computed by Novokshenov and Habibullin \cite{nh} and later made rigorous by Kamvissis \cite{km} under the additional assumption that no solitons are present. The full case (with solitons) was only recently presented by Kr\"uger and Teschl in \cite{krt} (for a review see also \cite{krt2}). Here 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\in\Z} n^6 (|a(n,t) - a_q(n,t)| + |b(n,t)- b_q(n,t)|) < \infty \ee for $t=0$ and hence for all (see e.g. \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{ft}, \cite{tjac} or \cite{ta}), while the more general case we want here was solved only recently in \cite{emt2} (see also \cite{mt}). To fix our background solution, consider a hyperelliptic Riemann surface of genus $g$ with real moduli $E_0, E_1, ...., E_{2g+1}$. Choose a Dirichlet divisor $\dimuz$ and introduce \be \ulz(n,t) = \hAmap(\infty_+) - \hamap(\dimuz) - n\ul{\hat A}_{\infty_-}(\infty_+) + t\ul{U}_0 - \hvrc \in \C^g, \ee where $\Amap$ ($\amap$) is Abel's map (for divisors) and $\vrc$, $\ul{U}_0$ are some constants defined in Section~\ref{secAG}. Then our background solution is given in terms of Riemann theta functions (defined in (2.14)) 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},\\ 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} where $\ti{a}$, $\tilde{b} \in\R$ are again some constants. We can of course view this hyperelliptic Riemann surface as formed by cutting and pasting two copies of the complex plane along bands. Having this picture in mind, we denote the standard projection to the complex plane by $\pi$. Assume for simplicity that the Jacobi operator \be H(t) f(n) = a(n,t) f(n+1) + a(n-1,t) f(n-1) + b(n,t) f(n), \quad f\in\ell^2(\Z), \ee corresponding to the perturbed problem \eqref{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) \om_{\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, $\zeta_\ell$ is a canonical basis of holomorphic differentials, $\om_{\infty_+\, \infty_-}$ is an Abelian differential of the third kind defined in \eqref{ominfpm}, and $C(n/t)$ is a contour on the Riemann surface. More specific, $C(n/t)$ is obtained by taking the spectrum of the unperturbed Jacobi operator $H_q$ between $-\infty$ and a special stationary phase point $z_j(n/t)$, for the phase of the underlying Riemann--Hilbert problem defined in the beginning of Section~\ref{secSPP}, and lifting it to the Riemann surface (oriented such that the upper sheet lies to its left). The point $z_j(n/t)$ will move from $-\infty$ to $+\infty$ as $n/t$ varies from $-\infty$ to $+\infty$. 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{secSPP} of this paper. \begin{remark} (i) It is easy to see how the asymptotic formula above describes the picture given by the numerics. Recall that the spectrum $\sig(H_q)$ of $H_q$ consists of $g+1$ bands whose band edges are the branch points of the underlying hyperelliptic Riemann surface. If $\frac{n}{t}$ is small enough, $z_j(n/t)$ is to the left of all bands implying that $C(n/t)$ is empty and thus $\delta_\ell(n,t)=0$; so we recover the purely periodic lattice. At some value of $\frac{n}{t}$ a stationary phase point first appears in the first band of $\sig(H_q)$ and begins to move form the left endpoint of the band towards the right endpoint of the band. (More precisely we have a pair of stationary phase points $z_j$ and $z_j^*$, one in each sheet of the hyperelliptic curve, with common projection $\pi(z_j)$ on the complex plane.) So $\delta_\ell(n,t)$ is now a non-zero quantity changing with $\frac{n}{t}$ and the asymptotic lattice has a slowly modulated non-zero phase. Also the factor given by the exponential of the integral is non-trivially changing with $\frac{n}{t}$ and contributes to a slowly modulated amplitude. Then, after the stationary phase point leaves the first band there is a range of $\frac{n}{t}$ for which no stationary phase point appears in the spectrum $\sig(H_q)$, hence the phase shift $\delta_\ell(n,t)$ and the integral remain constant, so the asymptotic lattice is periodic (but with a non-zero phase shift). Eventually a stationary phase point appears in the second band, so a new modulation appears and so on. Finally, when $\frac{n}{t}$ is large enough, so that all bands have been traversed by the stationary phase point(s), the asymptotic lattice is again periodic. Periodicity properties of theta functions easily show that phase shift is actually cancelled by the exponential of the integral and we recover the original periodic lattice with no phase shift at all. (ii) 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. In any case, our method works unaltered for such situations (cf.\ \cite{emt3}) as well. (iii) 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 $n \to \infty$. (iv) 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 $t \to \infty$. So the above theorem is actually true in $\{(n,t): |\frac{n}{t}| < C \}$. Near the resonances we expect both a "collisionless shock" phenomenon and a Painlev\'e region to appear (\cite{dvz}, \cite{dz}, \cite{km}, \cite{km2}). A proof of this can be given using the results of \cite{dvz} and \cite{dz}. (v) For the proof of Theorem~\ref{thmMain} and Theorem~\ref{thmMain2} it would suffice to assume \eqref{decay} with $n^6$ replaced by $|n|^3$ (or even $|n|$ plus the requirement that the associated reflection coefficient is H\"older continuous). Our stronger assumption is only required for the detailed decay estimates in Theorem~\ref{thmMain3} below. \end{remark} By dividing in \eqref{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 and $\Omega_0$ is an Abelian differential of the second kind defined in \eqref{Om0}. \end{theorem} The proof of this theorem will also be given in Section~\ref{secSPP} of this paper. The next question we address here concerns the higher order asymptotics. Namely, what is the rate at which the perturbed lattice approaches the limiting lattice? Even more, what is the exact asymptotic formula? \begin{theorem}\label{thmMain3} Let $D_j$ be the sector $D_j = \{ (n,t) ,:\, z_j(n/t)\in[E_{2j}+\eps, E_{2j+1}-\eps]$ for some $\eps>0$. Then one has \be \prod_{j=n}^{\infty} \left(\frac{a(j,t)}{a_l(j,t)}\right)^2 = 1 + \sqrt{\frac{\I}{\phi''(z_j(n/t))t}} 2 \re\left( \ol{\beta(n,t)} \I \Lambda_0(n,t) \right) + O(t^{-\alpha}) \ee and \be \sum_{j=n+1}^{\infty} \big(b(j,t) - b_l(j,t) \big) = \sqrt{\frac{\I}{\phi''(z_j(n/t))t}} 2 \re\left( \ol{\beta(n,t)} \I\Lambda_1(n,t)\right) + O(t^{-\alpha}) \ee for any $\alpha<1$ uniformly in $D_j$, as $t \to \infty$. Here \be \phi''(z_j) / \I = \frac{\prod_{k=0,k\ne j}^g (z_j -z_k)}{\I\Rg{z_j}} >0, \ee (where $\phi(p,n/t)$ is the phase function defined in \eqref{defsp} and $\Rg{z}$ the square root of the underlying Riemann surface), \begin{align}\nn \Lambda_0(n,t) & = \om_{\infty_-\, \infty_+}(z_j) + \sum_{k,\ell} c_{k\ell}(\uhnu) \int_{\infty_+}^{\infty_-} \om_{\hnu_\ell(n,t),0} \zeta_k(z_j),\\ \Lambda_1(n,t) &= \om_{\infty_-,0}(z_j) - \sum_{k,\ell} c_{k\ell}(\uhnu) \om_{\hnu_\ell(n,t),0}(\infty_+) \zeta_k(z_j), \end{align} with $c_{k\ell}(\uhnu)$ some constants defined in \eqref{defcjlnu}, $\om_{q,0}$ an Abelian differential of the second kind with a second order pole at $q$ (cf.\ Remark~\ref{remabdiff2k}), \begin{align}\nn \beta =& \sqrt{\nu} \E^{\I(\pi/4-\arg(R(z_j)))+\arg(\Gamma(\I\nu))-2 \nu \alpha(z_j))} \left(\frac{\phi''(z_j)}{\I}\right)^{\I\nu} \E^{-t \phi(z_j)} t^{-\I\nu} \times\\ \nn & \times \frac{\theta(\ulz(z_j,n,t)+ \ul{\delta}(n,t))}{\theta(\ulz(z_j,0,0))} \frac{\theta(\ulz(z_j^*,0,0))}{\theta(\ulz(z_j^*,n,t)+ \ul{\delta}(n,t))} \times\\ & \times \exp\left( \frac{1}{2\pi\I} \int_{C(n/t)} \log\left(\frac{1-|R|^2}{1-|R(z_j)|^2}\right) \om_{p\, p^*}\right), \end{align} where $\Gamma(z)$ is the gamma function, \be \nu = - \frac{1}{2\pi} \log (1-|R(z_j)|^2)>0, \ee and $\alpha(z_j)$ is a constant defined in \eqref{def:alzj}. \end{theorem} The proof of this theorem will be given in Section~\ref{secCROSS} of this paper. The idea of the proof is that even when a Riemann-Hilbert problem needs to be considered on an algebraic variety, a localized parametrix Riemann-Hilbert problem need only be solved in the complex plane and the local solution can then be glued to the global Riemann-Hilbert solution on the variety. The same idea can produce the asymptotics in the two resonance regions mentioned above: a "collisionless shock" phenomenon and a Painlev\'e region, for every resonance pint $E_s$, by simply using the results of (\cite{dvz}, \cite{dz}). We leave the details to the reader. \begin{remark} (i) The current work combines two articles that have appeared previously in the arXiv as arXiv:0705.0346 and arXiv:0805.3847 but have not been published otherwise. The necessary changes needed to include solitons are given in \cite{krt3} which was based on arXiv:0705.0346 (see also \cite{emt4}, \cite{krt}, and \cite{tag}). (ii) Combining our technique with the one from \cite{dz2} can lead to a complete asymptotic expansion. (iii) Finally, we note that the same proof works even if there are different spatial asymptotics as $n\to\pm\infty$ as long as they lie in the same isospectral class (cf.\ \cite{emt3}). \end{remark} \section{Algebro-geometric quasi-periodic finite-gap solutions} \label{secAG} 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}, \cite{ghmt}, \cite{tjac}. 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 $E_0$, $E_{2j-1}$ counterclockwise 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 \be\label{defcjk} 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. \ee 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{j+ \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 $\theta(\ulz(p,n,t))$ has precisely $g$ zeros $\hmu_j(n,t)$ (with $\hmu_j(0,0)=\hmu_j$), 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}\label{defpsiq} \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} \om_{\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 \begin{align}\nn H_q(t) \psi_q(p,n,t) &= \pi(p) \psi_q(p,n,t),\\ \frac{d}{dt} \psi_q(p,n,t) &= P_{q,2}(t) \psi_q(p,n,t), \end{align} where \begin{align} H_q(t) f(n) &= a_q(n,t) f(n+1) + a_q(n-1,t) f(n-1) + b_q(n,t) f(n),\\ P_{q,2}(t) f(n) &= a_q(n,t) f(n+1) - a_q(n-1,t) f(n-1) \end{align} 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 \be a(n,t) \psi_\pm(z,n+1,t) + a(n-1,t) \psi_\pm(z,n-1,t) + b(n,t) \psi_\pm(z,n,t) =z \psi_\pm(z,n,t) \ee defined by the asymptotic normalization \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} \om_{\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) =& \, \psi_{q,\pm}(z,0,t) \frac{z^{\mp n} \Big(\prod_{j=0}^{n-1} a_q(j,t)\Big)^{\pm 1}}{A_\pm(n,t)} \times\\ \label{asympsipm} & \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 where $T(z)$, $R_\pm(z)$ are the transmission respectively reflection coefficients. 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 $\hat{\mu}_j(0,0)$ 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 \eqref{relscat} and \eqref{reltrpm}, we obtain \be m_+(p,n,t) = m_-(p,n,t) \begin{pmatrix} 1-|R(p)|^2 & -\ol{R(p)} \\ R(p) & 1 \end{pmatrix}, \ee where we have extended our definition of $R$ to $\Sigma$ such that it is equal to $R(z)$ on $\Sigma_+$ and equal to $\ol{R(z)}$ on $\Sigma_-$. 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 \be\label{defTheta} \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))} \ee and \be\label{defsp} \phi(p,\frac{n}{t}) = 2 \int_{E_0}^p \Omega_0 + 2 \frac{n}{t} \int_{E_0}^p \om_{\infty_+\, \infty_-} \in \I \R \ee for $p\in\Sigma$. Note \[ \frac{\psi_q(p,n,t)}{\psi_q(p^*,n,t)} = \Theta(p,n,t) \E^{t\phi(p)}. \] Observe that \[ m^2(p) = \ol{m^2(\ol{p})} \] and \[ m^2(p^*) = m^2(\ol{p}) \sigI, \] which follow directly from the definition \eqref{defm2}. They are related to the symmetries \[ J^2(p)=\ol{J^2(\ol{p})} \quad\text{and}\quad J^2(p) = \sigI J^2(p^*)^{-1} \sigI. \] Now we come to the normalization condition at $\infty_+$. To this end note \be \label{m2infp} m(p,n,t) = \begin{pmatrix} A_+(n,t) (1 - B_+(n-1,t) \frac{1}{z}) & \frac{1}{A_+(n,t)}(1 + B_+(n,t) \frac{1}{z} ) \end{pmatrix} + O(\frac{1}{z^2}), \ee for $p=(z,+)\to\infty_+$, with $A_\pm(n,t)$ and $B_\pm(n,t)$ are defined in \eqref{defABpm}. The formula near $\infty_-$ follows by flipping the columns. Here we have used \be T(z) = A_-(n,t) A_+(n,t) \Big( 1 - \frac{B_+(n,t) + b_q(n,t) - b(n,t) + B_-(n,t)}{z} + O(\frac{1}{z^2})\Big). \ee Using the properties of $\psi(p,n,t)$ and $\psi_q(p,n,t)$ one checks that its divisor satisfies \be (m_1) \ge -\dimus{n,t}, \qquad (m_2) \ge -\dimu{n,t}. \ee Next we show how to normalize the problem at infinity. The use of the above symmetries is necessary and it makes essential use of the second sheet of the Riemann surface (see also the Conclusion of this paper). \begin{theorem}\label{thmm3} The function \be m^3(p)= \frac{1}{A_+(n,t)} m^2(p,n,t) \ee with $m^2(p,n,t)$ defined in \eqref{defm2} is meromorphic away from $\Sigma$ and satisfies: \begin{align}\nn & m^3_+(p) = m^3_-(p) J^3(p), \quad p\in\Sigma,\\ \label{rhpm3} & (m^3_1) \ge -\dimus{n,t}, \quad (m^3_2) \ge -\dimu{n,t},\\ \nn & m^3(p^*) = m^3(p) \sigI \\ & m^3(\infty_+) = \begin{pmatrix} 1 & * \end{pmatrix}, \end{align} where the jump is given by \be\label{rhpm3.1} J^3(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}. \ee \end{theorem} Setting $R(z)\equiv 0$ we clearly recover the purely periodic solution, as we should. Moreover, note \be \label{asymm3} m^3(p) = \begin{pmatrix} \frac{1}{A_+(n,t)^2} & 1 \end{pmatrix} + \begin{pmatrix} \frac{B_+(n,t)}{A_+(n,t)^2} & - B_+(n-1,t) \end{pmatrix} \frac{1}{z} + O(\frac{1}{z^2}). \ee for $p=(z,-)$ near $\infty_-$. While existence of a solution follows by construction, uniqueness follows from Theorem~\ref{thmrhpQ} and Remark~\ref{remrhpQ}. \begin{theorem} \label{thmuniq} The solution of the Riemann--Hilbert problem of Theorem~\ref{thmm3} is unique. \end{theorem} \section{The stationary phase points and corresponding contour deformations} \label{secSPP} The phase in the factorization problem \eqref{rhpm2.1} is $t\, \phi$ where $\phi$ was defined in \eqref{defsp}. Invoking \eqref{ominfpm} and \eqref{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 $\lam_j$, $1\le j \le g$, 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 $\Omega_0 + \eta \om_{\infty_+\, \infty_-}$ there is at least one stationary phase point in each gap and they are all different. Furthermore, \[ z_j' = - \frac{q(z_j)}{\ti{q}'(z_j)+ \eta q'(z_j)} = -\frac{\prod_{k=1}^g (z_j -\lam_k)}{\prod_{k=0,k\ne j}^g z_j-z_k}, \] where \[ \ti{q}(z) =\prod_{k=0}^g (z -\tilde\lam_k), \quad q(z)= \prod_{k=1}^g (z -\lam_k). \] Since the points $\lam_k$ are fixed points of this ordinary first order differential equation (note that the denominator cannot vanish since the $z_j$'s are always different), the numbers $z_j$ cannot cross these points. Combining the behavior as $\eta\to\pm\infty$ with the fact that there must always be at least one of them in each gap, we conclude that $z_j$ must stay between $\lam_j$ and $\lam_{j+1}$. This also shows $z_j' <0$ and thus $z_j(\eta)$ is monotone decreasing. \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} \subsection*{Case (i)} Note that in this case \be\label{phipp} \phi''(z_j) / \I = \frac{\prod_{k=0,k\ne j}^g (z_j -z_k)}{\I\Rg{z_j}} >0. \ee Let us 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 \[ \re(\phi) < (>) 0, \quad \text{in } D_k, \quad k > (<) j. \] Now observe that our jump condition \eqref{rhpm3.1} has the following important factorization \be J^3=(b_-)^{-1} b_+, \ee where \[ b_- =\begin{pmatrix} 1 & \ol{ R \Theta} \E^{-t\, \phi} \\ 0 &1 \end{pmatrix}, \qquad b_+= \begin{pmatrix} 1 & 0 \\ R \Theta \E^{t\, \phi} & 1 \end{pmatrix}. \] 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 \[ B_- =\begin{pmatrix} 1 & 0 \\ - \frac{R \Theta \E^{t\, \phi}}{1-|R|^2} &1 \end{pmatrix}, \qquad B_+= \begin{pmatrix} 1 & -\frac{\ol{ R \Theta} \E^{-t\, \phi}}{1-|R|^2} \\ 0 & 1 \end{pmatrix}. \] This is the right factorization for $z0. \ee Here $(z-z_j)^{\pm\I\nu}=\exp(\pm\I\nu\log(z-z_j))$, where the branch cut of the logarithm is along the negative real axis. 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 we first rewrite \eqref{defd} as \begin{align}\nn d(p,n,t) = & \exp\left( \I\nu \int_{C(n/t)} \om_{p\,\infty_+}\right) \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\\ \label{eq:dnearzj} & \times \exp\left( \frac{1}{2\pi\I} \int_{C(n/t)} \log\left(\frac{1-|R|^2}{1-|R(z_j)|^2}\right) \om_{p\,\infty_+}\right). \end{align} Next observe \be \frac{1}{2} \int_{C(n/t)} \om_{p\, p^*} = \pm\log(z-z_j) \pm \alpha(z_j) + O(z-z_j), \quad p=(z,\pm), \ee where $\alpha(z_j)\in\R$, and hence \be \int_{C(n/t)} \om_{p\,\infty_+} = \pm\log(z-z_j) \pm \alpha(z_j) + \frac{1}{2} \int_{C(n/t)} \om_{\infty_-\,\infty_+} +O(z-z_j), \quad p=(z,\pm), \ee from which the first claim follows. For the second claim note that \[ t(p) = \frac{1}{T(\infty)} \begin{cases} T(z), & p =(z,+)\in\Pi_+,\\ T(z)^{-1}, & p=(z,-)\in\Pi_-,\end{cases} \] 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} Furthermore, \begin{lemma} We have \be e^{\pm}(z) = \ol{e^\mp(z)}, \qquad p=(z,\pm)\in\Sigma\backslash C(n/t), \ee and \begin{align}\nn e^+(z_j) = & \exp\left( \I\nu \alpha(z_j) + \frac{\I\nu}{2} \int_{C(n/t)} \om_{\infty_-\,\infty_+}\right) \times\\ \nn & \times \frac{\theta(\ulz(n,t)+\ul{\delta}(n,t))}{\theta(\ulz(n,t))} \frac{\theta(\ulz(z_j,n,t))}{\theta(\ulz(z_j,n,t)+ \ul{\delta}(n,t))} \times\\ & \times \exp\left( \frac{1}{2\pi\I} \int_{C(n/t)} \log\left(\frac{1-|R|^2}{1-|R(z_j)|^2}\right) \left(\om_{z_j\, z_j^*} + \om_{\infty_-\,\infty_+}\right)\right), \end{align} where \be\label{def:alzj} \alpha(z_j) = \lim_{p\to z_j} \frac{1}{2} \int_{C(n/t)} \om_{p\, p^*} - \log(\pi(p)-z_j). \ee Here $\alpha(z_j)\in\R$ and $\om_{p\, p^*}$ is real whereas $\om_{\infty_-\,\infty_+}$ is purely imaginary on $C(n/t)$. \end{lemma} \begin{proof} The first claim follows since $d(p^*) = d(\ol{p}) = \ol{d(p)}$ for $p\in\Sigma\backslash C(n/t)$. The second claim follows from \eqref{eq:dnearzj} using $\int_{C(n/t)} f\, \om_{p\,\infty_+} = \frac{1}{2} \int_{C(n/t)} f\, (\om_{p\, p} + \om_{\infty_-\,\infty_+})$ for symmetric functions $f(q)=f(q^*)$. \end{proof} Having solved the scalar problem above for $d$ we can introduce the new Riemann--Hilbert problem \be \label{defm4} m^4(p) = d(\infty_-)^{-1} 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_1) \ge -\dinus{n,t}, \quad (m^4_2) \ge -\dinu{n,t},\\ \nn & m^4(p^*) = m^4(p) \sigI \\ \nn & m^4(\infty_+) = \begin{pmatrix} 1 & * \end{pmatrix}, \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 $\hat\nu_j(n,t)$. 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} R \Theta \E^{t\, \phi} & 1 \end{pmatrix} = (\ti{b}_-)^{-1} \ti{b}_+, \quad p\in\Sigma\setminus C(n/t), \ee where $\ti{b}_\pm = D^{-1} b_\pm D$, that is, \be\label{defbt} \ti{b}_- = \begin{pmatrix} 1 & \frac{d}{d^*}\ol{R \Theta} \E^{-t\, \phi} \\ 0 & 1 \end{pmatrix},\qquad \ti{b}_+ = \begin{pmatrix} 1 & 0 \\ \frac{d^*}{d} R \Theta \E^{t\, \phi} & 1 \end{pmatrix}, \ee 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_+} R \Theta \E^{t\, \phi} & 1-|R|^2 \end{pmatrix} = (\ti{B}_-)^{-1} \ti{B}_+, \quad p\in C(n/t), \ee where $\ti{B}_\pm = D_\pm^{-1} B_\pm D_\pm$, that is, \be\label{defBt} \ti{B}_- = \begin{pmatrix} 1 & 0 \\ -\frac{d^*_-}{d_-} \frac{R \Theta}{1-|R|^2} \E^{t\, \phi} & 1 \end{pmatrix},\qquad \ti{B}_+ = \begin{pmatrix} 1 & - \frac{d_+}{d^*_+} \frac{\ol{R \Theta}}{1-|R|^2} \E^{-t\, \phi} \\ 0 & 1 \end{pmatrix}, \ee 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. 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 in the spirit of \cite{dz}. We will provide the details in Section~\ref{secanap}. The crucial observation now is that the jumps $J^5$ on the oriented paths $C_k$, $C_k^*$ 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 $t \to \infty$) implying that the solution should asymptotically be given by the constant vector $\rI$. That this can in fact be done will be shown in the next section by explicitly computing the contribution of the stationary phase points thereby showing that they are of the order $O(t^{-1/2})$, that is, \[ m^5(p) = \rI + O(t^{-1/2}) \] uniformly for $p$ a way from the jump contour. Hence all which remains to be done to prove Theorem~\ref{thmMain} and Theorem~\ref{thmMain2} is to trace back the definitions of $m^4$ and $m^3$ and comparing with \eqref{asymm3}. First of all, since $m^5$ and $m^4$ coincide near $\infty_-$ we have \[ m^4(p) = \rI + O(t^{-1/2}) \] uniformly for $p$ in a neighborhood of $\infty_-$. Consequently, by the definition of $m^4$ from \eqref{defm4}, we have \[ m^3(p) = d(\infty_-) \begin{pmatrix} d(p^*)^{-1} & d(p)^{-1} \end{pmatrix} + O(t^{-1/2}) \] again uniformly for $p$ in a neighborhood of $\infty_-$. Finally, comparing this last identity with \eqref{asymm3} shows \be A_+(n,t)^2 = d(\infty_-,n,t) + O(t^{-1/2}), \quad B_+(n,t) = - d_1(n,t) + O(t^{-1/2}), \ee where $d_1$ is defined via \[ d(p) = 1 + \frac{d_1}{z} + O(\frac{1}{z^2}), \qquad p=(z,+) \text{ near } \infty_+ \] 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 \eqref{Om0}. \subsection*{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 expects different local asymptotics expressed in terms of 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 an extra "collisionless shock" phenomenon (on top of the Painlev\'e 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 ${R \over { 1-|R|^2}}$. An appropriate "local" Riemann--Hilbert problem however is still explicitly solvable and the actual contribution of the band edges is similar to the free case. All this can be studied as in Section~\ref{secCROSS} (see also our discussion of this in the Introduction). But in the present work, we will assume that the stationary phase points stay away from the $E_k$. \subsection*{Case (iii)} In the case where no stationary phase points lie in the spectrum the situation is similar to the case (i). In fact, it is much simpler since there is no contribution from the stationary phase points: 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 \eqref{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 \eqref{defdinfm} is still true. \begin{remark} We have asymptotically reduced our Riemann--Hilbert problem to one defined on two small crosses. If we are only interested in showing that the contribution of these crosses is small (i.e that the solution of the Riemann--Hilbert problem is uniformly small for large times) we can evoke the existence theorem in the second appendix as well as some rescaling argument. Since we are interested in actually computing the higher order asymptotic term, a more detailed analysis of the local parametrix Riemann--Hilbert problem is required. \end{remark} \section{The "local" Riemann--Hilbert problems on the small crosses} \label{secCROSS} In the previous section we have shown how the long-time asymptotics can be read off from the Riemann--Hilbert problem \begin{align}\nn & m^5_+(p,n,t) = m^5_-(p,n,t) J^5(p,n,t), \quad p\in\Sigma^5,\\ \nn & (m^5_1) \ge -\dinus{n,t}, \quad (m^5_2) \ge -\dinu{n,t},\\ \nn & m^5(p^*,n,t) = m^5(p,n,t) \sigI \\ & m^5(\infty_+,n,t) = \begin{pmatrix} 1 & * \end{pmatrix}. \end{align} In this section we are interested in the actual asymptotic rate at which $m^5(p) \to \begin{pmatrix} 1 & 1 \end{pmatrix}$. We have already seen in the previous section that the jumps $J^5$ on the oriented paths $C_k$, $C_k^*$ for $k\ne j$ are of the form $\id+ exponentially~small$ asymptotically as $t \to \infty$. The same is true for the oriented paths $C_{j1}, C_{j2}, C_{j1}^*, C_{j2}^*$ at least away from the stationary phase points $z_j$, $z^*_j$. On these paths, and in particular near the stationary phase points (see Figure~\ref{fig4}), \begin{figure} \begin{picture}(10,6) \put(1.4,5.4){$C_{j1}$}% $\ti B_+$ \put(8.0,5.4){$C_{j2}$} %$\ti b_-^{-1}$ \put(1.4,3.4){$C_{j1}^*$} %$\ti B_-^{-1}$ \put(8.0,3.4){$C_{j2}^*$} %$b_+$ \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(2,4.5){\line(1,0){6}} \put(5,4.5){\curve(-2.645, 0.951, 0, 0)} \put(5,4.5){\curve(2.645, 0.951, 0, 0)} \put(5,4.5){\curve(1, 0, 0.7, 0.7, 0, 1, -0.7, 0.7, -1, 0)} \curvedashes{0.05,0.05} \put(5,4.5){\curve(0, 0, -2.645, -0.951)} \put(5,4.5){\curve(0, 0, 2.645, -0.951)} \put(5,4.5){\curve(1, 0, 0.7, -0.7, 0, -1, -0.7, -0.7, -1, 0)} \curvedashes{} %%%%% \put(1.4,2.3){$C_{j1}^*$} %$\ti b_+$ \put(8.0,2.3){$C_{j2}^*$} % $\ti B_-^{-1}$ \put(1.4,0.3){$C_{j1}$} % $\ti B_+$ \put(8.0,0.3){$C_{j2}$} % $\ti b_-^{-1}$ \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(5,1.5){\curve(0, 0, 2.645, -0.951)} \put(5,1.5){\curve(0, 0, -2.645, -0.951)} \put(5,1.5){\curve(1, 0, 0.7, -0.7, 0, -1, -0.7, -0.7, -1, 0)} \curvedashes{0.05,0.05} \put(2,1.5){\curve(0,0,6,0)} \put(5,1.5){\curve(1, 0, 0.7, 0.7, 0, 1, -0.7, 0.7, -1, 0)} \put(5,1.5){\curve(2.645, 0.951, 0, 0)} \put(5,1.5){\curve(-2.645, 0.951, 0, 0)} \end{picture} \caption{The small cross containing the 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{fig4} \end{figure} the jumps read {\allowdisplaybreaks \begin{align}\nn J^5 &= \ti{B}_+ = \begin{pmatrix} 1 & - \frac{d}{d^*} \frac{R^* \Theta^*}{1-R^* R} \E^{-t\, \phi} \\ 0 & 1 \end{pmatrix}, \quad p \in C_{j1}, \\ \nn J^5 &= \ti{B}_-^{-1} = \begin{pmatrix} 1 & 0 \\ \frac{d^*}{d} \frac{R \Theta}{1-R^* R} \E^{t\, \phi} & 1 \end{pmatrix}, \quad p \in C_{j1}^{*}, \\ \nn J^5 &= \ti{b}_+ = \begin{pmatrix} 1 & 0 \\ \frac{d^*}{d} R \Theta \E^{t\, \phi} & 1 \end{pmatrix}, \quad p \in C_{j2}, \\ J^5 &= \ti{b}_-^{-1} = \begin{pmatrix} 1 & -\frac{d}{d^*} R^* \Theta^* \E^{-t\, \phi} \\ 0 & 1 \end{pmatrix}, \quad p \in C_{j2}^*. \end{align}}% Note that near the stationary phase points the jumps are given by (cf.\ Lemma~\ref{lemd}) {\allowdisplaybreaks \begin{align}\nn \hat{B}_+ & = \begin{pmatrix} 1 & - \left(\sqrt{\frac{\phi''(z_j)}{\I}}(z-z_j)\right)^{2\I\nu} \frac{\ol{r}}{1-|r|^2} \E^{-t\, \phi} \\ 0 & 1 \end{pmatrix}, \quad p \in L_{j1}, \\ \nn \hat{B}_-^{-1} & = \begin{pmatrix} 1 & 0 \\ \left(\sqrt{\frac{\phi''(z_j)}{\I}}(z-z_j)\right)^{-2\I\nu} \frac{r}{1-|r|^2} \E^{t\, \phi} & 1 \end{pmatrix}, \quad p \in L_{j1}^{*}, \\ \hat{b}_+ & = \begin{pmatrix} 1 & 0 \\ \nn \left(\sqrt{\frac{\phi''(z_j)}{\I}}(z-z_j)\right)^{-2\I\nu} r \E^{t\, \phi} & 1 \end{pmatrix}, \quad p \in L_{j2}, \\ \hat{b}_-^{-1} &= \begin{pmatrix} 1 & -\left(\sqrt{\frac{\phi''(z_j)}{\I}}(z-z_j)\right)^{2\I\nu} \ol{r} \E^{-t\, \phi} \\ 0 & 1 \end{pmatrix}, \quad p \in L_{j2}^*, \end{align}}% where (cf.\ \eqref{defTheta} and \eqref{defepm}) \be\label{defr} r=R(z_j) \Theta(z_j,n,t) \frac{\ol{e^+(z_j)}}{e^+(z_j)} \left(\frac{\phi''(z_j)}{\I}\right)^{\I\nu}. \ee Since the reflection coefficients are continuously differentiable by our decay assumption \eqref{decay} and by Lemma~\ref{lemd} the error terms will satisfy appropriate H\"older estimates, that is \be \| \ti{B}_+(p) - \hat{B}_+(p)\| \le C |z-z_j|^\alpha, \qquad p=(z,+) \in C_{j1}, \ee for any $\alpha<1$ and similarly for the other matrices. To reduce our Riemann--Hilbert problem to the one corresponding to the two crosses we proceed as follows: We take a small disc $D$ around $z_j(n/t)$ and project it to the complex plane using the canonical projection $\pi$. Now consider the (holomorphic) matrix Riemann--Hilbert problem {\em in the complex plane} with the very jump obtained by projection and normalize it to be $\id$ near $\infty$. Denote this solution by $M(z)$. Then, as is shown in \cite{dz} (see also \cite[Thm.~A.1]{krt2}), the solution of this matrix Riemann--Hilbert problem on a small cross in the complex plane is asymptotically of the form \be M(z) = \id + \frac{M_0}{z-z_j} \frac{1}{t^{1/2}} + O(t^{-\alpha}), \ee for any $\alpha<1$ and $z$ outside a neighborhood of $z_j$, where \begin{align} \nn M_0 &= \I \sqrt{\I/\phi''(z_j)} \begin{pmatrix} 0 & -\beta(t)\\ \ol{\beta(t)} & 0 \end{pmatrix},\\ \beta(t) &= \sqrt{\nu} \E^{\I(\pi/4-\arg(r)+\arg(\Gamma(\I\nu)))} \E^{-\I t \phi(z_j)} t^{-\I\nu}. \end{align} Now we lift this solution back to the small disc on our Riemann-surface by setting $M(p)=M(z)$ for $p\in D$ and $M(p)=\ol{M(\ol{z})}$ for $p\in D^*$. We define \be m^6(p) = \begin{cases} m^5(p) M^{-1}(p), & p\in D \cup D^*\\ m^5(p), & \text{else}. \end{cases} \ee \\ Note that $m^6$ has no jump inside $D\cup D^*$. Its jumps on the boundary are given by \be m^6_+(p) = m^6_-(p) M^{-1}(p), \qquad p\in\partial D \cup \partial D^* \ee and the remaining jumps are unchanged. In summary, all jumps outside $D\cup D^*$ are of the form $\id+ exponentially~small$ and the jump on $\partial D \cup \partial D^*$ is of the form $\id + O(t^{-1/2})$. In order to identify the leading behaviour it remains to rewrite the Riemann--Hilbert problem for $m^6$ as a singular integral equation following Section~\ref{secSIE}. Let the operator $C_{w^6}: L^2(\Sigma^6) \to L^2(\Sigma^6) $ be defined by \be C_{w^6} f = C_- (f w^6) \ee for a vector valued $f$, where $w^6 = J^6 - \id$ and \be (C_\pm f)(q) = \lim_{p \to q \in \Sigma^6} \frac{1}{2 \pi \I} \int_{\Sigma^6} f\, \ul{\Omega}_p^\uhnuz, \qquad \ul{\Omega}_p^\uhnuz = \begin{pmatrix} \Omega_p^{\uhnuz^*,\infty_+} & 0 \\ 0 & \Omega_p^{\uhnuz,\infty_-} \end{pmatrix}, \ee are the Cauchy operators for our Riemann surface. In particular, $\Omega_p^{\uhnuz,q}$ is the Cauchy kernel given by \be\label{defOm} \Omega_p^{\uhnuz,q} = \om_{p\, q} + \sum_{j=1}^g I_j^{\uhnuz,q}(p) \zeta_j, \ee where \be I_j^{\uhnuz,q}(p) = \sum_{\ell=1}^g c_{j\ell}(\uhnuz) \int_q^p \om_{\hnu_\ell,0}. \ee Here $\om_{q,0}$ is the (normalized) Abelian differential of the second kind with a second order pole at $q$ (cf.\ Remark~\ref{remabdiff2k} below). Note that $I_j^{\uhnuz,q}(p)$ has first order poles at the points $\uhnuz$. The constants $c_{j\ell}(\uhnuz)$ are chosen such that $\Omega_p^{\uhnuz,q}$ is single valued, that is, \be\label{defcjlnu} \left( c_{\ell k}(\uhnuz) \right)_{1 \le \ell,k \le g} = \left( \sum_{j=1}^g c_k(j) \frac{\mu_\ell^{j-1}d\pi}{R_{2g+2}^{1/2}(\hmu_\ell)} \right)_{1 \le \ell,k \le g}^{-1} \ee where $c_k(j)$ are defined in \eqref{defcjk} (cf.\ Lemma~\ref{lemck}). \begin{remark}\label{remabdiff2k} Note that the Abelian integral appearing in the previous lemma is explicitly given by \be \om_{\infty_-,0} = \frac{-\pi^{g+1} + \frac{1}{2} \sum_{j=0}^{2g+1} E_j \pi^g + P_{\infty_-,0}(\pi) + R_{2g+2}^{1/2}}{R_{2g+2}^{1/2}}d\pi, \ee with $P_{\infty_-,0}$ a polynomial of degree $g-1$ which has to be determined from the normalization. Similarly, \be \om_{\hat\nu,0} = \frac{R_{2g+2}^{1/2} + R_{2g+2}^{1/2}(\hat\nu) + \frac{R_{2g+2}'(\hat\nu)}{2R_{2g+2}^{1/2}(\hat\nu)} (\pi-\nu) + P_{\hat\nu,0}(\pi) \cdot (\pi-\nu)^2 }{2 (\pi-\nu)^2 R_{2g+2}^{1/2}}d\pi, \ee with $P_{\hat\nu,0}$ a polynomial of degree $g-1$ which has to be determined from the normalization. \end{remark} Consider the solution $\mu^6$ of the singular integral equation \be %\label{musie} \mu = \rI + C_{w^6} \mu \quad\text{ in }\quad L^2(\Sigma^6). \ee Then the solution of our Riemann--Hilbert problem is given by \be m^6(p) = \rI + \frac{1}{2\pi\I} \int_{\Sigma^6} \mu^6 \, w^6 \, \ul{\Omega}_{p}^\uhnuz. \ee Since $\|w^6\|_\infty = O(t^{-1/2})$ Neumann's formula implies \be \mu^6(q) = (\id - C_{w^6})^{-1} \rI = \rI + O(t^{-1/2}). \ee Moreover, \be w^6(p) = \begin{cases} -\frac{M_0}{z-z_j} \frac{1}{t^{1/2}} + O(t^{-\alpha}), & p \in \partial D,\\ -\frac{\ol{M_0}}{z-z_j} \frac{1}{t^{1/2}} + O(t^{-\alpha}), & p \in \partial D^*. \end{cases} \ee Hence we obtain \begin{align}\nn m^6(p) = & \rI - \frac{\rI M_0}{t^{1/2}} \frac{1}{2\pi\I} \int_{\partial D} \frac{1}{\pi-z_j} \, \ul{\Omega}_{p}^\uhnuz\\ \nn & - \frac{\rI \ol{M_0}}{t^{1/2}} \frac{1}{2\pi\I} \int_{\partial D^*} \frac{1}{\pi-z_j} \, \ul{\Omega}_{p}^\uhnuz + O(t^{-\alpha})\\ \nn = & \rI - \frac{\rI M_0}{t^{1/2}} \ul{\Omega}_{p}^\uhnuz(z_j) - \frac{\rI \ol{M_0}}{t^{1/2}} \ul{\Omega}_{p}^\uhnuz(z_j^*) + O(t^{-\alpha})\\ \nn = & \rI\\\nn &- \sqrt{\frac{\I}{\phi''(z_j)t}} \begin{pmatrix} \I \ol{\beta} \Omega_p^{\uhnuz^*,\infty_+}(z_j) - \I \beta\Omega_p^{\uhnuz^*,\infty_+}(z_j^*) & - \I \beta \Omega_p^{\uhnuz,\infty_-}(z_j) + \I \ol{\beta} \Omega_p^{\uhnuz,\infty_-}(z_j^*) \end{pmatrix}\\ & + O(t^{-\alpha}). \end{align} Note that the right-hand side is real-valued for $p\in\pi^{-1}(\R)\backslash\Sigma$ since $\ol{\Omega_p^{\uhnuz,\infty_\pm}(\ol{q})} = \Omega_{\ol{p}}^{\uhnuz,\infty_\pm}(q)$ implies \be \Omega_p^{\uhnuz,\infty_\pm}(z_j^*) = \ol{\Omega_p^{\uhnuz,\infty_\pm}(z_j)}, \qquad p\in\pi^{-1}(\R)\backslash\Sigma. \ee Since we need the asymptotic expansions around $\infty_-$ we note \begin{lemma} We have \be \Omega_p^{\uhnuz,\infty_+}(z_j)= \Lambda^\uhnuz_0 + \Lambda^\uhnuz_1 \frac{1}{z} + O(\frac{1}{z^2}) \ee for $p=(z,-)$ near $\infty_-$, where \be \Lambda^\uhnuz_0= \Omega_{\infty_-}^{\uhnuz,\infty_+}(z_j) = \Omega_{\infty_-}^{\uhnuz^*,\infty_+}(z_j) = \om_{\infty_-\, \infty_+}(z_j) + \sum_{k,\ell} c_{k\ell}(\uhnuz) \int_{\infty_+}^{\infty_-} \om_{\hnu_\ell,0} \zeta_k(z_j) \ee and \begin{align}\nn \Lambda^\uhnuz_1 &= \om_{\infty_-,0}(z_j) + \sum_{k,\ell} c_{k\ell}(\uhnuz) \om_{\hnu_\ell,0}(\infty_-) \zeta_k(z_j)\\ &= \om_{\infty_-,0}(z_j) - \sum_{k,\ell} c_{k\ell}(\uhnuz^*) \om_{\hnu_\ell^*,0}(\infty_+) \zeta_k(z_j). \end{align} \end{lemma} \begin{proof} To see $\Omega_{\infty_-}^\uhnuz(z_j) = \Omega_{\infty_-}^{\uhnuz^*}(z_j)$ note $c_{k\ell}(\uhnuz^*)= - c_{k\ell}(\uhnuz)$ and $\int_{\infty_+}^{\infty_-} \om_{\hnu_\ell^*,0}= \int_{\infty_-}^{\infty_+} \om_{\hnu_\ell,0}$. \end{proof} Observe that since $c_{k\ell}(\uhnuz)\in\R$ and $\int_{\infty_+}^{\infty_-} \om_{\hnu_\ell,0}\in\R$ we have $\Lambda^\uhnuz_0\in \I\R$. As in the previous section, the asymptotics can be read off by using \be m^3(p) = d(\infty_-) m^6(p) \begin{pmatrix} \frac{1}{d(p^*)} & 0 \\ 0 & \frac{1}{d(p)} \end{pmatrix} \ee for $p$ near $\infty_-$ and comparing with \eqref{asymm3}. We obtain \be A_+(n,t)^2 = \frac{1}{d(\infty_-)} \left( 1 + \sqrt{\frac{\I}{\phi''(z_j)t}} \left(\I\ol{\beta} \Lambda^\uhnuz_0 - \I\beta \ol{\Lambda^\uhnuz_0} \right)\right) + O(t^{-\alpha}) \ee and \be B_+(n,t) = - d_1 - \sqrt{\frac{\I}{\phi''(z_j)t}} \left(\I\ol{\beta} \Lambda^{\uhnuz^*}_1 - \I\beta \ol{\Lambda^{\uhnuz^*}_1} \right) + O(t^{-\alpha}), \ee for any $\alpha<1$. Theorem~\ref{thmMain3} and hence also Theorems~\ref{thmMain} and \ref{thmMain2} are now proved under the assumption that $R(p)$ admits an analytic extension (which will be true if in our decay assumption \eqref{decay} the weight $n^6$ is replaced by $\exp(-\eps |n|)$ for some $\eps>0$) to be able to make our contour deformations. We will show how to get rid of this assumption by analytic approximation in the next section. Summarizing, let us emphasize that the general significance of the method developed in this section is this: even when a Riemann-Hilbert problem needs to be considered on an algebraic variety, a localized parametrix Riemann-Hilbert problem need only be solved in the complex plane and the local solution can then be glued to the global Riemann-Hilbert solution on the variety. After this gluing procedure the resulting Riemann-Hilbert problem on the variety is asymptotically small and can be solved asymptotically (on the variety) by virtue of the associated singular integral equations. The method described in this section can thus provide the higher order asymptotics {\em also} in the collisonless shock and Painlev\'e regions mentioned in the Introduction, by using existing results in (\cite{dvz}, \cite{dz}). \section{Analytic Approximation} \label{secanap} In this section we want to show how to get rid of the analyticity assumption on the reflection coefficient $R(p)$. To this end we will split $R(p)$ into an analytic part $R_{a,t}$ plus a {\em small} residual term $R_{r,t}$ following the ideas of \cite{dz} (see also \cite[Sect.~6]{krt2}). The analytic part will be moved to regions of the Riemann surface while the residual term remains on $\Sigma=\pi^{-1}\big( \sigma(H_q)\big)$. This needs to be done in such a way that the residual term is of $O(t^{-1})$ and the growth of the analytic part can be controlled by the decay of the phase. In order to avoid problems when one of the poles $\nu_j$ hits $\Sigma$, we have to make the approximation in such a way that the nonanalytic residual term vanishes at the band edges. That is, split $R$ according to \begin{align} \nn R(p) =& R(E_{2j}) \frac{z-E_{2j}}{E_{2j+1}-E_{2j}} + R(E_{2j+1}) \frac{z-E_{2j+1}}{E_{2j}-E_{2j+1}}\\ \label{splitR} & \pm \sqrt{z-E_{2j}} \sqrt{z-E_{2j+1}} \ti{R}(p), \qquad p=(z,\pm), \end{align} and approximate $\ti{R}$. Note that if $R \in C^l(\Sigma)$, then $\ti{R} \in C^{l-1}(\Sigma)$. We will use different splittings for different bands depending on whether the band contains our stationary phase point $z_j(n/t)$ or not. We will begin with some preparatory lemmas. For the bands containing no stationary phase points we will use a splitting based on the following Fourier transform associated with the background operator $H_q$. Given $R \in C^l(\Sigma)$ we can write \be R(p) = \sum_{n\in\Z} \hat{R}(n)\psi_q(p,n,0), \ee where $\psi_q(p,x,t)$ denotes the time-dependent Baker--Akhiezer function and (cf.\ \cite{emt}, \cite{emt2}) \be \hat{R}(n) = \frac{1}{2\pi\I} \oint_\Sigma R(p) \psi_q(p^*,n,0) \frac{\I \prod_{j=1}^g (\pi(p)-\mu_j)}{\Rg{p}}d\pi(p). \ee If we make use of \eqref{defpsiq}, the above expression for $R(p)$ is of the form \be R(p)= \sum_{n\in\Z} \hat{R}(n) \theta_q(p,n,0)\exp \big( \I n k(p) \big). \ee where $k(p)= -\I \int_{E_0}^p \om_{\infty_+\, \infty_-}$ and $\theta_q(p,n,t)$ collects the remaining parts in \eqref{defpsiq}. Using $k(p)$ as a new coordinate and performing $l$ integration by parts one obtains \be |\hat{R}(n)| \le \frac{const}{1+|n|^l} \ee provided $R \in C^l(\Sigma)$. \begin{lemma}\label{lem:analapprox} Suppose $\hat{R} \in \ell^1(\Z)$, $n^l \hat{R}(n) \in \ell^1(\Z)$ and let $\beta>0$ be given. Then we can split $R(p)$ according to \[ R(p)= R_{a,t}(p) + R_{r,t}(p), \] such that $R_{a,t}(p)$ is analytic for in the region $0 < \im(k(p)) <\eps$ and \begin{align} |R_{a,t}(p) \E^{-\beta t} | &= O(t^{-l}), \quad 0 < \im(k(p)) <\eps,\\ |R_{r,t}(p)| &= O(t^{-l}), \quad p\in\Sigma. \end{align} \end{lemma} \begin{proof} We choose \[ R_{a,t}(p) = \sum_{n=-N(t)}^\infty \hat{R}(n) \theta_q(p,n,0)\exp \big( \I n k(p) \big) \] with $N(t) = \floor{\frac{\beta_0}{\eps} t}$ for some positive $\beta_0<\beta$. Then, for $0< \im(k(p)) <\eps$, \begin{align*} \left\vert R_{a,t}(k)\E^{-\beta t} \right\vert & \leq C \E^{-\beta t} \sum_{n=-N(t)}^\infty | \hat{R}(n) | \E^{-\im(k(p)) n}\\ & \leq C \E^{-\beta t}\E^{N(t)\eps}\| F \|_1 = \|\hat{R}\|_1 \E^{-(\beta-\beta_0)t}, \end{align*} which proves the first claim. Similarly, for $p\in\Sigma$, \[ \vert R_{r,t}(k) \vert \leq C \sum_{n=N(t)+1}^{\infty} \frac{n^l |\hat{R}(-n)|}{n^l} \leq C \frac{\|n^l \hat{R}(-n)\|_{\ell^1(\N)}}{N(t)^l} \leq \frac{\tilde{C}}{t^{l}} \] \end{proof} For the band which contains $z_j(n/t)$ we need to take the small vicinities of the stationary phase points into account. Since the phase is cubic near these points, we cannot use it to dominate the exponential growth of the analytic part away from $\Sigma$. Hence we will take the phase as a new variable and use the Fourier transform with respect to this new variable. Since this change of coordinates is singular near the stationary phase points, there is a price we have to pay, namely, requiring additional smoothness for $R(p)$. Without loss of generality we will choose the path of integration in our phase $\phi(p)$, defined in \eqref{defsp}, such that $\phi(p)$ is continuous (and thus analytic) in $D_{j,1}$ with continuous limits on the boundary (cf.\ Figure~\ref{fig2}). We begin with \begin{lemma} Suppose $R(p)\in C^5(\Sigma)$. Then we can split $R(p)$ according to \be R(p) = R_0(p) +(\pi(p)- \pi(z_j)) H(p), \qquad p \in \Sigma \cap D_{j,1}, \ee where $R_0(p)$ is a real rational function on $\M$ such that $H(p)$ vanishes at $z_j$, $z_j^*$ of order three and has a Fourier series \be H(p)=\sum_{n\in\Z} \hat{H}(n) \E^{n \om_0 \phi(p)}, \qquad \om_0= \frac{2\pi\I}{\phi(z_j)-\phi(z_j^*)}>0, \ee with $n\hat{H}(n)$ summable. Here $\phi$ denotes the phase defined in \eqref{defsp}. \end{lemma} \begin{proof} We begin by choosing a rational function $R_0(p) = a(z) + b(z) \Rg{p}$ with $p=(z,\pm)$ such that $a(z)$, $b(z)$ are real-valued polynomials which are chosen such that $a(z)$ matches the values of $\re(R(p))$ and its first four derivatives at $z_j$ and $\I^{-1} b(z) \Rg{p}$ matches the values of $\im(R(p))$ and its first four derivatives at $z_j$. Since $R(p)$ is $C^5$ we infer that $H(p)\in C^4(\Sigma)$ and it vanishes together with its first three derivatives at $z_j$, $z_j^*$. Note that $\phi(p)/\I$, where $\phi$ is defined in \eqref{defsp} has a maximum at $z_j^*$ and a minimum at $z_j$. Thus the phase $\phi(p)/\I$ restricted to $\Sigma \cap D_{j,1}$ gives a one to one coordinate transform $\Sigma \cap D_{j,1} \to [\phi(z_j^*)/\I, \phi(z_j)/\I]$ and we can hence express $H(p)$ in this new coordinate. The coordinate transform locally looks like a cube root near $z_j$ and $z_j^*$, however, due to our assumption that $H$ vanishes there, $H$ is still $C^2$ in this new coordinate and the Fourier transform with respect to this new coordinates exists and has the required properties. \end{proof} Moreover, as in Lemma~\ref{lem:analapprox} we obtain: \begin{lemma}\label{lem:analapprox2} Let $H(p)$ be as in the previous lemma. Then we can split $H(p)$ according to $H(p)= H_{a,t}(p) + H_{r,t}(p)$ such that $H_{a,t}(p)$ is analytic in the region $\re(\phi(p))<0$ and \be |H_{a,t}(p) \E^{\phi(p) t/2} | = O(1), \: p\in \ol{D_{j,1}}, \quad |H_{r,t}(p)| = O(t^{-1}), \: p\in\Sigma. \ee \end{lemma} \begin{proof} We choose $H_{a,t}(p) = \sum_{n=-K(t)}^{\infty}\hat{H}(n)\E^{n \om_0 \phi(p)}$ with $K(t) = \floor{t/(2 \om_0)}$. Then we can proceed as in Lemma~\ref{lem:analapprox}: \begin{align*} \vert H_{a,t}(p) \E^{\phi(p) t/2} \vert \leq \|\hat{H}\|_1 |\E^{-K(t) \om_0 \phi(p)+\phi(p) t/2}| \leq \|\hat{H}\|_1 \end{align*} and \[ |H_{r,t}(p)| \leq \frac{1}{K(t)} \sum_{n=K(t)+1}^\infty n |\hat{H}(-n)| \leq \frac{C}{t}. \] \end{proof} Clearly an analogous splitting exists for $p\in\Sigma \cap D_{j2}$. Now we are ready for our analytic approximation step. First of all recall that our jump is given in terms $\ti{b}_\pm$ and $\ti{B}_\pm$ defined in \eqref{defbt} and \eqref{defBt}, respectively. While $\ti{b}_\pm$ are already in the correct form for our purpose, this is not true for $\ti{B}_\pm$ since they contain the non-analytic expression $|T(p)|^2$. To remedy this we will rewrite $\ti{B}_\pm$ in terms of the left rather than the right scattering data. For this purpose let us use the notation $R_r(p) \equiv R_+(p)$ for the right and $R_l(p) \equiv R_-(p)$ for the left reflection coefficient. Moreover, let $d_r(p,x,t) = d(p,x,t)$ and $d_l(p,x,t) \equiv T(p)/d(p,x,t)$. With this notation we have \be J^4(p) = \begin{cases} \ti{b}_-(p)^{-1} \ti{b}_+(p), \qquad \pi(p)> z_j(n/t),\\ \ti{B}_-(p)^{-1} \ti{B}_+(p), \qquad \pi(p)< z_j(n/t),\\ \end{cases} \ee where \begin{align*} \ti{b}_- &= \begin{pmatrix} 1 & \frac{d_r(p,x,t)}{d_r(p^*,x,t)}R_r(p^*)\Theta(p^*)\E^{-t\phi(p)} \\ 0 & 1 \end{pmatrix}, \\ \ti{b}_+ &= \begin{pmatrix} 1 & 0 \\ \frac{d_r(p^*,x,t)}{d_r(p,x,t)}R_r(p)\Theta(p)\E^{-t\phi(p)}& 1 \end{pmatrix}, \end{align*} and \begin{align*} \ti{B}_- &= \begin{pmatrix} 1 & 0 \\ -\frac{d_{r,-}(p^*,x,t)}{d_{r,-}(p,x,t)} \frac{R_r(p) \Theta(p)}{|T(p)|^2} \E^{t\, \phi(p)} & 1 \end{pmatrix}, \\ \ti{B}_+ &= \begin{pmatrix} 1 & - \frac{d_{r,+}(p,x,t)}{d_{r,+}(p^*,x,t)} \frac{R_r(p^*) \Theta(p^*)}{|T(p)|^2} \E^{-t\, \phi(p)} \\ 0 & 1 \end{pmatrix}. \end{align*} Using \eqref{reltrpm} we can write \begin{align*} \ti{B}_- &= \begin{pmatrix} 1 & 0 \\ \frac{d_l(p^*,x,t)}{d_l(p,x,t)}R_l(p)\Theta(p)\E^{-t\phi(p)} & 1 \end{pmatrix}, \\ \ti{B}_+ &= \begin{pmatrix} 1 & \frac{d_l(p,x,t)}{d_l(p^*,x,t)}R_l(p^*)\Theta(p^*)\E^{-t\phi(p)} \\ 0 & 1 \end{pmatrix}. \end{align*} Now we split $R_r(p)=R_{a,t}(p)+R_{r,t}(p)$ by splitting $\ti{R}_r(p)$ defined via \eqref{splitR} according to Lemma~\ref{lem:analapprox} for $\pi(p)\in [E_{2k},E_{2k+1}]$ with $k< j$ (i.e., not containing $z_j(n/t)$) and according to Lemma~\ref{lem:analapprox2} for $\pi(p)\in [E_{2j},z_j(n/t)]$. In the same way we split $R_l(p)=R_{a,t}(p)+R_{r,t}(p)$ for $\pi(p)\in [z_j(n/t),E_{2j+1}]$ and $\pi(p)\in [E_{2k},E_{2k+1}]$ with $k> j$. For $\beta$ in Lemma~\ref{lem:analapprox} we can choose \be \beta=\left\{ \begin{array}{ll} \min_{p\in C_k} -\re(\phi(p))>0, & \pi(p)>z_j(n/t),\\ \min_{p\in C_k} \re(\phi(p))>0, & \pi(p)z_j(n/t), \\ \ti{b}_{a,t,-}(p)^{-1}, &p\in C_k^*, \quad \pi(p)>z_j(n/t),\\ \ti{b}_{r,t,-}(p)^{-1} \ti{b}_{r,t,+}(p), & p\in\Sigma, \quad \pi(p)>z_j(n/t), \\ \ti{B}_{a,t,+}(p), & p\in C_k, \quad \pi(p)