Content-Type: multipart/mixed; boundary="-------------0807310723341" This is a multi-part message in MIME format. ---------------0807310723341 Content-Type: text/plain; name="08-143.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="08-143.comments" 29 pages, one figure ---------------0807310723341 Content-Type: text/plain; name="08-143.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="08-143.keywords" Riemann-Hilbert problem, KdV equation, solitons ---------------0807310723341 Content-Type: application/x-tex; name="KdVRHP.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="KdVRHP.tex" %% @texfile{ %% filename="KdVRHP.tex", %% version="1.0", %% date="Jul-2008", %% cdate="20080707", %% filetype="LaTeX2e", %% pics="KdVRHP1", %% journal="Preprint", %% copyright="Copyright (C) K. Grunert and G. Teschl". %% } \documentclass{amsart} \usepackage{hyperref} \usepackage{graphicx} \usepackage{curves} \unitlength1cm %%%%%%%%%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{\T}{\mathbb{T}} %%%%%%%%%%%%%%%%%%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{\wti}{\widetilde} \newcommand{\wha}{\widehat} \newcommand{\norm}[1]{\lVert#1 \rVert} \newcommand{\abs}[1]{\lvert#1 \rvert} \newcommand{\spr}[2]{\langle #1 , #2 \rangle} \newcommand{\id}{\mathbb{I}} \newcommand{\I}{\mathrm{i}} \newcommand{\E}{\mathrm{e}} \newcommand{\ind}{\mathop{\mathrm{ind}}} \newcommand{\re}{\mathop{\mathrm{Re}}} \newcommand{\im}{\mathop{\mathrm{Im}}} \newcommand{\sech}{\mathop{\mathrm{sech}}} \DeclareMathOperator{\res}{Res} \newcommand{\db}{\mathfrak{D}} \newcommand{\lz}{\ell^2(\Z)} \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-} %%%%%%%%%%%%%%%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} %%%%%% \newcommand{\sigI}{\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}} \newcommand{\rI}{\begin{pmatrix} 1 & 1 \end{pmatrix}} \newcommand{\rN}{\begin{pmatrix} 0 & 0 \end{pmatrix}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \title[Long-Time Asymptotics for the KdV Equation]{Long-Time Asymptotics for the Korteweg--de Vries Equation via Nonlinear Steepest Descent} \author[K. Grunert]{Katrin Grunert} \address{Faculty of Mathematics\\ Nordbergstrasse 15\\ 1090 Wien\\ Austria} \email{\href{mailto:katrin.grunert@univie.ac.at}{katrin.grunert@univie.ac.at}} \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 by the Austrian Science Fund (FWF) under Grant No.\ Y330.} \keywords{Riemann--Hilbert problem, KdV equation, solitons} \subjclass[2000]{Primary 37K40, 35Q53; Secondary 37K45, 35Q15} \begin{abstract} We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Korteweg--de Vries equation for decaying initial data in the soliton and similarity region. \end{abstract} \maketitle \section{Introduction} One of the most famous examples of completely integrable wave equations is the Korteweg--de Vries (KdV) equation \be\label{kdv} q_t(x,t)=6q(x,t)q_x(x,t)-q_{xxx}(x,t), \quad (x,t)\in\R\times\R, \ee where, as usual, the subscripts denote the differentiation with respect to the corresponding variables. Following the seminal work of Gardner, Green, Kruskal, and Miura \cite{ggkm}, one can use the inverse scattering transform to establish existence and uniqueness of (real-valued) classical solutions for the corresponding initial value problem with rapidly decaying initial conditions. We refer to, for instance, the monographs by Marchenko \cite{mar} or Eckhaus and Van Harten \cite{evh}. Our concern here are the long-time asymptotics of such solutions. The classical result is that an arbitrary short-range solution of the above type will eventually split into a number of solitons travelling to the right plus a decaying radiation part travelling to the left, as illustrated in Figure~\ref{fig1}. \begin{figure} \includegraphics[width=8cm]{KdVRHP1} \caption{Numerically computed solution $q(x,t)$ of the KdV equation at time $t=5$, with initial condition $q(x,0)=\sech(x+3)-5\sech(x-1)$.} \label{fig1} \end{figure} This was first established numerically by Zabusky and Kruskal \cite{zakr} and later rigorously proved by Eckhaus and Schuur \cite{es}. Precise asymptotics for the radiation part were first formally derived by Zakharov and Manakov \cite{zama} in case of the nonlinear Schr\"odinger equation and in the KdV case by Ablowitz and Segur \cite{as}, \cite{as2} (assuming absence of solitons) with further extensions by Buslaev and Sukhanov \cite{bs}. To describe the asymptotics in more detail, we recall the well-known fact (see e.g.\ \cite{dt}, \cite{mar}) that $q(x,t)$ is uniquely determined by the (right) scattering data of the associated Schr\"odinger operator \be H(t)=-\frac{d^2}{dx^2}+q(x,t). \ee The scattering data consist of the reflection coefficient $R(k,t)$, a finite number of ($t$ independent) eigenvalues $-\kappa_j^2$ with $0<\kappa_1<\kappa_2<\dots<\kappa_N$, and norming constants $\gamma_j(t)$. We will write $R(k)=R(k,0)$ and $\gamma_j=\gamma_j(0)$ for the scattering data of the initial condition. Then the long-time asymptotics can be described by distinguishing the following main regions: (i).\ The soliton region, $x/t>C$ for some $C>0$, in which the solution is asymptotically given by a sum of one-soliton solutions \be q(x,t)\sim -2 \sum_{j=1}^N\frac{\kappa_j^2}{\cosh^2(\kappa_j x -4\kappa_j^3 t -p_j)}, \ee where the phase shifts are given by \be p_j=\frac{1}{2}\log\left(\frac{\gamma_j^2}{2\kappa_j}\prod_{l=j+1}^{N} \left(\frac{\kappa_l-\kappa_j}{\kappa_l+\kappa_j}\right)^2\right). \ee In the case of a pure $N$-soliton solution (i.e., $R(k,t)=0$) this was first established independently by Hirota \cite{hi}, Tanaka \cite{ta}, and Wadati and Toda \cite{wt}. In the general case it was first established rigorously by Eckhaus and Schuur \cite{es} (see also \cite{schuurbook}). (ii).\ The self-similar region, $|x/(3t)^{1/3}| \leq C$ for some $C>0$, in which the solution is connected with the Painl\'eve II transcendent. This was first established by Segur and Ablowitz \cite{as2}. (iii).\ The collisionless shock region, $x<0$ and $C^{-1}<\frac{-x}{(3t)^{1/3}(\log(t))^{2/3}}1$, which only occurs in the generic case (i.e., when $R(0)=-1$). Here the asymptotics can be given in terms of elliptic functions as was pointed out by Segur and Ablowitz \cite{as2} with further extensions in Deift, Venakides, and Zhou \cite{dvz}. (vi).\ The similarity region, $x/t<-C$ for some $C>0$, where \be q(x,t) \sim \left(\frac{4\nu(k_0) k_0}{3t}\right)^{1/2}\sin(16tk^3_0-\nu(k_0)\log(192tk^3_0)+\delta(k_0)), \ee with \begin{align*} \nu(k_0)= &-\frac{1}{2\pi}\log(1-\left\vert R(k_0) \right\vert^2),\\ \delta(k_0) =& \frac{\pi}{4}- \arg(R(k_0))+\arg(\Gamma(\I\nu(k_0))) + 4 \sum_{j=1}^N \arctan\big(\frac{\kappa_j}{k_0}\big)\\ & + \frac{1}{\pi}\int_{-k_0}^{k_0}\log(\left\vert \zeta-k_0 \right\vert) d\log(1-\left\vert R(\zeta) \right\vert^2). \end{align*} Here $k_0=\sqrt{-\frac{x}{12t}}$ denotes the stationary phase point, $R(k)=R(k,t=0)$ the reflection coefficient, and $\Gamma$ the Gamma function. Again this was found by Ablowitz and Segur \cite{as} with further extensions by Buslaev and Sukhanov \cite{bs}. However, to the best of our knowledge a full rigorous treatment of these results is still missing, except for the paper by Eckhaus and Schuur \cite{es}, which covers the soliton region together with the fact that the solution decays in the other regions. Our aim here is to use the nonlinear steepest descent method for oscillatory Riemann--Hilbert problems from Deift and Zhou \cite{dz} and apply it to rigorously establish the long-time asymptotics in the soliton and similarity regions (Theorem~\ref{thm:asym} respectively \ref{thm:asym2} below). Solitons will be added using the techniques from Deift, Kamvissis, Kriecherbauer, and Zhou \cite{dkkz} respectively Kr\"uger and Teschl \cite{kt}. Overall we closely follow the recent review article \cite{kt2}, where Kr\"uger and Teschl applied these methods to compute the long-time asymptotics for the Toda lattice. Finally, note that if $q(x,t)$ solves the KdV equation, then so does $q(-x,-t)$. Therefore it suffices to investigate the case $t\to\infty$. \section{The Inverse scattering transform and the Riemann--Hilbert problem} \label{sec:istrhp} In this section we want to derive the Riemann--Hilbert problem for the KdV equation from scattering theory. This is essentially classical (compare, e.g., \cite{bdt}) except for two points. The eigenvalues will be added by appropriate pole conditions which are then turned into jumps following Deift, Kamvissis, Kriecherbauer, and Zhou \cite{dkkz}. We will impose an additional symmetry conditions to ensure uniqueness later on following Kr\"uger and Teschl \cite{kt}. For the necessary results from scattering theory respectively the inverse scattering transform for the KdV equation we refer to \cite{mar} (see also \cite{bdt} and \cite{dt}). We consider real-valued classical solutions $q(x,t)$ of the KdV equation (\ref{kdv}), which decay rapidly, that is \be\label{decay} \max_{|t|\leq T} \int_{\R} (1+|x|)^{1+l} |q(x,t)| dx < \infty, \qquad \text{for all } T>0, \ee for some $l\in\N_0$. Existence of such solutions can for example be established via the inverse scattering transform if one assumes (cf.\ \cite[Sect.~4.2]{mar}) that the initial condition satisfies \be \int_{\R} (1+|x|) \left(|q(x,0)|+|q_x(x,0)|+|q_{xx}(x,0)|+|q_{xxx}(x,0)|\right)dx < \infty. \ee Associated with $q(x,t)$ is a self-adjoint Schr{\"o}dinger operator \begin{equation} \label{defjac} H(t) = -\frac{d^2}{dx^2}+q(.,t), \qquad \db(H)=H^2(\R) \subset L^2(\R). \end{equation} Here $L^2(\R)$ denotes the Hilbert space of square integrable (complex-valued) functions over $\R$ and $H^k(\R)$ the corresponding Sobolev spaces. By our assumption (\ref{decay}) the spectrum of $H$ consists of an absolutely continuous part $[0,\infty)$ plus a finite number of eigenvalues $-\kappa_j^2\in\R\backslash(-\infty,0]$, $1\le j \le N$. In addition, there exist two Jost solutions $\psi_\pm(k,x,t)$ which solve the differential equation \be H(t) \psi_\pm(k,x,t) = k^2 \psi_\pm(k,x,t), \qquad \im (k)> 0, \ee and asymptotically look like the free solutions \be \lim_{x \to \pm \infty} \E^{\mp \I kx} \psi_{\pm}(k,x,t) =1. \ee Both $\psi_\pm(k,x,t)$ are analytic for $\im (k) > 0$ and continuous for $\im (k)\geq 0$. The asymptotics of the two Jost solutions are \be\label{eq:psiasym} \psi_\pm(k,x,t) = \E^{\pm \I kx} \Big(1 + Q_{\pm}(x,t) \frac{1}{2\I k} + O\big(\frac{1}{k^2}\big) \Big), \ee as $k \to \infty$ with $\im(k)> 0$, where \be \label{defQ} \aligned Q_+(x,t) &= -\int_x^{\infty} q(y,t) dy , \quad Q_-(x,t)= -\int_{-\infty}^x q(y,t) dy. \endaligned \ee Furthermore, one has the scattering relations \be \label{relscat} T(k) \psi_\mp(k,x,t) = \ol{\psi_\pm(k,x,t)} + R_\pm(k,t) \psi_\pm(k,x,t), \qquad k \in \R, \ee where $T(k)$, $R_\pm(k,t)$ are the transmission respectively reflection coefficients. They have the following well-known properties: \begin{lemma} The transmission coefficient $T(k)$ is meromorphic for $\im (k) > 0$ with simple poles at $\I \kappa _1 , \dots, \I \kappa_N$ and is continuous up to the real line. The residues of $T(k)$ are given by \be\label{eq:resT} \res_{\I \kappa_j} T(k) = \I \mu _j(t) \gamma _{+,j}(t)^2 = \I \mu _j \gamma _{+,j}^2, \ee where \be \gam_{+,j}(t)^{-1} = \lVert \psi _+(\I \kappa_j,.,t)\rVert_2 \ee and $\psi_+ (\I \kappa_j,x,t) = \mu_j(t) \psi_-(\I \kappa_j,x,t)$. Moreover, \be \label{reltrpm} T(k) \ol{R_+(k,t)} + \ol{T(k)} R_-(k,t)=0, \qquad |T(k)|^2 + |R_\pm(k,t)|^2=1. \ee \end{lemma} In particular one reflection coefficient, say $R(k,t)=R_+(k,t)$, and one set of norming constants, say $\gam_j(t)= \gam_{+,j}(t)$, suffices. Moreover, the time dependence is given by: \begin{lemma} The time evolutions of the quantities $R(k,t)$ and $\gam_j(t)$ are given by \begin{align} R(k,t) &= R(k) \E^{8 \I k^3 t},\\ \gam_j(t) &= \gam_j \E^{4 \kappa_j^3 t}, \end{align} where $R(k)=R(k,0)$ and $\gam_j=\gam_j(0)$. \end{lemma} We will set up a Riemann--Hilbert problem as follows: \be\label{defm} m(k,x,t)= \left\{\begin{array}{c@{\quad}l} \begin{pmatrix} T(k) \psi_-(k,x,t) \E^{\I kx} & \psi_+(k,x,t) \E^{-\I kx} \end{pmatrix}, & \im (k) > 0,\\ \begin{pmatrix} \psi_+(-k,x,t) \E^{\I kx} & T(-k) \psi_-(-k,x,t) \E^{-\I kx} \end{pmatrix}, & \im(k) < 0. \end{array}\right. \ee We are interested in the jump condition of $m(k,x,t)$ on the real axis $\R$ (oriented from negative to positive). To formulate our jump condition we use the following convention: When representing functions on $\R$, the lower subscript denotes the non-tangential limit from different sides. By $m_+(k)$ we denote the limit from above and by $m_-(k)$ the one from below. Using the notation above implicitly assumes that these limits exist in the sense that $m(k)$ extends to a continuous function on the real axis. \begin{theorem}[Vector Riemann--Hilbert problem]\label{thm:vecrhp} Let $\mathcal{S}_+(H(0))=\{ R(k),\; k\geq 0; \: (\kappa_j, \gam_j), \: 1\le j \le N \}$ be the right scattering data of the operator $H(0)$. Then $m(k)=m(k,x,t)$ defined in (\ref{defm}) is meromorphic away from the real axis with simple poles at $\I\kappa_j$, $-\I\kappa_j$ and satisfies: \begin{enumerate} \item The jump condition \be \label{eq:jumpcond} m_+(k)=m_-(k) v(k), \qquad v(k)=\begin{pmatrix} 1-|R(k)|^2 & - \ol{R(k)} \E^{-t\Phi(k)} \\ R(k) \E^{t\Phi(k)} & 1 \end{pmatrix}, \ee for $k \in \R$, \item the pole conditions \be\label{eq:polecond} \aligned \res_{\I\kappa_j} m(k) &= \lim_{k\to\I\kappa_j} m(k) \begin{pmatrix} 0 & 0\\ \I \gam_j^2 \E^{t\Phi(\I \kappa_j)} & 0 \end{pmatrix},\\ \res_{-\I\kappa_j} m(k) &= \lim_{k\to -\I\kappa_j} m(k) \begin{pmatrix} 0 & - \I \gam_j^2 \E^{t\Phi(\I \kappa_j)} \\ 0 & 0 \end{pmatrix}, \endaligned \ee \item the symmetry condition \be \label{eq:symcond} m(-k) = m(k) \sigI, \ee \item and the normalization \be\label{eq:normcond} \lim_{\kappa\to\infty} m(\I\kappa) = (1\quad 1). \ee \end{enumerate} Here the phase is given by \begin{equation} \Phi(k)= 8 \I k^3+2\I k \frac {x}{t}. \end{equation} \end{theorem} \begin{proof} The jump condition (\ref{eq:jumpcond}) is a simple calculation using the scattering relations (\ref{relscat}) plus (\ref{reltrpm}). The pole conditions follow since $T(k)$ is meromorphic for $\im (k) > 0$ with simple poles at $\I \kappa_j$ and residues given by (\ref{eq:resT}). The symmetry condition holds by construction and the normalization (\ref{eq:normcond}) is immediate from the following lemma below. \end{proof} Observe that the pole condition at $\I \kappa_j$ is sufficient since the one at $-\I \kappa_j$ follows by symmetry. Moreover, using \be\label{eq:asygf} T(k)\psi_-(k,x,t)\psi_+(k,x,t)=1+\frac{q(x,t)}{2k^2} + O(\frac{1}{k^4}) \ee as $k \to \infty$ with $\im(k)> 0$ (observe that the right-hand side is just the diagonal Green's functions of $H(t)$ divided by the free one) we obtain from (\ref{eq:psiasym}) \begin{lemma}\label{lem:asymp} The function $m(k,x,t)$ defined in (\ref{defm}) satisfies \be\label{eq:asym} m(k,x,t)=\begin{pmatrix} 1 & 1 \end{pmatrix}+ Q(x,t)\frac{1}{2\I k} \begin{pmatrix} -1 & 1 \end{pmatrix} +O\left(\frac{1}{k^2}\right). \ee Here $Q(x,t)=Q_+(x,t)$ is defined in (\ref{defQ}). \end{lemma} For our further analysis it will be convenient to rewrite the pole condition as a jump condition and hence turn our meromorphic Riemann--Hilbert problem into a holomorphic Riemann--Hilbert problem following \cite{dkkz}. Choose $\eps$ so small that the discs $\left\vert k- \I \kappa_j \right\vert<\eps$ lie inside the upper half plane and do not intersect. Then redefine $m(k)$ in a neighborhood of $\I \kappa_j$ respectively $- \I \kappa_j$ according to \be\label{eq:redefm} m(k) = \begin{cases} m(k) \begin{pmatrix} 1 & 0 \\ -\frac{\I \gamma_j^2 \E^{t\Phi(\I \kappa_j)} }{k- \I \kappa_j} & 1 \end{pmatrix}, & |k- \I \kappa_j|< \eps,\\ m(k) \begin{pmatrix} 1 & \frac{\I \gamma_j^2 \E^{t\Phi(\I \kappa_j)} }{k+ \I \kappa_j} \\ 0 & 1 \end{pmatrix}, & |k+ \I \kappa_j|< \eps,\\ m(k), & \text{else}.\end{cases} \ee Note that for $\im (k) <0$ we redefined $m(k)$ such that it respects our symmetry (\ref{eq:symcond}). Then a straightforward calculation using $\res_{\I \kappa} m(k) = \lim_{k\to\I\kappa} (k-\I \kappa)m(k)$ shows: \begin{lemma}\label{lem:holrhp} Suppose $m(k)$ is redefined as in (\ref{eq:redefm}). Then $m(k)$ is holomorphic away from the real axis and the small circles around $\I \kappa_j$ and $-\I\kappa_j$. Furthermore it satisfies (\ref{eq:jumpcond}), (\ref{eq:symcond}), (\ref{eq:normcond}) and the pole condition is replaced by the jump condition \be \label{eq:jumpcond2} \aligned m_+(k) &= m_-(k) \begin{pmatrix} 1 & 0 \\ -\frac{\I \gamma_j^2 \E^{t\Phi(\I\kappa_j)}}{k-\I \kappa_j} & 1 \end{pmatrix},\quad |k-\I \kappa_j|=\eps,\\ m_+(k) &= m_-(k) \begin{pmatrix} 1 & \frac{\I \gamma_j^2 \E^{t\Phi(\I \kappa_j)}}{k+ \I \kappa_j} \\ 0 & 1 \end{pmatrix},\quad |k+ \I \kappa_j|=\eps, \endaligned \ee where the small circle around $\I \kappa_j$ is oriented counterclockwise. \end{lemma} Next we turn to uniqueness of the solution of this vector Riemann--Hilbert problem. This will also explain the reason for our symmetry condition. We begin by observing that if there is a point $k_1\in\C$, such that $m(k_1)=\rN$, then $n(k)=\frac{1}{k-k_1} m(k)$ satisfies the same jump and pole conditions as $m(k)$. However, it will clearly violate the symmetry condition! Hence, without the symmetry condition, the solution of our vector Riemann--Hilbert problem will not be unique in such a situation. Moreover, a look at the one-soliton solution verifies that this case indeed can happen. \begin{lemma}[One-soliton solution]\label{lem:singlesoliton} Suppose there is only one eigenvalue and that the reflection coefficient vanishes, that is, $\mathcal{S}_+(H(t))=\{ R(k,t)\equiv 0,\; k\in\R; \: (\kappa, \gam (t)), \kappa>0, \gamma>0 \}$. Then the unique solution of the Riemann--Hilbert problem (\ref{eq:jumpcond})--(\ref{eq:normcond}) is given by \begin{align}\label{eq:oss} m_0(k) &= \begin{pmatrix} f(k) & f(-k) \end{pmatrix} \\ \nn f(k) &= \frac{1}{1+(2\kappa)^{-1} \gamma^2 \E^{t\Phi(\I\kappa)}} \left(1+\frac{k+\I\kappa}{k-\I\kappa} (2\kappa)^{-1} \gamma^2 \E^{t\Phi(\I\kappa)}\right). \end{align} In particular, \be Q(x,t)=\frac{2 \gamma^2 \E^{t\Phi(\I \kappa)}}{1+(2\kappa)^{-1} \gamma^2 \E^{t\Phi(\I\kappa)}}. \ee \end{lemma} \begin{proof} By assumption the reflection coefficient vanishes and so the jump along the real axis disappears. Therefore and by the symmetry condition, we know that the solution is of the form $m_0(k) = \begin{pmatrix} f(k) & f(-k) \end{pmatrix}$ where $f(z)$ is meromorphic. Furthermore the function $f(k)$ has only a simple pole at $\I\kappa$, so that we can make the ansatz $f(k)=C+D\frac{k+\I\kappa}{k-\I\kappa}$. Then the constants $C$ and $D$ are uniquely determined by the pole conditions and the normalization. \end{proof} In fact, observe $f(k_1)=f(-k_1)=0$ if and only if $k_1=0$ and $2\kappa=\gamma^2 \E^{t\Phi (\I\kappa)} $. Furthermore, even in the general case $m(k_1)=\rN$ can only occur at $k_1=0$ as the following lemma shows. \begin{lemma}\label{lem:resonant} If $m(k_1) = \rN$ for $m$ defined as in (\ref{defm}), then $k_1 = 0$. Moreover, the zero of at least one component is simple in this case. \end{lemma} \begin{proof} By (\ref{defm}) the condition $m(k_1) = \rN$ implies that the Jost solutions $\psi_-(k,x)$ and $\psi_+(k,x)$ are linearly dependent or that the transmission coefficient $T(k_1)=0$. This can only happen, at the band edge, $k_1 = 0$ or at an eigenvalue $k_1=\I\kappa_j$. We begin with the case $k_1=\I\kappa_j$. In this case the derivative of the Wronskian $W(k)= (\psi_+(k,x)\psi_-'(k,x)-\psi_+'(k,x)\psi_-(k,x))$ does not vanish by the well-known formula $\frac{d}{dk} W(k) |_{k=k_1} = - 2k_1\int_\R \psi_+(k_1,x) \psi_-(k_1,x) dx \ne 0$. Moreover, the diagonal Green's function $g(z,x)= W(k)^{-1} \psi_+(k,x) \psi_-(k,x)$ is Herglotz as a function of $z=-k^2$ and hence can have at most a simple zero at $z=-k_1^2$. Since $z\to-k^2$ is conformal away from $z=0$ the same is true as a function of $k$. Hence, if $\psi_+(\I\kappa_j,x) = \psi_-(\I\kappa_j,x) =0$, both can have at most a simple zero at $k=\I\kappa_j$. But $T(k)$ has a simple pole at $\I\kappa_j$ and hence $T(k) \psi_-(k,x)$ cannot vanish at $k=\I\kappa_j$, a contradiction. It remains to show that one zero is simple in the case $k_1=0$. In fact, one can show that $\frac{d}{dk} W(k) |_{k=k_1} \ne 0$ in this case as follows: First of all note that $\dot{\psi}_\pm(k)$ (where the dot denotes the derivative with respect to $k$) again solves $H\dot{\psi}_\pm(k_1) = -k_1^2 \dot{\psi}_\pm(k_1)$ if $k_1=0$. Moreover, by $W(k_1)=0$ we have $\psi_+(k_1) = c\, \psi_-(k_1)$ for some constant $c$ (independent of $x$). Thus we can compute \begin{align*} \dot{W}(k_1) &= W(\dot{\psi}_+(k_1),\psi_-(k_1)) + W(\psi_+(k_1),\dot{\psi}_-(k_1))\\ &= c^{-1} W(\dot{\psi}_+(k_1),\psi_+(k_1)) + c W(\psi_-(k_1),\dot{\psi}_-(k_1)) \end{align*} by letting $x\to+\infty$ for the first and $x\to-\infty$ for the second Wronskian (in which case we can replace $\psi_\pm(k)$ by $\E^{\pm \I k x}$), which gives \[ \dot{W}(k_1) = -\I(c+c^{-1}). \] Hence the Wronskian has a simple zero. But if both functions had more than simple zeros, so would the Wronskian, a contradiction. \end{proof} \section{A uniqueness result for symmetric vector Riemann--Hilbert problems} \label{sec:uni} In this chapter we want to investigate uniqueness for the holomorphic vector Riemann--Hilbert problem \begin{align}\nn & m_+(k) = m_-(k) v(k), \qquad k\in \Sigma,\\ \label{eq:rhp4m} & m(-k) = m(k) \sigI,\\ \nn & \lim_{\kappa\to\infty} m(\I\kappa) = \begin{pmatrix} 1 & 1\end{pmatrix}. \end{align} \begin{hypothesis}\label{hyp:sym} Let $\Sigma$ consist of a finite number of smooth oriented curves in $\C$ such that the distance between $\Sigma$ and $\{ \I y | y\ge y_0\}$ is positive for some $y_0>0$. Assume that the contour $\Sigma$ is invariant under $k\mapsto -k$. Moreover, suppose the jump matrix $v$ can be factorized according to $v = b_-^{-1} b_+ = (\id-w_-)^{-1}(\id+w_+)$, where $w_\pm$ are continuous, bounded, square integrable and satisfy \be w_\pm(-k) = \sigI w_\mp(k) \sigI,\quad k\in\Sigma. \ee \end{hypothesis} Now we are ready to show that the symmetry condition in fact guarantees uniqueness. \begin{theorem} Suppose there exists a solution $m(k)$ of the Riemann--Hilbert problem (\ref{eq:rhp4m}) for which $m(k)=\begin{pmatrix} 0 & 0\end{pmatrix}$ can happen at most for $k=0$ in which case $\limsup_{k\to 0} \frac{k}{m_j(k)}$ is bounded from any direction for $j=1$ or $j=2$. Then the Riemann--Hilbert problem (\ref{eq:rhp4m}) with norming condition replaced by \be\label{eq:rhp4ma} \lim_{\kappa\to\infty} m(\I\kappa) = \begin{pmatrix} \alpha & \alpha \end{pmatrix} \ee for given $\alpha\in\C$, has a unique solution $m_\alpha(k) = \alpha\, m(k)$. \end{theorem} \begin{proof} Let $m_\alpha(k)$ be a solution of (\ref{eq:rhp4m}) normalized according to (\ref{eq:rhp4ma}). Then we can construct a matrix valued solution via $M=(m, m_\alpha)$ and there are two possible cases: Either $\det M(k)$ is nonzero for some $k$ or it vanishes identically. We start with the first case. By lemma 2.5, we can rewrite all poles as jumps with determinant one. Hence, the determinant of this modified Riemann--Hilbert problem has no jump. Since, it is bounded at infinity, the determinant is constant. But taking determinants in \[ M(-k) = M(k) \sigI. \] gives a contradiction. It remains to investigate the case where $\det(M)\equiv 0$. In this case we have $m_\alpha(k) = \delta(k) m(k)$ with a scalar function $\delta$. Moreover, $\delta(k)$ must be holomorphic for $k\in\C\backslash\Sigma$ and continuous for $k\in\Sigma$ except possibly at the points where $m(k_1) = \rN$. Since it has no jump across $\Sigma$, \[ \delta_+(k) m_+(k) = m_{\alpha,+}(k) = m_{\alpha,-}(k) v(k) = \delta_-(k) m_-(k) v(k) = \delta_-(k) m_+(k), \] it is even holomorphic in $\C\backslash\{0\}$ with at most a simple pole at $k=0$. Hence it must be of the form \[ \delta(k) = A + \frac{B}{k}. \] Since $\delta$ has to be symmetric, $\delta(k) = \delta(-k)$, we obtain $B = 0$. Now, by the normalization we obtain $\delta(k) = A = \alpha$. This finishes the proof. \end{proof} Furthermore, the requirements cannot be relaxed to allow (e.g.) second order zeros in stead of simple zeros. In fact, if $m(k)$ is a solution for which both components vanish of second order at, say, $k=0$, then $\ti{m}(k)=\frac{1}{k^2} m(k)$ is a nontrivial symmetric solution of the vanishing problem (i.e.\ for $\alpha=0$). By Lemma~\ref{lem:resonant} we have \begin{corollary}\label{cor:unique} The function $m(k,x,t)$ defined in (\ref{defm}) is the only solution of the vector Riemann--Hilbert problem (\ref{eq:jumpcond})--(\ref{eq:normcond}). \end{corollary} Observe that there is nothing special about $k \to\infty$ where we normalize, any other point would do as well. However, observe that for the one-soliton solution (\ref{eq:oss}), $f(k)$ vanishes at \[ k =\I\kappa \frac{1-(2\kappa)^2 \gamma^2 \E^{t\Phi(\I\kappa)}}{1+(2\kappa)^2 \gamma^2 \E^{t\Phi(\I\kappa)}} \] and hence the Riemann--Hilbert problem normalized at this point has a nontrivial solution for $\alpha=0$ and hence, by our uniqueness result, no solution for $\alpha=1$. This shows that uniqueness and existence are connected, a fact which is not surprising since our Riemann--Hilbert problem is equivalent to a singular integral equation which is Fredholm of index zero (see Appendix~\ref{sec:sieq}). \section{Conjugation and Deformation} \label{sec:condef} This section demonstrates how to conjugate our Riemann-Hilbert problem and how to deform our jump contour, such that the jumps will be exponentially close to the identity away from the stationary phase points. Throughout this and the following section, we will assume that the $R(k)$ has an analytic extension to a small neighborhood of the real axis. This is for example the case if we assume that our solution is exponentially decaying. In Section~\ref{sec:analapprox} we will show how to remove this assumption. For easy reference we note the following result: \begin{lemma}[Conjugation]\label{lem:conjug} Assume that $\wti{\Sigma}\subseteq\Sigma$. Let $D$ be a matrix of the form \be D(k) = \begin{pmatrix} d(k)^{-1} & 0 \\ 0 & d(k) \end{pmatrix}, \ee where $d: \C\backslash\wti{\Sigma}\to\C$ is a sectionally analytic function. Set \be \ti{m}(k) = m(k) D(k), \ee then the jump matrix transforms according to \be \ti{v}(k) = D_-(k)^{-1} v(k) D_+(k). \ee If $d$ satisfies $d(-k) = d(k)^{-1}$ and $d(k)=1+O(\frac{1}{|k|})$ as $k\to\infty$. Then the transformation $\ti{m}(k) = m(k) D(k)$ respects our symmetry, that is, $\ti{m}(k)$ satisfies (\ref{eq:symcond}) if and only if $m(k)$ does, and our normalization condition. \end{lemma} In particular, we obtain \be \ti{v} = \begin{pmatrix} v_{11} & v_{12} d^{2} \\ v_{21} d^{-2} & v_{22} \end{pmatrix}, \qquad k\in\Sigma\backslash\wti{\Sigma}, \ee respectively \be \ti{v} = \begin{pmatrix} \frac{d_-}{d_+} v_{11} & v_{12} d_+ d_- \\ v_{21} d_+^{-1} d_-^{-1} & \frac{d_+}{d_-} v_{22} \end{pmatrix}, \qquad k\in\Sigma\cap\wti{\Sigma}. \ee In order to remove the poles there are two cases to distinguish. If $k_0\in\I\R$ and $\kappa_j<\kappa_0$ then the corresponding jump is exponentially close to the identity and there is nothing to do. Otherwise we use conjugation to turn the jumps into one with exponentially decaying off-diagonal entries, again following Deift, Kamvissis, Kriecherbauer, and Zhou \cite{dkkz}. It turns out that we will have to handle the poles at $\I \kappa_j$ and $-\I\kappa_j$ in one step in order to preserve symmetry and in order to not add additional poles elsewhere. \begin{lemma}\label{lem:twopolesinc} Assume that the Riemann--Hilbert problem for $m$ has jump conditions near $\I\kappa$ and $-\I\kappa$ given by \be \aligned m_+(k)&=m_-(k)\begin{pmatrix}1&0\\ -\frac{\I\gamma^2}{k-\I\kappa}&1\end{pmatrix}, && \left\vert k-\I\kappa \right\vert=\eps, \\ m_+(k)&=m_-(k)\begin{pmatrix}1&\frac{\I\gamma^2}{k+\I\kappa}\\0&1\end{pmatrix}, && \left\vert k+\I\kappa \right\vert=\eps. \endaligned \ee Then this Riemann--Hilbert problem is equivalent to a Riemann--Hilbert problem for $\ti{m}$ which has jump conditions near $\I\kappa$ and $-\I\kappa$ given by \begin{align*} \ti{m}_+(k)&= \ti{m}_-(k)\begin{pmatrix}1& -\frac{(k+\I\kappa)^2}{\I\gamma^2(k-\I\kappa)}\\ 0 &1\end{pmatrix}, && \left\vert k-\I\kappa \right\vert=\eps, \\ \ti{m}_+(k)&= \ti{m}_-(k)\begin{pmatrix}1& 0 \\ \frac{(k-\I\kappa)^2}{\I\gamma^2(k+\I\kappa)}&1\end{pmatrix}, && \left\vert k+\I\kappa \right\vert=\eps, \end{align*} and all remaining data conjugated (as in Lemma~\ref{lem:conjug}) by \be D(k) = \begin{pmatrix} \frac{k-\I\kappa}{k+\I\kappa} & 0 \\ 0 & \frac{k+\I\kappa}{k-\I\kappa} \end{pmatrix}. \ee \end{lemma} \begin{proof} To turn $\gam^2$ into $\gam^{-2}$, introduce $D$ by \[ D(k) = \begin{cases} \begin{pmatrix} 1 & -\frac{k-\I\kappa}{\I\gamma^2} \\ \frac{\I\gamma^2}{k-\I\kappa} & 0 \end{pmatrix} \begin{pmatrix} \frac{k-\I\kappa}{k+\I\kappa} & 0 \\ 0 & \frac{k+\I\kappa}{k-\I\kappa} \end{pmatrix}, & \left\vert k-\I\kappa \right\vert <\eps, \\ \begin{pmatrix} 0 & -\frac{\I\gamma^2}{k+\I\kappa} \\ \frac{k+\I\kappa}{\I\gamma^2} & 1 \end{pmatrix} \begin{pmatrix} \frac{k-\I\kappa}{k+\I\kappa} & 0 \\ 0 & \frac{k+\I\kappa}{k-\I\kappa} \end{pmatrix}, & \left\vert k+\I\kappa \right\vert <\eps, \\ \begin{pmatrix} \frac{k-\I\kappa}{k+\I\kappa} & 0 \\ 0 & \frac{k+\I\kappa}{k-\I\kappa} \end{pmatrix}, & \text{else}, \end{cases} \] and note that $D(k)$ is analytic away from the two circles. Now set $\ti{m}(k) = m(k) D(k)$and note that $D(k)$ is also symmetric. Therefore the jump conditions can be verified by straightforward calculations and Lemma \ref{lem:conjug}. \end{proof} The jump along the real axis is of oscillatory type and our aim is to apply a contour deformation following \cite{dz} such that all jumps will be moved into regions where the oscillatory terms will decay exponentially. Since the jump matrix $v$ contains both $\exp(t\Phi)$ and $\exp(-t\Phi)$ we need to separate them in order to be able to move them to different regions of the complex plane. We recall that the phase of the associated Riemann--Hilbert problem is given by \begin{equation} \label{eq:Phi} \Phi(k)=8\I k^3+2\I k\frac{x}{t} \end{equation} and the stationary phase points, $\Phi'(k)=0$, are denoted by \be k_0= \sqrt{-\frac{x}{12t}},\quad -k_0= - \sqrt{-\frac{x}{12t}}, \qquad \lam_0=\frac{x}{12t}. \ee For $\frac{x}{t}>0$ we have $k_0\in\I\R$, and for $\frac{x}{t}<0$ we have $k_0\in\R$. For $\frac{x}{t}>0$ we will also need the value $\I\kappa_0\in\I\R$ defined via $\re(\Phi(\I\kappa_0))=0$, that is, \be \frac{x}{t} = 4\kappa_0^2. \ee We will set $\kappa_0=0$ if $\frac{x}{t}<0$ for notational convenience. A simple analysis shows that for $\frac{x}{t}>0$ we have $00$ \item $\ol{T(\ol{k},k_0)}=T(k,k_0)^{-1}$. \end{enumerate} \end{theorem} \begin{proof} That $\I\kappa_j$ are simple poles and $-\I\kappa_j$ are simple zeros is obvious from the Blaschke factors and that $T(k,k_0)$ has the given jump follows from Plemelj's formulas. (i) and (ii) are straightforward to check. \end{proof} Now we are ready to perform our conjugation step. Introduce \[ D(k) = \begin{cases} \begin{pmatrix} 1 & -\frac{k-\I\kappa_j}{\I\gamma_j^2 \E^{t\Phi (\I\kappa_j)}}\\ \frac{\I\gamma_j^2 \E^{t\Phi(\I\kappa_j)}}{k-\I\kappa_j} & 0 \end{pmatrix} D_0(k), & |k-\I\kappa_j|<\eps, \: \kappa_0 < \kappa_j,\\ \begin{pmatrix} 0 & -\frac{\I\gamma_j^2 \E^{t\Phi (\I\kappa_j)}}{k+\I\kappa_j} \\ \frac{k+\I\kappa_j}{\I\gamma_j^2 \E^{t\Phi(\I\kappa_j)}} & 1 \end{pmatrix} D_0(k), & |k+\I\kappa_j|<\eps, \: \kappa_0 < \kappa_j,\\ D_0(k), & \text{else}, \end{cases} \] where \[ D_0(k) = \begin{pmatrix} T(k,k_0)^{-1} & 0 \\ 0 & T(k,k_0) \end{pmatrix}. \] Note that we have \[ D(-k)= \sigI D(k) \sigI. \] Now we conjugate our problem using $D(k)$ and set \be\label{def:mti} \ti{m}(k)=m(k) D(k). \ee Then using Lemma~\ref{lem:conjug} and Lemma~\ref{lem:twopolesinc} the jump corresponding to $\kappa_0 < \kappa_j$ (if any) is given by \be \aligned \ti{v}(k) &= \begin{pmatrix}1& -\frac{k-\I\kappa_j} {\I\gamma_j^2 \E^{t\Phi (\I\kappa_j)}T(k,k_0)^{-2}}\\ 0 &1\end{pmatrix}, \qquad |k-\I\kappa_j|=\eps, \\ \ti{v}(k) &= \begin{pmatrix}1& 0 \\ \frac{k+\I\kappa_j} {\I\gamma_j^2 \E^{t\Phi(\I\kappa_j)} T(k,k_0)^2}&1\end{pmatrix}, \qquad |k+\I\kappa_j|=\eps, \endaligned \ee and corresponding to $\kappa_0>\kappa_j$ (if any) by \be \aligned \ti{v}(k) &= \begin{pmatrix} 1 & 0 \\ -\frac{\I\gamma_j^2 \E^{t\Phi(\I\kappa_j)} T(k,k_0)^{-2}}{k-\I\kappa_j} & 1 \end{pmatrix}, \qquad |k-\I\kappa_j|=\eps, \\ \ti{v}(k) &= \begin{pmatrix} 1 & \frac{\I\gamma_j^2 \E^{t\Phi(\I\kappa_j)} T(k,k_0)^2}{k+\I\kappa_j} \\ 0 & 1 \end{pmatrix}, \qquad |k+\I\kappa_j|=\eps. \endaligned \ee In particular, all jumps corresponding to poles, except for possibly one if $\kappa_j=\kappa_0$, are exponentially close to the identity. In this case we will keep the pole condition which now reads \be \aligned \res_{\I\kappa_j} \ti{m}(k) &= \lim_{k\to\I\kappa_j} \ti{m}(k) \begin{pmatrix} 0 & 0\\ \I\gamma_j^2 \E^{t\Phi(\I\kappa_j)} T(\I\kappa_j,k_0)^{-2} & 0 \end{pmatrix},\\ \res_{-\I\kappa_j} \ti{m}(k) &= \lim_{k\to -\I\kappa_j} \ti{m}(k) \begin{pmatrix} 0 & -\I\gamma_j^2 \E^{t\Phi(\I\kappa_j)} T(\I\kappa_j,k_0)^{-2} \\ 0 & 0 \end{pmatrix}. \endaligned \ee Furthermore, the jump along $\R$ is given by \be \ti{v}(k) = \begin{cases} \ti{b}_-(k)^{-1} \ti{b}_+(k), \qquad \re(k_0)< |k|,\\ \ti{B}_-(k)^{-1} \ti{B}_+(k), \qquad \re(k_0)> |k|,\\ \end{cases} \ee where \be \label{eq:deftib} \ti{b}_-(k) = \begin{pmatrix} 1 & \frac{R(-k) \E^{-t\Phi(k)}}{T(-k,k_0)^2} \\ 0 & 1 \end{pmatrix}, \quad \ti{b}_+(k) = \begin{pmatrix} 1 & 0 \\ \frac{R(k) \E^{t\Phi(k)}}{T(k,k_0)^2} & 1 \end{pmatrix}, \ee and \begin{align*} \ti{B}_-(k) =\begin{pmatrix} 1 & 0 \\ -\frac{T_-(k,k_0)^{-2}}{1-\left\vert R(k) \right\vert^2}R(k)\E^{t\Phi(k)} & 1\end{pmatrix} =\begin{pmatrix} 1 & 0 \\ - \frac{T_-(-k,k_0)}{T_-(k,k_0)} R(k) \E^{t\Phi(k)} & 1 \end{pmatrix}, \end{align*} \begin{align*} \ti{B}_+(k) =\begin{pmatrix} 1 & -\frac{T_+(k,k_0)^2}{1-\left\vert R(k)\right\vert^2}R(-k)\E^{-t\Phi(k)} \\ 0 & 1 \end{pmatrix} =\begin{pmatrix} 1 & - \frac{T_+(k,k_0)}{T_+(-k,k_0)} R(-k) \E^{-t\Phi(k)} \\ 0 & 1 \end{pmatrix}. \end{align*} Here we have used \[ T_\pm(-k,k_0)=T_\mp(k,k_0)^{-1}, \qquad k\in\Sigma(k_0) \] and the jump condition for the partial transmission coefficient $T(k,k_0)$ along $\Sigma(k_0)$ in the last step, which shows that the matrix entries are bounded for $k\in\R$. Note also that we have used $T(k,k_0)^{-1}=\ol{T(k,k_0)}$ and $R(-k)=\ol{R(k)}$ for $k\in\R$ to show that there exists an analytic continuation into the neighborhood of the real axis. Now we deform the jump along $\R$ to move the oscillatory terms into regions where they are decaying. There are two cases to distinguish: \begin{figure}\centering \begin{picture}(3.75,3) \put(0.3,1.25){\line(1,0){2.9}} \put(1.5,3){$k_0\in\I\R$} \put(0.3,1.75){$-$} \put(0.3,0.65){$+$} \put(1.65,2.25){$+$} \put(1.65,0.25){$-$} \curve(0.934,2.5, 1.05,2.3125, 1.3,2., 1.55,1.8125, 1.8,1.75, 2.05,1.8125, 2.3,2., 2.55,2.3125, 2.666,2.5) \curve(0.934,0., 1.05,0.1875, 1.3,0.5, 1.55,0.6875, 1.8,0.75, 2.05,0.6875, 2.3,0.5, 2.55,0.1875, 2.666,0.) \end{picture}\quad \begin{picture}(3.75,3) \put(0,1.25){\line(1,0){3.5}} \put(1.5,3){$k_0\in\R$} \put(0.15,0.5){$+$} \put(0.15,1.9){$-$} \put(2.95,0.5){$+$} \put(2.95,1.9){$-$} \put(1.45,2.25){$+$} \put(1.45,0.25){$-$} \put(1.3,1.3){$\scriptstyle -k_0$} \put(2.35,1.3){$\scriptstyle k_0$} \curve(0.,0.025, 0.425,0.25, 0.775,0.5, 1.025,0.75, 1.225,1.25, 1.025,1.75, 0.775,2., 0.425,2.25, 0.,2.47) \curve(3.5,0.025, 3.075,0.25, 2.725,0.5, 2.475,0.75, 2.275,1.25, 2.475,1.75, 2.725,2., 3.075,2.25, 3.5,2.47) \end{picture} \caption{Sign of $\re(\Phi(k))$ for different values of $k_0$}\label{fig:signRePhi} \end{figure} Case 1: $k_0\in\I\R$, $k_0\not=0$: \begin{figure}\centering \begin{picture}(7,5.2) \put(0.6,2.5){\line(1,0){5.8}} \put(2,2.5){\vector(1,0){0.4}} \put(5,2.5){\vector(1,0){0.4}} \put(6,2.2){$\R$} \put(0.6,2){\line(1,0){5.8}} \put(2,2){\vector(1,0){0.4}} \put(5,2){\vector(1,0){0.4}} \put(6,1.6){$\Sigma_-$} \put(0.6,3){\line(1,0){5.8}} \put(2,3){\vector(1,0){0.4}} \put(5,3){\vector(1,0){0.4}} \put(6,3.2){$\Sigma_+$} \put(0.6,3.5){$\re(\Phi)<0$} \put(0.6,1.3){$\re(\Phi)>0$} \put(2.9,4.5){$\re(\Phi)>0$} \put(2.9,0.5){$\re(\Phi)<0$} \curve(1.868,5., 2.1,4.625, 2.6,4., 3.1,3.625, 3.6,3.5, 4.1,3.625, 4.6,4., 5.1,4.625, 5.332,5.) \curve(1.868,0., 2.1,0.375, 2.6,1., 3.1,1.375, 3.6,1.5, 4.1,1.375, 4.6,1., 5.1,0.375, 5.332,0.) \end{picture} \caption{Deformed contour for $k_0\in\I\R^+$} \label{figure:solreg} \end{figure} We set $\Sigma_\pm = \lbrace k\in\C \vert \im(k) = \pm \varepsilon\rbrace$ for some small $\varepsilon$ such that $\Sigma_\pm$ lies in the region with $\pm \re(k) < 0$ and such that we don't intersect any other contours. Then we can split our jump by redefining $\ti{m}(k)$ according to \be \wha{m}(k) = \begin{cases} \ti{m}(k)\ti{b}_+(k)^{-1} , & 0<\im(k)<\varepsilon,\\ \ti{m}(k) \ti{b}_-(k)^{-1}, & -\varepsilon < \im(k) < 0,\\ \ti{m}(k), & \text{else}.\end{cases} \ee Thus the jump along the real axis disappears and the jump along $\Sigma_\pm$ is given by \be\label{eq:jumpsolreg} \wha{v}(k) = \begin{cases} \ti{b}_+(k) , & k\in\Sigma_+ \\ \ti{b}_-(k)^{-1} , & k\in\Sigma_-.\end{cases} \ee All other jumps are unchanged. Note that the resulting Riemann-Hilbert problem still satisfies our symmetry condition (\ref{eq:symcond}), since we have \be \ti{b}_\pm(-k)= \sigI \ti{b}_\mp(k)\sigI. \ee By construction the jump along $\Sigma_{\pm}$ is exponentially close to the identity as $t\to\infty$. Case 2: $k_0\in\R$, $k_0\not=0$: \begin{figure}\centering \begin{picture}(7,5.2) \put(0,2.5){\line(1,0){7.0}} \put(2,2.5){\vector(1,0){0.4}} \put(5,2.5){\vector(1,0){0.4}} \put(6.6,2.2){$\R$} \put(0,2){\line(1,0){2.0}} \put(2.9,3){\line(1,0){1.2}} \put(5,2){\line(1,0){2.0}} \put(1.1,2){\vector(1,0){0.4}} \put(5.5,2){\vector(1,0){0.4}} \put(3.3,3){\vector(1,0){0.4}} \put(1.3,1.5){$\Sigma_-^1$} \put(5.7,1.5){$\Sigma_-^1$} \put(3.5,3.3){$\Sigma_+^2$} \curve(2.,2., 2.2,2.1, 2.4,2.4, 2.45,2.5, 2.5,2.6, 2.7,2.9, 2.9,3.) \curve(4.1,3., 4.3,2.9, 4.5,2.6, 4.55,2.5, 4.6,2.4, 4.8,2.1, 5.,2.) \put(0,3){\line(1,0){2.0}} \put(2.9,2){\line(1,0){1.2}} \put(5,3){\line(1,0){2.0}} \put(1.1,3){\vector(1,0){0.4}} \put(5.5,3){\vector(1,0){0.4}} \put(3.3,2){\vector(1,0){0.4}} \curve(2.,3., 2.2,2.9, 2.4,2.6, 2.45,2.5, 2.5,2.4, 2.7,2.1, 2.9,2.) \curve(4.1,2., 4.3,2.1, 4.5,2.4, 4.55,2.5, 4.6,2.6, 4.8,2.9, 5.,3.) \put(1.3,3.3){$\Sigma_+^1$} \put(5.7,3,3){$\Sigma_+^1$} \put(3.5,1.5){$\Sigma_-^2$} \put(0.3,1.0){$\scriptstyle\re(\Phi)>0$} \put(0.3,3.8){$\scriptstyle\re(\Phi)<0$} \put(5.9,1.0){$\scriptstyle\re(\Phi)>0$} \put(5.9,4.0){$\scriptstyle\re(\Phi)<0$} \put(2.9,4.5){$\scriptstyle\re(\Phi)>0$} \put(2.9,0.5){$\scriptstyle\re(\Phi)<0$} \put(2.6,2.6){$\scriptstyle -k_0$} \put(4.7,2.6){$\scriptstyle k_0$} \curve(0.,0.05, 0.85,0.5, 1.55,1., 2.05,1.5, 2.45,2.5, 2.05,3.5, 1.55,4., 0.85,4.5, 0.,4.94) \curve(7.,0.05, 6.15,0.5, 5.45,1., 4.95,1.5, 4.55,2.5, 4.95,3.5, 5.45,4., 6.15,4.5, 7.,4.94) \end{picture} \caption{Deformed contour for $k_0\in\R^+$} \label{figure:simreg} \end{figure} We set $\Sigma_{\pm}=\Sigma_{\pm}^1 \cup\Sigma_{\pm}^2$. Again note that $\Sigma_{\pm}^1$ respectively $\Sigma_{\pm}^2$ lie in the region with $\pm\re(\Phi(k))<0$. Then we can split our jump by redefining $\ti{m}(k)$ according to \be \wha{m}(k)= \begin{cases} \ti{m}(k)\ti{b}_+(k)^{-1} , & k\textnormal{ between } \R \textnormal{ and } \Sigma_+^1, \\ \ti{m}(k)\ti{b}_-(k)^{-1} , & k \textnormal{ between } \R \textnormal{ and } \Sigma_-^1,\\ \ti{m}(k)\ti{B}_+(k)^{-1} , & k \textnormal{ between } \R \textnormal{ and } \Sigma_+^2,\\ \ti{m}(k)\ti{B}_-(k)^{-1} , & k \textnormal{ between } \R \textnormal{ and } \Sigma_-^2,\\ \ti{m}(k) , & \textnormal{else}. \end{cases} \ee One checks that the jump along $\R$ disappears and the jump along $\Sigma_{\pm}$ is given by \be \wha{v}(k)=\begin{cases} \ti{b}_+(k) , & k\in\Sigma_+^1,\\ \ti{b}_-(k)^{-1} , & k\in\Sigma_-^1,\\ \ti{B}_+(k) , & k\in\Sigma_+^2,\\ \ti{B}_-(k)^{-1} , & k\in\Sigma_-^2. \end{cases} \ee All other jumps are unchanged. Again the resulting Riemann-Hilbert problem still satisfies our symmetry condition (\ref{eq:symcond}) and the jump along $\Sigma_{\pm}\backslash\{k_0,-k_0\}$ is exponentially decreasing as $t\to\infty$ \begin{theorem}\label{thm:asym} Assume (\ref{decay}) and abbreviate by $c_j= 4 \kappa_j^2$ the velocity of the $j$'th soliton determined by $\re(\Phi(\I \kappa_j))=0$. Then the asymptotics in the soliton region, $x/t \geq C $ for some $C>0$, are as follows: Let $\eps > 0$ sufficiently small such that the intervals $[c_j-\eps,c_j+\eps]$, $1\le j \le N$, are disjoint and lie inside $\R^+$. If $|\frac{x}{t} - c_j|<\eps$ for some $j$, one has \begin{align} \int_x^{\infty} q(y,t) dy &= -4 \sum_{i=j+1}^N \kappa_i - \frac{2\gamma_j^2(x,t)}{1+ (2\kappa_j)^{-1} \gamma_j^2(x,t)} + O(t^{-l}), \end{align} respectively \begin{align} q(x,t)& = \frac{-4\kappa_j\gamma_j^2(x,t)}{(1+(2\kappa_j)^{-1}\gamma_j^2(x,t))^2} +O(t^{-l}), \end{align} where \be \gam_j^2(x,t) = \gamma_j^2 \E^{-2\kappa_j x + 8 \kappa_j^3 t} \prod_{i=j+1}^N \left(\frac{\kappa_i-\kappa_j}{\kappa_i+\kappa_j}\right)^2. \ee If $|\frac{x}{t} -c_j| \geq \eps$, for all $j$, one has \begin{align}\nn \int_x^{\infty} q(y,t) dy &= -4 \sum_{\kappa_i \in (\kappa_0,\infty)} \kappa_i + O(t^{-l}), \qquad \kappa_0=\sqrt{\frac{x}{4t}}, \end{align} respectively \begin{align} q(x,t)& = O(t^{-l}). \end{align} \end{theorem} \begin{proof} Since $\wha{m}(k) = \wha{m}(k)$ for $k$ away from $\R$ equations (\ref{eq:asyt}) and (\ref{eq:asym}) imply the following asymptotics \begin{align}\label{eq:asyhm} \wha{m}(k) = \begin{pmatrix} 1 & 1 \end{pmatrix} + \left(-2T_1(k_0) + Q(x,t)\right) \frac{1}{2\I k} \begin{pmatrix}-1 & 1 \end{pmatrix} + O\left(\frac{1}{k^2}\right). \end{align} By construction, the jump along $\Sigma_\pm$ is exponentially decreasing as $t\to\infty$. Hence we can apply Theorem~\ref{thm:remcontour} as follows: If $\left\vert \frac{x}{t}-c_j\right\vert>\varepsilon$ for all $j$ we can choose $\gamma_0=0$ and $\wha{w}_0^t \equiv 0$ in Theorem~\ref{thm:remcontour}. Since the error between $\wha{w}_0^t$ and $\wha{w}^t$ is exponentially small as $t\to\infty$, the solutions of the associated Riemann-Hilbert problems only differ by $O(t^{-l})$ for any $l\ge 1$. Comparing $\wha{m}_0= \begin{pmatrix} 1 & 1 \end{pmatrix}$ with the above asymptotics shows $Q_+(x,t)= 2T_1(k_0) +O(t^{-l})$. If $\left\vert \frac{x}{t} -c_j \right\vert < \varepsilon$ for some $j$, we choose $\gamma_0^t=\gamma_k(x,t)$ and $\wha{w}_0^t\equiv 0$. As before we conclude that the error between the solutions of $\wha{w}^t$ and $\wha{w}_0^t$ is exponentially small and so the associated solutions of the Riemann-Hilbert problems only differ by $O(t^{-l})$. From Lemma \ref{lem:singlesoliton}, we have the one-soliton solution $\wha{m}_0(k) = \begin{pmatrix} \wha{f}(k) & \wha{f}(-k) \end{pmatrix}$ with $\wha{f}(k)= \frac{1}{1+(2\kappa_j)^{-1}\gamma_j^2(x,t)}\big(1+\frac{k+\I\kappa_j}{k-\I\kappa_j}(2\kappa_j)^{-1} \gamma_j^2(x,t)\big)$, where \[ \gamma_j^2(x,t)=\gamma_j^2 \E^{t\Phi(\I\kappa_j)}T(\I\kappa_j, \I\frac{\kappa_j}{\sqrt{3}})^{-2} = \gamma_j^2 \E^{-2\kappa_j x + 8 \kappa_j^3 t} \prod_{i=j+1}^N \left(\frac{\kappa_i-\kappa_j}{\kappa_i+\kappa_j}\right)^2. \] As before, comparing with the above asymptotics shows \[ Q(x,t)= 2 T_1(k_0) + \frac{2\gamma_j^2(x,t)}{1+(2\kappa_j)^{-1}\gamma_j^2(x,t)}+O(t^{-l}). \] To see the second part just use (\ref{eq:asygf}) in place of (\ref{eq:asym}). This finishes the proof in the case where $R(z)$ has an analytic extensions. We will remove this assumption in Section~\ref{sec:analapprox} thereby completing the proof. \end{proof} Since the one-soliton solution is exponentially decaying away from its minimum, we also obtain the form stated in the introduction: \begin{corollary} Assume (\ref{decay}), then the asymptotic in the soliton region, $x/t \geq C$ for some $C>0$, is given by \begin{align} q(x,t)= -2\sum_{j=1}^{N}\frac{\kappa_j^2}{\cosh^2(\kappa_j x-4\kappa_j^3-p_j)} + O(t^{-l}), \end{align} where \be p_j=\frac{1}{2}\log\left(\frac{\gamma_j^2}{2\kappa_j}\prod_{i=j+1}^{N} \left(\frac{\kappa_i-\kappa_j}{\kappa_i+\kappa_j}\right)^2\right). \ee \end{corollary} \section{Reduction to a Riemann--Hilbert problem on a small cross} \label{sec:simrhp} In the previous section we have seen that for $k_0\in\R$ we can reduce everything to a Riemann-Hilbert problem for $\wha{m}(k)$ such that the jumps are exponentially close to the identity except in small neighborhoods of the stationary phase points $k_0$ and $-k_0$. Hence we need to continue our investigation of this case in this chapter. Denote by $\Sigma^c(\pm k_0)$ the parts of $\Sigma_+\cup \Sigma_-$ inside a small neighborhood of $\pm k_0$. We will now show that solving the two problems on the small crosses $\Sigma^c(k_0)$ respectively $\Sigma^c(-k_0)$ will lead us to the solution of our original problem. Introduce the cross $\Sigma = \Sigma_1 \cup\dots\cup \Sigma_4$ (see Figure~\ref{fig:contourcross}) by \begin{align} \nn \Sigma_1 & = \{u \E^{-\I\pi/4},\,u\in [0,\infty)\} & \Sigma_2 & = \{u \E^{\I\pi/4}, \,u\in [0,\infty)\} \\ \Sigma_3 & = \{u \E^{3\I\pi/4}, \,u\in [0,\infty)\} & \Sigma_4 & = \{u \E^{-3\I\pi/4}, \,u\in [0,\infty)\}. \end{align} \begin{figure} \begin{picture}(7,5.2) \put(1,5){\line(1,-1){5}} \put(2,4){\vector(1,-1){0.4}} \put(4.7,1.3){\vector(1,-1){0.4}} \put(1,0){\line(1,1){5}} \put(2,1){\vector(1,1){0.4}} \put(4.7,3.7){\vector(1,1){0.4}} \put(6.0,0.1){$\Sigma_1$} \put(5.3,4.8){$\Sigma_2$} \put(1.3,4.8){$\Sigma_3$} \put(1.4,0.1){$\Sigma_4$} \put(2.8,0.5){$\scriptsize\begin{pmatrix} 1 & - R_1(z) \cdots\\ 0 & 1 \end{pmatrix}$} \put(4.5,3.1){$\scriptsize\begin{pmatrix} 1 & 0 \\ R_2(z) \cdots & 1 \end{pmatrix}$} \put(1.9,4.5){$\scriptsize\begin{pmatrix} 1 & - R_3(z) \cdots \\ 0 & 1 \end{pmatrix}$} \put(0.5,1.8){$\scriptsize\begin{pmatrix} 1 & 0 \\ R_4(z) \cdots & 1 \end{pmatrix}$} \end{picture} \caption{Contours of a cross} \label{fig:contourcross} \end{figure}% Orient $\Sigma$ such that the real part of $z$ increases in the positive direction. Denote by $\mathbb{D} = \{z,\,|z|<1\}$ the open unit disc. Throughout this section $z^{\I\nu}$ will denote the function $\E^{\I \nu \log(z)}$, where the branch cut of the logarithm is chosen along the negative real axis $(-\infty,0)$. Introduce the following jump matrices ($v_j$ for $z\in\Sigma_j$) \begin{align} \nn v_1 &= \begin{pmatrix} 1 & - R_1(z) z^{2\I\nu} \E^{- t \Phi(z)} \\ 0 & 1 \end{pmatrix}, & v_2 &= \begin{pmatrix} 1 & 0 \\ R_2(z) z^{-2\I\nu} \E^{t \Phi(z)} & 1 \end{pmatrix}, \\ v_3 &= \begin{pmatrix} 1 & - R_3(z) z^{2\I\nu} \E^{- t \Phi(z)} \\ 0 & 1 \end{pmatrix}, & v_4 &= \begin{pmatrix} 1 & 0 \\ R_4(z) z^{-2\I\nu} \E^{t \Phi(z)} & 1 \end{pmatrix} \end{align} and consider the RHP given by \begin{align}\label{eq:rhpcross} m_+(z) &= m_-(z) v_j(z), && z\in\Sigma_j,\quad j=1,2,3,4,\\ \nn m(z) &\to \id, && z\to \infty. \end{align} The solution is given in the following theorem of Deift and Zhou \cite{dz} (for a proof of the version stated below see Kr\"uger and Teschl \cite{kt2}). \begin{theorem}[\cite{dz}]\label{thm:solcross} Assume there is some $\rho_0>0$ such that $v_j(z)=\id$ for $|z|>\rho_0$. Moreover, suppose that within $|z|\le\rho_0$ the following estimates hold: \begin{enumerate} \item The phase satisfies $\Phi(0)=\I\Phi_0\in\I\R$, $\Phi'(0) = 0$, $\Phi''(0) = \I$ and \begin{align}\label{estPhi} \pm \re\big(\Phi(z)-\Phi(0)\big) &\geq \frac{1}{4} |z|^2,\quad \begin{cases} + & \mbox{for } z\in\Sigma_1\cup\Sigma_3,\\ - &\mbox{else},\end{cases}\\ \label{estPhi2} |\Phi(z) - \Phi(0) - \frac{\I z^2}{2}| &\leq C |z|^3. \end{align} \item There is some $r\in\mathbb{D}$ and constants $(\alpha, L) \in (0,1] \times (0,\infty)$ such that $R_j$, $j=1,\dots,4$, satisfy H\"older conditions of the form \begin{align}\nn \abs{R_1(z) - \ol{r}} &\leq L |z|^\alpha, & \abs{R_2(z) - r} &\leq L |z|^\alpha, \\\label{holdcondrj} \abs{R_3(z) - \frac{\ol{r}}{1-\abs{r}^2}} &\leq L |z|^\alpha, & \abs{R_4(z) - \frac{r}{1-\abs{r}^2}} &\leq L |z|^\alpha. \end{align} \end{enumerate} Then the solution of the RHP (\ref{eq:rhpcross}) satisfies \be m(z) = \id + \frac{1}{z} \frac{\I}{t^{1/2}} \begin{pmatrix} 0 & -\beta \\ \ol{\beta} & 0 \end{pmatrix} + O(t^{- \frac{1 + \alpha}{2}}), \ee for $|z|>\rho_0$, where \be \beta = \sqrt{\nu} \E^{\I(\pi/4-\arg(r)+\arg(\Gamma(\I\nu)))} \E^{-\I t \Phi_0} t^{-\I\nu}, \qquad \nu = - \frac{1}{2\pi} \log(1 - |r|^2). \ee Furthermore, if $R_j(z)$ and $\Phi(z)$ depend on some parameter, the error term is uniform with respect to this parameter as long as $r$ remains within a compact subset of $\mathbb{D}$ and the constants in the above estimates can be chosen independent of the parameters. \end{theorem} \begin{theorem}[Decoupling]\label{thm:decoup} Consider the Riemann--Hilbert problem \be \aligned m_+(k)=m_-(k)v(k), \qquad k\in\Sigma, \\ m(\infty)=\begin{pmatrix} 1 & 1 \end{pmatrix}, \endaligned \ee and let $0<\alpha<\beta \leq 2\alpha, \rho(t)\to\infty$ be given. Suppose that for every sufficiently small $\varepsilon >0$ both the $L^2$ and the $L^{\infty}$ norms of $v$ are $O(t^{-\beta})$ away from some $\varepsilon$ neighborhoods of some points $k_j, 1\leq j\leq n$. Moreover, suppose that the solution of the matrix problem with jump $v(k)$ restricted to the $\frac{\varepsilon}{2}$ neighborhood has a solution given by \be \aligned M_j(k)=\id +\frac{1}{\rho(t)^{\alpha}}\frac{M_j}{k-k_j}+O(\rho(t)^{-\beta}), \qquad \left\vert k-k_j \right\vert >\frac{\varepsilon}{2}. \endaligned \ee Then the solution $m(k)$ is given by \be \aligned m(k)=\begin{pmatrix} 1 & 1 \end{pmatrix} + \frac{1}{\rho(t)^{\alpha}}\begin{pmatrix} 1 & 1 \end{pmatrix} \sum_{j=1}^n \frac{M_j}{k-k_j} + O(\rho(t)^{-\beta}), \endaligned \ee where the error term depends on the distance of $k$ to $\Sigma$. \end{theorem} \begin{proof} In this proof we will use the theory developed in Appendix \ref{sec:sieq} with $m_0(k)=\id$ and the usual Cauchy kernel $\Omega_{\infty} (s,k)=\id \frac{ds}{s-k}$. Assume that $m(k)$ exists. Introduce $\ti{m}(k)$ by \be \aligned \ti{m}(k)=\begin{cases} m(k)M_j(k)^{-1}, & \left\vert k-k_j \right\vert \leq \varepsilon,\\ m(k), & \text{else}. \end{cases} \endaligned \ee The Riemann--Hilbert problem for $\ti{m}(k)$ has jumps given by \be \aligned \ti{v}(k)=\begin{cases} M_j(k)^{-1}, & \left\vert k-k_j \right\vert =\varepsilon,\\ M_j(k)v(k)M_j(k)^{-1}, & k\in\Sigma, \frac{\varepsilon}{2} < \left\vert k-k_j \right\vert < \varepsilon,\\ \id, & k\in\Sigma, \left\vert k-k_j \right\vert < \frac{\varepsilon}{2},\\ v(k), & \text{else}. \end{cases} \endaligned \ee By assumption the jumps are $\id+O(\rho(t)^{-\alpha})$ on the circles $\left\vert k-k_j \right\vert = \varepsilon$ and even $\id +O(\rho(t)^{-\beta})$ on the rest (both in the $L^2$ and $L^{\infty}$ norms). In particular, we infer \be \left\Vert \ti{\mu}-\begin{pmatrix} 1 & 1 \end{pmatrix} \right\Vert_2=O (\rho(t)^{-\alpha}), \ee because $\mu$ satisfies $(\mu-\begin{pmatrix} 1 & 1 \end{pmatrix})=(\id-C_{w})^{-1}C_{w} \begin{pmatrix} 1 & 1 \end{pmatrix}$ and therefore we can estimate $\mu-\begin{pmatrix} 1 & 1 \end{pmatrix}$ by using the Neumann series to obtain $\Vert \mu-\begin{pmatrix} 1 & 1 \end{pmatrix} \Vert_2 \leq \frac{c\Vert w \Vert_2}{1-c\Vert w \Vert_{\infty}}$. Thus we conclude \be \aligned m(k) & = \begin{pmatrix} 1 & 1 \end{pmatrix} + \frac{1}{2\pi\I}\int_{\ti{\Sigma}} \ti{\mu}(s)\ti{w}(s) \frac{ds}{s-k}\\ & = \begin{pmatrix} 1 & 1\end{pmatrix} + \frac{1}{2\pi\I}\sum_{j=1}^n \int_{\left\vert k-k_j \right\vert = \varepsilon}\ti{\mu}(s)(M_j(s)^{-1}-\id)\frac{ds}{s-k}+O(\rho(t)^{-\beta})\\ & = \begin{pmatrix} 1 & 1 \end{pmatrix}- \rho(t)^{-\alpha}\begin{pmatrix} 1 & 1 \end{pmatrix} \frac{1}{2\pi\I} \sum_{j=1}^n M_j \int_{\left\vert k-k_j \right\vert = \varepsilon}\frac{1}{s-k_j}\frac{ds}{s-k}+O (\rho(t)^{-\beta})\\ & = \begin{pmatrix} 1 & 1 \end{pmatrix} + \rho(t)^{-\alpha} \begin{pmatrix} 1 & 1 \end{pmatrix} \sum_{j=1}^n \frac{M_j}{k-k_j} + O(\rho(t)^{-\beta}), \endaligned \ee and hence the claim is proven. \end{proof} Now let us turn to the solution of the problem on $\Sigma^c(k_0)=(\Sigma_+\cup\Sigma_-)\cap\{k \vert \left\vert k-k_0 \right\vert<\varepsilon\}$ for some small $\varepsilon > 0$. Without loss we can also deform our contour slightly such that $\Sigma^c(k_0)$ consists of two straight lines. Next, note \[ \Phi(k_0)=-16 \I k_0^3, \qquad \Phi''(k_0)=48\I k_0. \] As a first step we make a change of coordinates \be\label{eq:zeta} \zeta=\sqrt{48 k_0}(k-k_0) , \qquad k=k_0+\frac{\zeta}{\sqrt{48 k_0}} \ee such that the phase reads $\Phi(k)=\Phi(k_0)+\frac{\I}{2}\zeta^2+O(\zeta^3)$. Next we need the behavior of our jump matrix near $k_0$, that is, the behavior of $T(k,k_0)$ near $k_0$. \begin{lemma} Let $k_0\in\R$, then \be T(k,k_0)=\left(\frac{k-k_0}{k+k_0}\right)^{\I\nu} \ti{T}(k,k_0), \ee where $\nu=-\frac{1}{\pi}\log(\left\vert T(k_0) \right\vert)$ and the branch cut of the logarithm is chosen along the negative real axis. Here \be\label{tiT} \ti{T}(k,k_0)=\prod_{j=1}^{N} \frac{k+\I\kappa_j}{k-\I\kappa_j} \exp\left(\frac{1}{2\pi\I}\int\limits_{-k_0}^{k_0} \log\left( \frac{\left\vert T(\zeta) \right\vert ^2}{\left\vert T(k_0) \right\vert ^2} \right) \frac{1}{\zeta -k}{d\zeta}\right) \ee is H\"older continuous of any exponent less then $1$ at the stationary phase point $k=k_0$ and satisfies $\ti{T}(k_0,k_0)\in\T$. \end{lemma} \begin{proof} First of all observe that \be \exp\left(\frac{1}{2\pi \I}\int_{-k_0}^{k_0} \log(\left\vert T(k_0) \right\vert^2) \frac{1}{\zeta-k}d\zeta \right) = \left(\frac{k-k_0}{k+k_0}\right)^{\I \nu}. \ee H\"older continuity of any exponent less than $1$ is well-known (cf.\ \cite{mu}). \end{proof} If $k(\zeta)$ is defined as in (\ref{eq:zeta}) and $0<\alpha<1$, then there is an $L>0$ such that \be \left\vert T(k(\zeta ),k_0)-\zeta^{\I\nu}\ti{T}(k_0,k_0)\E^{-\I\nu \log(2 k_0\sqrt{48 k_0})} \right\vert\leq L\left\vert \zeta \right\vert^{\alpha}, \ee where the branch cut of $\zeta^{\I\nu}$ is chosen along the negative real axis. We also have \be \left\vert R(k(\zeta))-R(k_0) \right\vert \leq L\left\vert \zeta \right\vert^{\alpha} \ee and thus the assumptions of Theorem~\ref{thm:solcross} are satisfied with \be r=R(k_0)\ti{T}(k_0,k_0)^{-2}\E^{2\I \nu \log(2k_0 \sqrt{48k_0})}. \ee Therefore we can conclude that the solution on $\Sigma^c(k_0)$ is given by \be \aligned M_1^c(k) & =\id + \frac{1}{\zeta}\frac{\I}{t^{1/2}}\begin{pmatrix} 0 & -\beta \\ \ol{\beta} & 0 \end{pmatrix} + O(t^{-\alpha})\\ & = \id+ \frac{1}{\sqrt{48k_0}(k-k_0)}\frac{\I}{t^{1/2}} \begin{pmatrix} 0 & -\beta \\ \ol{\beta} & 0 \end{pmatrix} + O(t^{-\alpha}), \endaligned \ee where $\beta$ is given by \be \aligned \beta & = \sqrt{\nu}\E^{\I (\pi/4 -\arg(r) +\arg(\Gamma (\I\nu)))}\E^{-t\Phi(k_0)}t^{-\I\nu}\\ & = \sqrt{\nu}\E^{\I(\pi/4- \arg(R(k_0)) +\arg(\Gamma(\I\nu)))}\ti{T}(k_0,k_0)^2 (192k_0^3)^{-\I\nu}\E^{-t\Phi(k_0)}t^{-\I\nu}. \endaligned \ee and $1/2<\alpha<1$. We also need the solution $M_2^c(k)$ on $\Sigma^c(-k_0)$. We make the following ansatz, which is inspired by the symmetry condition for the vector Riemann--Hilbert problem, outside the two small crosses: \begin{align}\nn M_2^c(k) &= \sigI M_1^c(-k) \sigI\\ &= \id - \frac{1}{\sqrt{48k_0}(k+k_0)}\frac{\I}{t^{1/2}}\begin{pmatrix} 0 & \ol{\beta} \\ -\beta & 0 \end{pmatrix} +O(t^{-\alpha}). \end{align} Applying Theorem~\ref{thm:decoup} yields the following result: \begin{theorem}\label{thm:asym2} Assume (\ref{decay}) with $l=5$, then the asymptotics in the similarity region, $x/t \leq -C$ for some $C>0$, are given by \begin{align}\nn \int_x^{\infty} q(y,t)dy= & -4 \sum_{\kappa_j\in (\kappa_0,\infty)} \kappa_j - \frac{1}{\pi} \int_{-k_0}^{k_0} \log(\left\vert T(\zeta) \right\vert^2)d\zeta\\\label{eq:simasymp2} & -\sqrt{\frac{\nu(k_0)}{3 k_0 t}} \cos (16tk_0^3 -\nu(k_0) \log(192 t k_0^3)+\delta(k_0) )+O(t^{-\alpha}) \end{align} respectively \be\label{eq:simasymp} \aligned q(x,t)=\sqrt{\frac{4\nu(k_0) k_0}{3t}}\sin(16tk_0^3-\nu(k_0)\log(192 t k_0^3)+\delta(k_0))+O(t^{-\alpha}) \endaligned \ee for any $1/2<\alpha <1$. Here $k_0= \sqrt{-\frac{x}{12t}}$ and \begin{align} \nu(k_0)= & -\frac{1}{\pi} \log(\left\vert T(k_0) \right\vert),\\ \nn \delta(k_0)= & \frac{\pi}{4}- \arg(R(k_0))+\arg(\Gamma(\I\nu(k_0)))+4 \sum_{j=1}^N \arctan\big(\frac{\kappa_j}{k_0}\big)\\ & -\frac{1}{\pi}\int_{-k_0}^{k_0}\log\left(\frac{\left\vert T(\zeta) \right\vert^2}{\left\vert T(k_0) \right\vert^2}\right)\frac{1}{\zeta-k_0}d\zeta. \end{align} \end{theorem} \begin{proof} By Theorem~\ref{thm:decoup} we have \begin{align*} \wha{m}(k) = & \begin{pmatrix} 1 & 1 \end{pmatrix}+\frac{1}{\sqrt{48k_0}}\frac{\I}{t^{1/2}} \left(\frac{1}{k-k_0}\begin{pmatrix} \ol{\beta} & -\beta \end{pmatrix}-\frac{1}{k+k_0} \begin{pmatrix} -\beta & \ol{\beta} \end{pmatrix}\right) +O(t^{-\alpha})\\ = & \begin{pmatrix} 1 & 1 \end{pmatrix} +\frac{1}{\sqrt{48k_0}}\frac{\I}{t^{1/2}}\frac{1}{k} \left( \sum_{l=0}^{\infty}\left(\frac{k_0}{k}\right)^l \begin{pmatrix} \ol{\beta} & -\beta \end{pmatrix}-\sum_{l=0}^{\infty}\left(-\frac{k_0}{k}\right)^l \begin{pmatrix} -\beta & \ol{\beta} \end{pmatrix}\right) \\ & +O(t^{-\alpha}), \end{align*} which leads to \[ Q(x,t) = 2T_1(k_0)+\frac{4}{\sqrt{48k_0}}\frac{1}{t^{1/2}}(\re(\beta))+ O(t^{-\alpha}) \] upon comparison with (\ref{eq:asyhm}). Using the fact that $\left\vert \beta/\sqrt{\nu} \right\vert =1$ proves the first claim. To see the second part, as in the proof of Theorem~\ref{thm:asym}, just use (\ref{eq:asygf}) in place of (\ref{eq:asym}), which shows \[ q(x,t)=\sqrt{\frac{4k_0}{3t}}\im(\beta) + O(t^{-\alpha}). \] This finishes the proof in the case where $R(z)$ has an analytic extensions. We will remove this assumption in Section~\ref{sec:analapprox} thereby completing the proof. \end{proof} Equivalence of the formula for $\delta(k_0)$ given in the previous theorem with the one given in the introduction follows after a simple integration by parts. \begin{remark} Formally the equation (\ref{eq:simasymp}) for $q$ can be obtained by differentiating the equation (\ref{eq:simasymp2}) for $Q$ with respect to $x$. That this step is admissible could be shown as in Deift and Zhou \cite{dz2}, however our approach avoids this step. \end{remark} \begin{remark} Note that Theorem~\ref{thm:decoup} does not require uniform boundedness of the associated integral operator in contradistinction to Theorem~\ref{thm:remcontour}. We only need the knowledge of the solution in some small neighborhoods. However it cannot be used in the soliton region, because our solution is not of the form $\id+o(1)$. \end{remark} \section{Analytic Approximation} \label{sec:analapprox} In this chapter we want to present the necessary changes in the case where the reflection coefficient does not have an analytic extension. The idea is to use an analytic approximation and to split the reflection in an analytic part plus a small rest. The analytic part will be moved to the complex plane while the rest remains on the real axis. This needs to be done in such a way that the rest is of $O(t^{-1})$ and the growth of the analytic part can be controlled by the decay of the phase. In the soliton region a straightforward splitting based on the Fourier transform \be R(k) =\int_{\R}\E^{\I kx}\hat{R}(x)dx \ee will be sufficient. It is well-known that our decay assumption (\ref{decay}) implies $\hat{R} \in L^1(\R)$ and the estimate (cf.\ \cite[Sect.~3.2]{mar}) \be |\hat{R}(-2x)| \le const \int_x^\infty q(r) dr, \qquad x \ge 0, \ee implies $x^l \hat{R}(-x) \in L^1(0,\infty)$. \begin{lemma}\label{lem:analapprox} Suppose $\hat{R} \in L^1(\R)$, $x^l \hat{R}(-x) \in L^1(0,\infty)$ and let $\varepsilon, \beta>0$ be given. Then we can split the reflection coefficient according to $R(k)= R_{a,t}(k) + R_{r,t}(k)$ such that $R_{a,t}(k)$ is analytic in $0 <\im(k)< \varepsilon$ and \be |R_{a,t}(k) \E^{-\beta t} | = O(t^{-l}), \quad 0< \im(k) < \varepsilon, \qquad |R_{r,t}(k)| = O(t^{-l}), \quad k\in\R. \ee \end{lemma} \begin{proof} We choose $R_{a,t}(k) = \int_{-K(t)}^\infty \E^{\I kx}\hat{R}(x)dx $ with $K(t) = \frac{\beta_0}{\varepsilon} t$ for some positive $\beta_0<\beta$. Then, for $0< \im(k) <\varepsilon$, \begin{align*} \left\vert R_{a,t}(k)\E^{-\beta t} \right\vert \leq \E^{-\beta t} \int_{-K(t)}^\infty \vert \hat{R}(x) \vert \E^{-\im(k) x}dx \leq \E^{-\beta t}\E^{K(t)\varepsilon}\Vert \hat{R} \Vert_1 = \Vert \hat{R} \Vert_1 \E^{-(\beta-\beta_0)t}, \end{align*} which proves the first claim. Similarly, for $\im(k)=0$, \be\nn\aligned \vert R_{r,t}(k) \vert =\int_{K(t)}^{\infty} \frac{x^l \vert \hat{R}(-x)\vert}{x^l} dx \leq \frac{\|x^l \hat{R}(-x)\|_{L^1(0,\infty)}}{K(t)^l} \leq \frac{const }{t^{l}} \endaligned \ee \end{proof} To apply this lemma in the soliton region $k_0\in\I\R^+$ we choose \be \beta= \min_{\im(k)=-\varepsilon} -\re(\Phi(k))>0. \ee and split $R(k) = R_{a,t}(k) + R_{r,t}(k)$ according to Lemma~\ref{lem:analapprox} to obtain \be \ti{b}_\pm(k) = \ti{b}_{a,t,\pm}(k) \ti{b}_{r,t,\pm}(k) = \ti{b}_{r,t,\pm}(k) \ti{b}_{a,t,\pm}(k). \ee Here $\ti{b}_{a,t,\pm}(k)$, $\ti{b}_{r,t,\pm}(k)$ denote the matrices obtained from $\ti{b}_\pm(k)$ as defined in (\ref{eq:deftib}) by replacing $R(k)$ with $R_{a,t}(k)$, $R_{r,t}(k)$, respectively. Now we can move the analytic parts into the complex plane as in Chapter~\ref{sec:condef} while leaving the rest on the real axis. Hence, rather then (\ref{eq:jumpsolreg}), the jump now reads \be \hat{v}(k) = \begin{cases} \ti{b}_{a,t,+}(k), & k\in\Sigma_+, \\ \ti{b}_{a,t,-}(k)^{-1}, & k\in\Sigma_-,\\ \ti{b}_{r,t,-}(k)^{-1} \ti{b}_{r,t,+}(k), & k\in\R. \end{cases} \ee By construction we have $\hat{v}(k)= \id + O(t^{-l})$ on the whole contour and the rest follows as in Section~\ref{sec:condef}. In the similarity region not only $\ti{b}_{\pm}$ occur as jump matrices but also $\ti{B}_{\pm}$. These matrices $\ti{B}_{\pm}$ have at first sight more complicated off diagonal entries, but a closer look shows that they have indeed the same form. To remedy this we will rewrite them in terms of left rather then right scattering data. For this purpose let us use the notation $R_r(z) \equiv R_+(z)$ for the right and $R_l(z) \equiv R_-(z)$ for the left reflection coefficient. Moreover, let $T_r(z,z_0) \equiv T(z,z_0)$ be the right and $T_l(z,z_0) \equiv T(z)/T(z,z_0)$ be the left partial transmission coefficient. With this notation we have \be \ti{v}(k) = \begin{cases} \ti{b}_-(k)^{-1} \ti{b}_+(k), \qquad \re(k_0)< \left\vert k \right\vert,\\ \ti{B}_-(k)^{-1} \ti{B}_+(k), \qquad \re(k_0) > \left\vert k \right\vert,\\ \end{cases} \ee where \[ \ti{b}_-(k) = \begin{pmatrix} 1 & \frac{R_r(-k) \E^{-t\Phi(k)}}{T_r(-k,k_0)^2} \\ 0 & 1 \end{pmatrix}, \quad \ti{b}_+(k) = \begin{pmatrix} 1 & 0 \\ \frac{R_r(k) \E^{t\Phi(k)}}{T_r(k,k_0)^2}& 1 \end{pmatrix}, \] and \begin{align*} \ti{B}_-(k) &= \begin{pmatrix} 1 & 0 \\ - \frac{T_{r,-}(k,k_0)^{-2}}{|T(k)|^2} R_r(k) \E^{t\Phi(k)} & 1 \end{pmatrix}, \\ \ti{B}_+(k) &= \begin{pmatrix} 1 & - \frac{T_{r,+}(k,k_0)^2}{|T(k)|^2} R_r(-k) \E^{-t\Phi(k)} \\ 0 & 1 \end{pmatrix}. \end{align*} Using (\ref{reltrpm}) we can further write \be \ti{B}_-(k)= \begin{pmatrix} 1 & 0 \\ \frac{R_l(-k) \E^{t\Phi(k)}}{T_l(-k,k_0)^2} & 1 \end{pmatrix}, \quad \ti{B}_+(k)= \begin{pmatrix} 1 & \frac{R_l(k) \E^{-t\Phi(k)}}{T_l(k,k_0)^2} \\ 0 & 1 \end{pmatrix}. \ee Now we can proceed as before with $\ti{B}_\pm(z)$ as with $\ti{b}_\pm(z)$ by splitting $R_l(z)$ rather than $R_r(z)$. In the similarity region 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 the unit circle. 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 $R(k)$. In this respect note that (\ref{decay}) implies $R(k)\in C^l(\R)$ (cf.\ \cite{kl}). We begin with \begin{lemma} Suppose $R(k)\in C^5(\R)$. Then we can split $R(k)$ according to \be R(k) = R_0(k) + (k-k_0)(k+k_0) H(k), \qquad k \in \Sigma(k_0), \ee where $R_0(k)$ is a real rational function in $k$ such that $H(k)$ vanishes at $k_0$, $-k_0$ of order three and has a Fourier transform \be H(k)=\int_{\R}\hat{H}(x) \E^{x\Phi(k)}dx, \ee with $x\hat{H}(x)$ integrable. \end{lemma} \begin{proof} We can construct a rational function, which satisfies $f_n(-k)=\ol{f_n(k)}$ for $k\in\R$, by making the ansatz $f_n(k)=\frac{k_0^{2n+4}+1}{k^{2n+1}}\sum_{j=0}^n\frac{1}{j!(2 k_0)^j}(\alpha_j+\I\beta_j \frac{k}{k_0})(k-k_0)^j(k+k_0)^j$. Furthermore we can choose $\alpha_j$, $\beta_j\in\R$ for $j=1,\dots,n$, such that we can match the values of $R$ and its first four derivatives at $k_0$, $-k_0$ at $f_n(k)$. Thus we will set $R_0(k)=f_4(k)$ with $\alpha_0=\re(R(k_0))$, $\beta_0=\im(R(k_0))$, and so on. Since $R_0(k)$ is integrable we infer that $H(k)\in C^4(\R)$ and it vanishes together with its first three derivatives at $k_0$, $-k_0$. Note that $\Phi(k)/\I=8 (k^3-3 k_0^2 k)$ is a polynomial of order three which has a maximum at $-k_0$ and a minimum at $k_0$. Thus the phase $\Phi(k)/\I$ restricted to $ \Sigma(k_0)$ gives a one to one coordinate transform $\Sigma(k_0) \to [\Phi(k_0)/\I, \Phi(-k_0)/\I]=[-16k_0^3,16k_0^3]$ and we can hence express $H(k)$ in this new coordinate (setting it equal to zero outside this interval). The coordinate transform locally looks like a cube root near $k_0$ and $-k_0$, 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} Let $H(k)$ be as in the previous lemma. Then we can split $H(k)$ according to $H(k)= H_{a,t}(k) + H_{r,t}(k)$ such that $H_{a,t}(k)$ is analytic in the region $\re(\Phi(k))<0$ and \be |H_{a,t}(k) \E^{\Phi(k) t/2} | = O(1), \: \re(\Phi(k))<0, \im(k) \le 0, \quad |H_{r,t}(k)| = O(t^{-1}), \: k\in\R. \ee \end{lemma} \begin{proof} We choose $H_{a,t}(k) = \int_{-K(t)}^{\infty}\hat{H}(x)\E^{x \Phi(k)}dx$ with $K(t) = t/2$. Then we can conclude as in Lemma~\ref{lem:analapprox}: \begin{align*} \vert H_{a,t}(k) \E^{\Phi(k) t/2} \vert \leq \Vert \hat{H}(x) \Vert_1 \vert \E^{-K(t) \Phi(k)+\Phi(k) t/2} \vert \leq\Vert \hat{H}(x) \Vert_1 \leq const \end{align*} and \begin{align*} \vert H_{r,t}(k)\vert & \leq \int_{-\infty}^{-K(t)} \vert \hat{H}(x) \vert dx \leq const \sqrt{\int_{-\infty}^{-K(t)} \frac{1}{x^4} dx} \leq const \frac{1}{K(t)^{3/2}} \leq const \frac{1}{t}. \end{align*} \end{proof} By construction $R_{a,t}(k) = R_0(k) + (k-k_0)(k+k_0) H_{a,t}(k)$ will satisfy the required Lipschitz estimate in a vicinity of the stationary phase points (uniformly in $t$) and all jumps will be $\id+O(t^{-1})$. Hence we can proceed as in Chapter~\ref{sec:simrhp}. \appendix \section{Singular integral equations} \label{sec:sieq} In this section we show how to transform a meromorphic vector Riemann--Hilbert problem with simple poles at $\I\kappa$, $-\I\kappa$, \begin{align}\nn & m_+(k) = m_-(k) v(k), \qquad k\in \Sigma,\\ \label{eq:rhp5m} & \res_{\I\kappa} m(k) = \lim_{k\to\I\kappa} m(k) \begin{pmatrix} 0 & 0\\ \I\gamma^2 & 0 \end{pmatrix}, \quad \res_{-\I\kappa} m(k) = \lim_{k\to -\I\kappa} m(k) \begin{pmatrix} 0 & -\I\gamma^2 \\ 0 & 0 \end{pmatrix},\\ \nn & m(-k) = m(k) \sigI,\\ \nn & \lim_{\kappa\to\infty} m(\I\kappa) = \begin{pmatrix} 1 & 1\end{pmatrix} \end{align} into a singular integral equation. Since we require the symmetry condition (\ref{eq:symcond}) for our Riemann--Hilbert problem we need to adapt the usual Cauchy kernel to preserve this symmetry. Moreover, we keep the single soliton as an inhomogeneous term which will play the role of the leading asymptotics in our applications. The classical Cauchy-transform of a function $f:\Sigma\to \C$ which is square integrable is the analytic function $C f: \C\backslash\Sigma\to\C$ given by \be C f(k) = \frac{1}{2\pi\I} \int_{\Sigma} \frac{f(s)}{s - k} ds,\qquad k\in\C\backslash\Sigma. \ee Denote the tangential boundary values from both sides (taken possibly in the $L^2$-sense --- see e.g.\ \cite[eq.\ (7.2)]{deiftbook}) by $C_+ f$ respectively $C_- f$. Then it is well-known that $C_+$ and $C_-$ are bounded operators $L^2(\Sigma)\to L^2(\Sigma)$, which satisfy $C_+ - C_- = \id$ (see e.g. \cite{deiftbook}). Moreover, one has the Plemelj--Sokhotsky formula (\cite{mu}) \[ C_\pm = \frac{1}{2} (\I H \pm \id), \] where \be H f(k) = \frac{1}{\pi} \dashint_\Sigma \frac{f(s)}{k-s} ds,\qquad k\in\Sigma, \ee is the Hilbert transform and $\dashint$ denotes the principal value integral. In order to respect the symmetry condition we will restrict our attention to the set $L^2_{s}(\Sigma)$ of square integrable functions $f:\Sigma\to\C^{2}$ such that \be\label{eq:sym} f(-k) = f(k) \sigI. \ee Clearly this will only be possible if we require our jump data to be symmetric as well (i.e., Hypothesis~\ref{hyp:sym} holds). Next we introduce the Cauchy operator \be (C f)(k) = \frac{1}{2\pi\I} \int_\Sigma f(s) \Omega_\kappa(s,k) \ee acting on vector-valued functions $f:\Sigma\to\C^{2}$. Here the Cauchy kernel is given by \be \Omega_\kappa(s,k) = \begin{pmatrix} \frac{k+\I\kappa}{s+\I\kappa} \frac{1}{s-k} & 0 \\ 0 & \frac{k-\I\kappa}{s-\I\kappa} \frac{1}{s-k} \end{pmatrix} ds = \begin{pmatrix} \frac{1}{s-k} - \frac{1}{s+\I\kappa} & 0 \\ 0 & \frac{1}{s-k} - \frac{1}{s-\I\kappa} \end{pmatrix} ds, \ee for some fixed $\I\kappa\notin\Sigma$. In the case $\kappa=\infty$ we set \be \Omega_\infty(s,k) = \begin{pmatrix} \frac{1}{s-k} & 0 \\ 0 & \frac{1}{s-k} \end{pmatrix} ds. \ee and one easily checks the symmetry property: \be\label{eq:symC} \Omega_\kappa(-s,-k) = \sigI \Omega_\kappa(s,k) \sigI. \ee The properties of $C$ are summarized in the next lemma. \begin{lemma} Assume Hypothesis~\ref{hyp:sym}. The Cauchy operator $C$ has the properties, that the boundary values $C_\pm$ are bounded operators $L^2_s(\Sigma) \to L^2_s(\Sigma)$ which satisfy \be\label{eq:cpcm} C_+ - C_- = \id \ee and \be\label{eq:Cnorm} (Cf)(-\I\kappa) = (0\quad\ast), \qquad (Cf)(\I\kappa) = (\ast\quad 0). \ee Furthermore, $C$ restricts to $L^2_{s}(\Sigma)$, that is \be (C f) (-k) = (Cf)(k) \sigI,\quad k\in\C\backslash\Sigma \ee for $f\in L^2_{s}(\Sigma) \text{ or } L^{\infty}_{s}(\Sigma)$ and if $w_\pm$ satisfy (H.\ref{hyp:sym}) we also have \be \label{eq:symcpm} C_\pm(f w_\mp)(-k) = C_\mp(f w_\pm)(k) \sigI,\quad k\in\Sigma. \ee \end{lemma} \begin{proof} Everything follows from (\ref{eq:symC}) and the fact that $C$ inherits all properties from the classical Cauchy operator. \end{proof} We have thus obtained a Cauchy transform with the required properties. Following Section 7 and 8 of \cite{bc}, we can solve our Riemann--Hilbert problem using this Cauchy operator. Introduce the operator $C_w: L_s^2(\Sigma)\to L_s^2(\Sigma)$ by \be C_w f = C_+(fw_-) + C_-(fw_+),\quad f\in L^2_s(\Sigma) \ee and this operator is also well-defined for $f\in L_s^{\infty}(\Sigma)$ and $C_w f\in L_s^2(\Sigma)$. Furthermore recall from Lemma~\ref{lem:singlesoliton} that the unique solution corresponding to $v\equiv \id$ is given by \begin{align*} & m_0(k)= \begin{pmatrix} f(k) & f(-k) \end{pmatrix}, \\ & f(k) = \frac{1}{1+(2\kappa)^{-1}\gamma^2\E^{t\Phi(\I\kappa)}} \left( 1+\frac{k+\I\kappa}{k-\I\kappa} (2\kappa)^{-1} \gamma^2 \E^{t\Phi(\I\kappa)}\right). \end{align*} Observe that for $\gam=0$ we have $f(k)=1$ and for $\gam=\infty$ we have $f(k)= \frac{k+\I\kappa}{k-\I\kappa}$. In particular, $m_0(k)$ is uniformly bounded away from $\I\kappa$ for all $\gam\in[0,\infty]$. Then we have the next result. \begin{theorem}\label{thm:cauchyop} Assume Hypothesis~\ref{hyp:sym}. Suppose $m$ solves the Riemann--Hilbert problem (\ref{eq:rhp5m}). Then \be\label{eq:mOm} m(k) = (1-c_0)m_0(k) + \frac{1}{2\pi\I} \int_\Sigma \mu(s) (w_+(s) + w_-(s)) \Omega_\kappa(s,k), \ee where $$ \mu = m_+ b_+^{-1} = m_- b_-^{-1} \quad\mbox{and}\quad c_0= \left( \frac{1}{2\pi\I} \int_\Sigma \mu(s) (w_+(s) + w_-(s)) \Omega_\kappa(s,\infty) \right)_{\!1}. $$ Here $(m)_j$ denotes the $j$'th component of a vector. Furthermore, $\mu$ solves \be\label{eq:sing4muc} (\id - C_w) (\mu(k)-(1-c_0)m_0(k) ) = C_w(1-c_0) m_0(k) \ee Conversely, suppose $\ti{\mu}$ solves \be\label{eq:sing4mu} (\id - C_w) (\ti{\mu}(k)-m_0(k)) =C_w m_0(k), \ee and $$ \ti{c}_0= \left( \frac{1}{2\pi\I} \int_\Sigma \ti{\mu}(s) (w_+(s) + w_-(s)) \Omega_\kappa(s,\infty) \right)_{\!1} \ne 1, $$ then $m$ defined via (\ref{eq:mOm}), with $(1-c_0)=(1-\ti{c}_0)^{-1}$ and $\mu=(1-\ti{c}_0)^{-1}\ti{\mu}$, solves the Riemann--Hilbert problem (\ref{eq:rhp5m}) and $\mu= m_\pm b_\pm^{-1}$. \end{theorem} \begin{proof} First of all note that by (\ref{eq:symcpm}) $(\id - C_w)$ satisfies the symmetry condition and hence so do $m_0+(\id - C_w)^{-1} C_w m_0$ and $m$. So if $m$ solves (\ref{eq:rhp5m}) and we set $\mu = m_\pm b_\pm^{-1}$, then $m$ satisfies an additive jump given by $$ m_+ - m_- = \mu (w_+ + w_-). $$ Hence, if we denote the left hand side of (\ref{eq:mOm}) by $\ti{m}$, both functions satisfy the same additive jump. So $m-\ti{m}$ has no jump and must thus solve (\ref{eq:rhp5m}) with $v\equiv \id$. By uniqueness (Corollary~\ref{cor:unique}) $m-\ti{m} = \alpha m_0$ for some $\alpha\in\C$ and by looking at the first component at $k \to\infty$ we see $\alpha=0$, that is $m=\ti{m}$. Moreover, if $m$ is given by (\ref{eq:mOm}), then (\ref{eq:cpcm}) implies \begin{align} \label{eq:singtorhp} m_\pm &= (1-c_0)m_0 + C_\pm(\mu w_-) + C_\pm(\mu w_+) \\ \nn &= (1-c_0)m_0 + C_w(\mu) \pm \mu w_\pm \\ \nn &= (1-c_0)m_0 - (\id - C_w) \mu + \mu b_\pm. \end{align} From this we conclude that $\mu = m_\pm b_\pm^{-1}$ solves (\ref{eq:sing4muc}). Conversely, if $\ti{\mu}$ solves (\ref{eq:sing4mu}) then (\ref{eq:singtorhp}) implies $m_\pm= \mu b_\pm$ which shows that $m$ defined via (\ref{eq:mOm}) solves the Riemann--Hilbert problem (\ref{eq:rhp5m}). \end{proof} \begin{remark} In our case $m_0(k)\in L^{\infty}(\Sigma)$, but $m_0(k)$ is not square integrable and so $\mu\in L^2(\Sigma)+L^{\infty}(\Sigma)$ in general. In the case where the contour $\Sigma$ is bounded $m_0(k) \in L^{\infty}(\Sigma)$ implies that $m_0(k)$ square integrable and we can directly apply $(\id-C_w)^{-1}$ to $m_0(k)$. \end{remark} Note also that in the special case $\gamma=0$ we have $m_0(k)= \rI$ and we can choose $\kappa$ as we please, say $\kappa=\infty$ such that $c_0=\ti{c}_0=0$ in the above theorem. Hence we have a formula for the solution of our Riemann--Hilbert problem $m(k)$ in terms of $m_0+(\id - C_w)^{-1} C_w m_0$ and this clearly raises the question of bounded invertibility of $\id - C_w$ as a map from $L_s^2(\Sigma)\to L_s^2(\Sigma)$. This follows from Fredholm theory (cf.\ e.g. \cite{zh}): \begin{lemma} Assume Hypothesis~\ref{hyp:sym}. The operator $\id-C_w$ is Fredholm of index zero, \be \ind(\id-C_w) =0. \ee \end{lemma} By the Fredholm alternative, it follows that to show the bounded invertibility of $\id-C_w$ we only need to show that $\ker (\id-C_w) =0$. The latter being equivalent to unique solvability of the corresponding vanishing Riemann--Hilbert problem. \begin{corollary} Assume Hypothesis~\ref{hyp:sym}. A unique solution of the Riemann--Hilbert problem (\ref{eq:rhp5m}) exists if and only if the corresponding vanishing Riemann--Hilbert problem, where the normalization condition is replaced by $m(k)= \begin{pmatrix} 0 & 0\end{pmatrix}$ as $k \to\infty$, has at most one solution. \end{corollary} We are interested in comparing two Riemann--Hilbert problems associated with respective jumps $w_0$ and $w$ with $\|w-w_0\|_\infty$ and $\left\Vert w-w_0 \right\Vert_2$ small, where \be \|w\|_\infty= \|w_+\|_{L^\infty(\Sigma)} + \|w_-\|_{L^\infty(\Sigma)} \ee and \be \left\Vert w \right\Vert_2 = \left\Vert w_+ \right\Vert_{L^2(\Sigma)} + \left\Vert w_- \right\Vert_{L^2(\Sigma)}. \ee For such a situation we have the following result: \begin{theorem}\label{thm:remcontour} Assume that for some data $w_0^t$ the operator \be \id-C_{w_0^t}: L^2_s(\Sigma) \to L^2_s(\Sigma) \ee has a bounded inverse, where the bound is independent of $t$, and let $\kappa=\kappa_0$, $\gam^t=\gam_0^t$. Furthermore, assume $w^t$ satisfies \be \|w^t - w_0^t\|_{\infty} \leq \alpha(t) \textnormal{ and } \left\Vert w^t-w_0^t \right\Vert_2 \leq \alpha(t) \ee for some function $\alpha(t) \to 0$ as $t\to\infty$. Then $(\id-C_{w^t})^{-1}: L^2_s(\Sigma)\to L^2_s(\Sigma)$ also exists for sufficiently large $t$ and the associated solutions of the Riemann--Hilbert problems (\ref{eq:rhp5m}) only differ by $O(\alpha(t))$. \end{theorem} \begin{proof} By the boundedness of the Cauchy transform we conclude that \be\nn \left\Vert C_{w^t}-C_{w_0^t} \right\Vert_{L^2(\Sigma )\to L^2(\Sigma )}=O(\alpha(t)). \ee Thus by the second resolvent identity, we infer that $(\id-C_{w^t})^{-1}$ exists for large $t$ and \be \left\Vert (\id -C_{w_0^t})^{-1}-(\id -C_{w^t})^{-1}\right\Vert_{L^2(\Sigma)\to L^2(\Sigma)}=O(\alpha(t)). \ee Next we observe that $$\mu_0^t - \mu^t=(\id-C_{w_0^t})^{-1}C_{w_0^t}(1-c_0)m_0(k)-(\id-C_{w^t})^{-1}C_{w^t}(1-c_0)m_0(k) \in L^2_{s}(\Sigma)$$ and we can therefore conclude \be \left\Vert \mu_0^t - \mu^t \right\Vert_{L^2(\Sigma )} = O(\alpha(t) ). \ee This now implies that for $k$ away from $\Sigma$ \be \left\vert m^t(k)-m_0^t(k)\right\vert = O(\alpha(t) ). \ee For $k \in\Sigma$ we can conclude \be \left\Vert m_{\pm}^t(k)-m_{0,\pm}^t(k) \right\Vert_{L^2(\Sigma )} =O(\alpha(t) ), \ee Note that in all conclusions we have used that $\mu\in L^2(\Sigma)+L^{\infty}(\Sigma)$. \end{proof} \begin{thebibliography}{XXX} \bibitem{as} M. J. Ablowitz and H. Segur, {\em Asymptotic solutions of the Korteweg-de Vries equation}, Stud. Appl. Math {\bf 57}, 13--44 (1977). \bibitem{bc} R. Beals and R. Coifman, {\em Scattering and inverse scattering for first order systems}, Comm. in Pure and Applied Math. {\bf 37}, 39--90 (1984). \bibitem{bdt} R. Beals, P. Deift, and C. Tomei, {\em Direct and Inverse Scattering on the Real Line}, Math. Surv. and Mon. {\bf 28}, Amer. Math. Soc., Rhode Island, 1988. \bibitem{bs} V. S. Buslaev and V. V. Sukhanov, {\em Asymptotic behavior of solutions of the Korteweg-de Vries equation}, Jour. Sov. Math. {\bf 34}, 1905--1920 (1986). \bibitem{deiftbook} P. Deift, {\em Orthogonal Polynomials and Random Matrices: A Riemann--Hilbert Approach}, Courant Lecture Notes {\bf 3}, Amer. Math. Soc., Rhode Island, 1998. \bibitem{dkkz} P. Deift, S. Kamvissis, T. Kriecherbauer, and X. Zhou, {\em The Toda rarefaction problem}, Comm. Pure Appl. Math. {\bf 49}, no. 1, 35--83 (1996). \bibitem{dt} P. Deift, E. Trubowitz, {\em Inverse scattering on the line}, Commun. Pure Appl. Math. {\bf 32}, 121--251 (1979). \bibitem{dvz} P. Deift, S. Venakides and X. Zhou, {\em The collisionless shock region for the long-time behavior of solutions of the KdV equation}, Comm. in Pure and Applied Math. {\bf 47}, 199--206 (1994). \bibitem{dz} P. Deift and X. Zhou, {\em A steepest descent method for oscillatory Riemann--Hilbert problems}, Ann. of Math. (2) {\bf 137}, 295--368 (1993). \bibitem{dz2} P. Deift and X. Zhou, {\em Long time Asymptotics for Integrable Systems. Higher Order Theory}, Commun. Math. Phys. {\bf 165}, 175--191 (1994). \bibitem{es} W. Eckhaus and P. Schuur, {\em The emergence of solitons of the Korteweg-de Vries equation from arbitrary initial conditions}, Math. Meth. in the Appl. Sci. {\bf 5}, 97--116 (1983). \bibitem{evh} W. Eckhaus and A. Van Harten, {\em The Inverse Scattering Transformation and Solitons: An Introduction}, Math. Studies 50, North-Holland, Amsterdam, 1984. \bibitem{ggkm} C. S. Gardner, J. M. Green, M. D. Kruskal, and R. M. Miura, {\em A method for solving the Korteweg-de Vries equation}, Phys. Rev. Letters {\bf 19}, 1095--1097 (1967). \bibitem{hi} R. Hirota, {\em Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons}, Phys. Rev. Letters, {\bf 27} 1192--1194 (1971). \bibitem{kl} M. Klaus, {\em Low-energy behaviour of the scattering matrix for the Schr\"odinger equation on the line}, Inverse Problems {\bf 4}, 505--512 (1988). \bibitem{kt} H. Kr\"uger and G. Teschl, {\em Long-time asymptotics for the Toda lattice in the soliton region}, Math. Z. (to appear). \bibitem{kt2} H. Kr\"uger and G. Teschl, {\em Long-time asymptotics of the Toda lattice for decaying initial data revisited}, arXiv:0804.4693. \bibitem{mar} V. A. Marchenko, {\em Sturm--Liouville Operators and Applications}, Birkh\"auser, Basel, 1986. \bibitem{mu} N. I. Muskhelishvili, {\em Singular Integral Equations}, P. Noordhoff Ltd., Groningen, 1953. \bibitem{proe} S. Pr\"ossdorf, {\em Some Classes of Singular Equations}, North-Holland, Amsterdam, 1978. \bibitem{schuurbook} P. Schuur, {\em Asymptotic Analysis of Soliton problems; An Inverse Scattering Approach}, Lecture Notes in Mathematics {\bf 1232}, Springer, 1986. \bibitem{as2} H. Segur and M. J. Ablowitz, {\em Asymptotic solutions of nonlinear evolution equations and a Painl\'eve transcendent}, Phys. D {\bf 3}, 165--184 (1981). \bibitem{ta} S. Tanaka, {\em On the $N$-touple wave solutions of the Korteweg-de Vries equation}, Publ. Res. Inst. Math. Sci. {\bf 8}, 419--427 (1972/73). \bibitem{wt} M. Wadati and M. Toda, {\em The exact $N$-soliton solution of the Korteweg-de Vries equation}, Phys. Soc. Japan {\bf 32}, 1403--1411 (1972). \bibitem{zh} X. Zhou, {\em The Riemann--Hilbert problem and inverse scattering}, SIAM J. Math. Anal. {\bf 20-4}, 966--986 (1989). \bibitem{zama} V. E. Zakharov and S. V. Manakov, {\em Asymptotic behavior of nonlinear wave systems integrated by the inverse method}, Sov. Phys. JETP {\bf 44}, 106--112 (1976). \bibitem{zakr} N. J. Zabusky and M. D. Kruskal, {\em Interaction of solitons in a collisionless plasma and the recurrence of initial states}, Phys. Rev. Lett. {\bf 15}, 240--243 (1963). \end{thebibliography} \end{document} ---------------0807310723341 Content-Type: application/postscript; name="KdVRHP1.eps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="KdVRHP1.eps" %!PS-Adobe-2.0 EPSF-1.2 %%BoundingBox: 0 0 300 183 %%HiResBoundingBox: 0 0 300 183 %%Creator: (Wolfram Mathematica 6.0 for Mac OS X x86 (32-bit) (June 19, 2007)) %%CreationDate: (Wednesday, July 2, 2008) (15:40:11) %%Title: Clipboard %%DocumentNeededResources: font Times-Roman %%DocumentSuppliedResources: font Mathematica1 %%+ font Times-Roman-MISO %%DocumentNeededFonts: Times-Roman %%DocumentSuppliedFonts: Mathematica1 %%+ Times-Roman-MISO %%DocumentFonts: Times-Roman %%+ Mathematica1 %%+ Times-Roman-MISO %%EndComments /g { setgray} bind def /k { setcmykcolor} bind def /p { gsave} bind def /r { setrgbcolor} bind def /w { setlinewidth} bind def /C { curveto} bind def /F { fill} bind def /L { lineto} bind def /rL { rlineto} bind def /P { grestore} 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543 L 1979 538 L 1979 535 L 1980 533 L 1981 528 L 1982 525 L 1983 523 L 1984 521 L 1984 519 L 1985 517 L 1986 516 L 1986 515 L 1987 514 L 1988 513 L 1988 512 L 1989 512 L 1990 511 L 1991 511 L 1991 512 L 1992 512 L 1993 512 L 1993 513 L 1994 514 L 1995 514 L 1995 515 L 1996 516 L 1997 517 L 1998 519 L 1999 520 L 2000 521 L 2000 522 L 2001 523 L 2002 524 L 2002 525 L 2003 526 L 2004 528 L 2004 530 L 2005 531 L 2006 533 L 2007 535 L 2007 538 L 2008 540 L 2009 546 L 2010 548 L 2011 551 L 2012 558 L 2013 561 L 2013 564 L 2015 570 L 2015 573 L 2016 576 L 2017 579 L 2018 581 L 2018 584 L 2019 586 L 2020 588 L 2020 590 L 2021 592 L 2022 593 L 2022 595 L 2023 596 L 2024 597 L 2024 598 L 2025 599 L 2026 600 L 2026 601 L 2027 602 L 2028 602 L 2028 603 L 2029 604 L 2030 604 L 2030 605 L 2031 605 L 2032 606 L 2033 606 L 2033 606 L 2034 606 L 2035 606 L 2035 606 L 2036 605 L 2037 605 L 2037 604 L 2038 603 L 2039 601 L 2039 600 L 2040 598 L 2041 596 L 2041 594 L 2042 591 L 2043 589 L 2043 586 L 2045 580 L 2048 569 L 2048 566 L 2049 563 L 2050 558 L 2053 548 L 2054 546 L 2054 543 L 2056 539 L 2059 530 L 2060 528 L 2060 526 L 2062 522 L 2063 519 L 2063 517 L 2065 513 L 2066 512 L 2066 510 L 2067 509 L 2068 508 L 2069 507 L 2069 506 L 2070 506 L 2071 505 L 2072 506 L 2072 506 L 2073 507 L 2074 507 L 2074 508 L 2075 510 L 2076 511 L 2077 513 L 2077 514 L 2078 516 L 2080 520 L 2083 527 L 2083 529 L 2084 531 L 2086 535 L 2086 537 L 2087 539 L 2089 543 L 2089 546 L 2090 549 L 2091 554 L 2094 567 L 2095 571 L 2096 574 L 2097 581 L 2098 584 L 2099 587 L 2100 593 L 2101 596 L 2102 598 L 2102 600 L 2103 602 L 2104 603 L 2105 605 L 2105 606 L 2106 607 L 2107 608 L 2107 608 L 2108 609 L 2109 609 L 2109 609 L 2110 609 L 2111 609 L 2111 609 L 2112 609 L 2113 609 L 2114 609 L 2114 609 L 2115 608 L 2116 608 L 2116 607 L 2117 607 L 2118 606 L 2118 605 L 2119 604 L 2120 602 L 2120 601 L 2121 599 L 2122 597 L 2122 595 L 2123 593 L 2124 591 L 2125 585 L 2126 582 L 2127 579 L 2128 573 L 2129 570 L 2129 567 L 2131 560 L 2134 549 L 2134 546 L 2135 543 L 2136 539 L 2139 530 L 2140 528 L 2140 526 L 2142 523 L 2145 515 L 2145 514 L 2146 512 L 2147 510 L 2148 508 L 2148 507 L 2149 505 L 2150 504 L 2151 503 L 2151 502 L 2152 501 L 2153 501 L 2154 501 L 2154 501 L 2155 501 L 2156 502 L 2157 503 L 2157 504 L 2158 505 L 2159 507 L 2160 509 L 2160 511 L 2161 513 L 2163 517 L 2163 519 L 2164 521 L 2166 526 L 2169 535 L 2169 537 L 2170 539 L 2172 544 L 2172 547 L 2173 550 L 2175 555 L 2175 559 L 2176 562 L 2178 568 L 2181 582 L 2181 586 L 2182 589 L 2183 592 L 2184 595 L 2184 598 L 2185 601 L 2186 603 L 2187 605 L 2187 607 L 2188 609 L 2189 610 L 2190 611 L 2190 612 L 2191 613 L 2192 614 L 2193 614 L 2193 614 L 2194 615 L 2195 615 L 2195 615 L 2196 614 L 2197 614 L 2198 614 L 2198 613 L 2199 613 L 2200 612 L 2201 611 L 2201 610 L 2202 609 L 2203 608 L 2204 606 L 2204 605 L 2205 603 L 2206 601 L 2206 598 L 2207 596 L 2208 593 L 2209 590 L 2210 583 L 2211 580 L 2212 577 L 2213 570 L 2216 556 L 2217 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583 L 2301 569 L 2307 545 L 2308 542 L 2309 540 L 2310 534 L 2313 523 L 2314 521 L 2315 518 L 2316 513 L 2317 510 L 2318 508 L 2319 503 L 2320 501 L 2321 499 L 2321 498 L 2322 496 L 2323 495 L 2324 494 L 2324 493 L 2325 493 L 2326 492 L 2327 492 L 2327 492 L 2328 493 L 2329 493 L 2330 493 L 2330 494 L 2331 495 L 2332 496 L 2332 496 L 2334 498 L 2334 499 L 2335 500 L 2337 503 L 2337 504 L 2338 505 L 2339 507 L 2339 508 L 2340 510 L 2341 512 L 2342 516 L 2343 518 L 2343 520 L 2345 525 L 2346 528 L 2346 531 L 2348 537 L 2348 540 L 2349 544 L 2350 550 L 2353 564 L 2354 567 L 2355 570 L 2356 576 L 2359 587 L 2359 590 L 2360 593 L 2362 597 L 2364 607 L 2365 609 L 2366 611 L 2367 615 L 2368 617 L 2368 619 L 2369 621 L 2370 622 L 2371 624 L 2371 625 L 2372 626 L 2373 627 L 2373 628 L 2374 628 L 2375 629 L 2375 629 L 2376 629 L 2377 628 L 2377 627 L 2378 627 L 2379 625 L 2380 624 L 2380 623 L 2381 621 L 2382 619 L 2382 617 L 2384 613 L 2386 604 L 2387 601 L 2388 599 L 2389 594 L 2392 583 L 2397 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626 L 2480 625 L 2481 623 L 2481 622 L 2482 620 L 2483 618 L 2483 615 L 2484 613 L 2486 608 L 2488 596 L 2489 593 L 2490 590 L 2491 583 L 2494 571 L 2499 547 L 2500 544 L 2501 542 L 2502 536 L 2505 524 L 2505 521 L 2506 519 L 2508 513 L 2508 511 L 2509 508 L 2510 503 L 2511 501 L 2512 498 L 2513 496 L 2513 494 L 2514 493 L 2515 491 L 2516 490 L 2516 488 L 2517 487 L 2518 486 L 2519 485 L 2519 485 L 2520 484 L 2521 484 L 2521 483 L 2522 483 L 2523 483 L 2524 483 L 2524 483 L 2525 483 L 2526 483 L 2527 483 L 2527 483 L 2528 483 L 2529 484 L 2530 484 L 2530 485 L 2531 485 L 2532 486 L 2533 487 L 2533 488 L 2534 489 L 2535 491 L 2536 492 L 2536 494 L 2537 496 L 2538 498 L 2539 500 L 2540 505 L 2541 507 L 2542 510 L 2543 515 L 2546 527 L 2547 530 L 2548 533 L 2549 539 L 2552 552 L 2558 579 L 2559 582 L 2559 585 L 2561 592 L 2562 595 L 2562 599 L 2564 605 L 2564 608 L 2565 611 L 2566 613 L 2566 616 L 2567 618 L 2568 621 L 2569 625 L 2570 626 L 2571 628 L 2571 630 L 2572 631 L 2573 632 L 2573 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s P 3 w [ ] 0 setdash p 0 setlinecap 910 559 m 910 503 L s P p newpath 736 591 m 736 783 L 1083 783 L 1083 591 L closepath clip newpath p 16 w 0 g [ ] 0 setdash 2 setlinecap 0 setlinejoin 10 setmiterlimit [1 0 0 1 0 -1083.494] concat 16 w [ ] 0 setdash %%BeginResource: font Mathematica1 %%BeginFont: Mathematica1 %!PS-AdobeFont-1.0: Mathematica1 001.000 %%CreationDate: 8/26/01 at 4:07 PM %%VMusage: 1024 31527 % Mathematica typeface design by Andre Kuzniarek, with Gregg Snyder and Stephen Wolfram. Copyright \(c\) 1996-2001 Wolfram Research, Inc. [http://www.wolfram.com]. All rights reserved. [Font version 2.00] % ADL: 800 200 0 %%EndComments FontDirectory/Mathematica1 known{/Mathematica1 findfont dup/UniqueID known{dup /UniqueID get 5095641 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 20 dict begin /FontInfo 16 dict dup begin /version (001.000) readonly def /FullName (Mathematica1) readonly def /FamilyName (Mathematica1) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /Notice (Mathematica typeface design by Andre Kuzniarek, with Gregg Snyder and Stephen Wolfram. Copyright \(c\) 1996-2001 Wolfram Research, Inc. [http://www.wolfram.com]. All rights reserved. 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