%&amslatex \documentstyle[ctagsplt]{memo-l} \newtheorem{lem}{Lemma}[chapter] \newtheorem{thm}[lem]{Theorem} \newtheorem{defn}[lem]{Definition} \theoremstyle{definition} \newtheorem{exmp}[lem]{Example} \newtheorem{ack}{Acknowledgements} \renewcommand{\theack}{} \theoremstyle{remark} \newtheorem{rem}[lem]{Remark} \numberwithin{equation}{section} % \cprime is used in Russian names for soft sign'. The % references section uses this. \newcommand{\cprime}{$'\mathsurround=0pt$} \newcommand{\seq}[1]{ \setcounter{equation}{#1} } \newcommand{\ssize}{\scriptstyle} \newcommand{\sssize}{\scriptscriptstyle} \newcommand{\hf}{\hspace*{-5mm}} \newcommand{\ho}{\hspace*{-8mm}} \newcommand{\lb}{\label} \newcommand{\ti}{\tilde} \newcommand{\bs}{\backslash} \def\ds{\baselineskip 20pt plus 2pt} \def\ss{\baselineskip 10pt plus 1pt} \ds \voffset=1in \font\bigbold=cmbx10 scaled \magstep2 \newcommand{\ans}{\vrule height.1pt width400pt depth0pt} \overfullrule=0pt %\font\misc=mcyr7 scaled\magstep1 \newcommand{\N}{\Bbb{N}} \newcommand{\Z}{\Bbb{Z}} \newcommand{\C}{\Bbb{C}} \newcommand{\R}{\Bbb{R}} \newcommand{\D}{\cal{D}} \newcommand{\M}{\cal{M}} \newcommand{\E}{\cal{E}} \newcommand{\F}{\cal{F}} \renewcommand{\H}{\cal{H}} \renewcommand{\O}{\cal{O}} \renewcommand{\=}{\operatornamewithlimits{=}} \newcommand{\Div}{\operatorname{Div}} \newcommand{\ess}{\operatorname{ess}} \renewcommand{\Im}{\operatorname{Im}} \newcommand{\loc}{\operatorname{loc}} %%%%%%%%%%%%%%%%%% \font\fg=cmsy5 scaled\magstep 1 \newcommand{\DOT}{\kern-.75ex\raise 2ex \hbox {\fg\char 15}\kern -1.55ex} \renewcommand{\theequation}{\arabic{chapter}.\arabic{equation}} \begin{document} \pagenumbering{roman} \title{ (m)KdV Solitons on the Background of Quasi-Periodic Finite-Gap Solutions } \address{ Department of Mathematics, University of Missouri, Columbia, MO 65211, USA } % Research address for author one \email{ MATHFG@@MIZZOU1.MISSOURI.EDU } \address{ Department of Mathematics, University of Tennessee, Knoxville, TN 37916, USA } %address for % author two \curraddr{} \email{} \subjclass{Primary 35Q51, 35Q53; Secondary 34L40, 58F07} \date{Received by the editor November 7, 1991.} % \thanks will become a 1st page footnote. %\thanks{The first author was supported in part by NSF %Grant \#000000.} \keywords{(m)KdV equation, solitons, algebro-geometric solutions, commutation methods} \maketitle %\tableofcontents \pagestyle{plain} \pagenumbering{roman} \addtocounter{page}{2} \chapter*{Contents} \begin{tabbing} \= Abstract \ 1\\[3mm] \> 1. Introduction \ 2\\[3mm] \>\quad Acknowledgements \ 9\\[3mm] \> 2. Quasi-Periodic Finite-Gap (m)KdV-Solutions \ 10\\[3mm] \> 3. (m)KdV-Soliton Solutions on Quasi-Periodic Finite-Gap Backgrounds.\\[3mm] \> \hspace{2cm} \= I. The Single Commutation Method \ 19\\[3mm] \> 4. (m)KdV-Soliton Solutions on Quasi-Periodic Finite-Gap Backgrounds.\\[3mm] \> \> II. The Double Commutation Method \ 25\\[3mm] \> Appendix A. Single Commutation Methods \ 36\\[3mm] \> Appendix B. Double Commutation Methods \ 45\\[3mm] \> Appendix C. Lax Pairs, $\tau$-Functions and B\" acklund Transformations \58\\[3mm] \> Appendix D. (m)KdV-Soliton Solutions Relative to General Backgrounds \ 64\\[3mm] \> Appendix E. Hyperelliptic Curves and Theta Functions \ 75\\[3mm] \> References \ 84 \end{tabbing} %\input{mkdv-toc.tex} \clearpage \pagenumbering{arabic} \setcounter{page}{0} \begin{abstract} Using commutation methods we present a general formalism to construct Kor\-te\-weg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) $N$-soliton solutions relative to arbitrary (m)KdV background solutions. As an illustration of these techniques we combine them with algebro-geometric methods and Hirota's $\tau$-function approach to systematically derive the (m)KdV $N$-soliton solutions on quasi-periodic finite-gap backgrounds. \end{abstract} \pagestyle{myheadings} \markboth {\centerline{F.~GESZTESY\,\, AND \,\, R.~SVIRSKY}} {\centerline{(M)KDV SOLITONS ON QUASI-PERIODIC FINITE-GAP BACKGROUNDS}} \pagenumbering{arabic} %\setcounter{page}{1} \chapter*{1. Introduction} \addtocounter{page}{-1} \renewcommand{\theequation}{1.\arabic{equation}} \renewcommand{\thelem}{1.\arabic{lem}} In a certain sense, to be made precise below, this paper is a continuation of the earlier work \cite{39}, where a systematic use of commutation methods to transfer solutions of the Korteweg-de Vries (KdV) equation to solutions of the modified Korteweg-de Vries (mKdV) equation was first undertaken. In addition to presenting the general framework of this transfer based on Miura (respectively B\"acklund-Crum-Darboux) type transformations and the notion of (sub)critical one-dimensional Schr\"odinger operators in reference \cite{39}, we also gave a complete treatment of mKdV soliton (and soliton-like) solutions relative to a trivial (i.e., constant) background mKdV solution including the corresponding (inverse) scattering and spectral theory, the essentials of Floquet theory associated with transferring spatially periodic solutions, and hinted at the construction of singular mKdV solutions. However, due to lack of space, the particularly interesting case of quasi-periodic finite-gap mKdV solutions and more generally, mKdV soliton solutions relative to such quasi-periodic finite-gap (m)KdV background solutions (except for the simplest case associated with an elliptic curve) was not treated in \cite{39}. In this paper, among a variety of other things, we shall fill this gap and present a systematic derivation of (m)KdV N-soliton solutions relative to arbitrary (m)KdV background solutions with special emphasis to the quasi-periodic finite-gap case. Before we substantiate these statements in an informal manner and then turn to a detailed description of the content of this paper, we would like to stress that our treatment of $N$-soliton solutions relative to general background solutions is not only new in the mKdV case but also novel in the KdV context. This remark applies in particular to the case of $N$-solitons relative to quasi-periodic finite-gap backgrounds where so far, despite several existing treatments of the problem at hand (referred to in the course of a short historical account at the end of Section~4), no complete discussion of the KdV case has yet been offered in the literature. In order to start our informal discussion of the methods applied in this paper, we first recall the Lax formulation for the (m)KdV equations $$\text {KdV}(V)\equiv V_t-6V\,V_x+V_{xxx}=0, \tag{1.1}$$ $$\text {mKdV}(\phi;\lambda_0)\equiv \phi_t-6(\phi^2+\lambda_0)\phi_x+\phi_{xxx}=0 \tag{1.2}$$ for some $\lambda_0\in \R$, choosing $V,\,\phi\in C^\infty(\R^2)$ for simplicity. (The mKdV equation itself is then defined by $\lambda_0=0$ in (1.2) and simply abbreviated by mKdV$(\phi)=0$.) One has ($[.,.]$ denotes the commutator) \begin{align*} \dfrac d{dt}H-[B_V,H]&=\text { KdV}(V), \tag{1.3}\\ \dfrac d{dt}Q-[B,Q]&=\text { mKdV}(\phi;\lambda_0)\pmatrix 0&1\\ 1&0\endpmatrix, \tag{1.4} \end{align*} where $H,B_V, Q, B$ are differential expressions of the type \begin{align*} H(t)&=-\partial^2_x+V(t,x),\tag{1.5}\\ B_V(t)&=-4\partial^3_x+6V(t,x)\partial_x+3V_x(t,x),\ (t,x)\in \R^2 \tag{1.6} \end{align*} in the KdV case and \begin{align*} Q(t)&=\pmatrix 0&A(t)^+\\ A(t)&0\endpmatrix, \tag{1.7}\\ B(t)&=\pmatrix B_{\tilde V_0}(t)&0\\ 0&B_{V_0}(t)\endpmatrix, \tag{1.8} \end{align*} \begin{align*} A(t)&=\partial_x+\phi(t,x),\ A(t)^+=-\partial_x+\phi(t,x), \tag{1.9}\\ V_0(t,x)&=\phi(t,x)^2+\phi_x(t,x)+\lambda_0, \tag{1.10}\\ \tilde V_0(t,x)&=\phi(t,x)^2-\phi_x(t,x)+\lambda_0\tag{1.11} \end{align*} in the mKdV$(.;\,\lambda_0)$ case. Equation (1.8) together with $$Q(t)^2=\pmatrix A(t)^+A(t)&0\\ 0&A(t)A(t)^+\endpmatrix=\pmatrix \tilde H_0(t)-\lambda_0&0\\ 0&H_0(t)-\lambda_0\endpmatrix, \tag{1.12}$$ where $$H_0(t)=-\partial^2_x+V_0(t,x),\ \tilde H_0(t)=-\partial^2_x+\tilde V_0(t,x), \tag {1.13}$$ illustrates why Miura's transformation \cite{70} in (1.10) and (1.11) effects a close connection between the KdV and mKdV equation. These remarks may further be enhanced by the following observations. If $\psi_0$ is a solution of \begin {equation} (H_0(t)-\lambda_0)\psi(t)=0 \tag {1.14} then $$\text {KdV}(V_0)=0\text { iff }(H_0-\lambda_0)(\partial_t-B_{V_0}) \psi_0=0. \tag {1.15}$$ Therefore, in order to guarantee KdV$(V_0)=0$, we assume $$(\partial_t-B_{V_0}(t))\psi_0(t,x)=\alpha(t)\psi_0(t,x)+ \beta(t)\psi_0(t,x)\int\limits^xdx'\psi_0(t,x')^{-2} \tag {1.16}$$ for some $\alpha,\beta\in C^\infty(\R)$. Then, as shown in Appendix C, $$\text {KdV}(\tilde V_0)=0,\ \text { mKdV}(\phi;\,\lambda_0)=0 \text { iff }\beta=0\text { in (1.16) or }V_0=\tilde V_0=\phi^2=\lambda_0, \tag {1.17}$$ where $$\phi(t,x)=\partial_x\ln \psi_0(t,x) \tag {1.18}$$ according to (1.10). In this way a KdV solution $V_0(t,x)$ gives rise to an mKdV-type solution $\phi(t,x)$ and another KdV solution $\tilde V_0(t,x)$ (in general different from $V_0$). In particular, this formalism yields an auto-B\"acklund transformation for the KdV equation. (We also note that a simple Galilei transformation (see (D.37)--(D.40)) in $\phi$ eliminates $\lambda_0$ in (1.2) and hence yields (proper) mKdV solutions.) Conversely, mKdV-type solutions $\phi (t,x)$ produce KdV solutions $V_0(t,x),\ \tilde V_0(t,x)$ because of Miura's identities $$\text {KdV}(V_0)=(2\phi+\partial_x)\text { mKdV}(\phi;\lambda_0),\ \text {KdV}(\tilde V_0)=(2\phi-\partial_x)\text { mKdV}(\phi;\lambda_0). \tag {1.19}$$ We remark that $\phi$ in (1.18) is real-valued and nonsingular iff the one-dimensional Schr\"odinger operator in $L^2(\R)$ associated with the differential expression (1.5) is nonnegative and hence guarantees the existence of at least one positive solution $\psi_0$ of (1.14). Having sketched the connection between KdV and mKdV solutions, we now turn to a short description of how to introduce soliton solutions relative to general (m)KdV backgrounds. For brevity we shall restrict ourselves here mainly to the one-soliton solution in the KdV case. By definition (see the paragraphs following (D.46), (D.69), and (D.100) in Appendix~D for precise statements) $N$-soliton solutions relative to a fixed background KdV solution $V_0(t,x)$ are in a one-to-one correspondence with the insertion of $N$ additional eigenvalues into spectral gaps of the Schr\"odinger operator in $L^2(\R)$ associated with the differential expression $H_0(t)$ in (1.13). (For simplicity we do not distinguish between the Schr\"odinger operator and its differential expression and denote both by $H_0(t)$ in the following.) This approach of inserting eigenvalues into spectral gaps coincides with the familiar notion of solitons as reflectionless potentials whenever the latter is applicable but also works under considerably more general situations as indicated later on. The two main methods to insert eigenvalues into spectral gaps for Schr\"odinger-type operators which are called the single and double commutation method in the literature, will now be quickly reviewed. In the single commutation method (also called the Crum-Darboux method) one factorizes $H_0(t)$ as \begin{align*} H_0(t)&=A_{\sigma_1}(t)A_{\sigma_1}(t)^*+\lambda_1=-\dfrac {d^2}{dx^2}+V_0(t,x), \tag {1.20}\\ V_0(t,x)&=\phi_{\sigma_1}(t,x)^2+\phi_{\sigma_1,x}(t,x)+\lambda_1, \tag {1.21} \end{align*} where $$A_{\sigma_1}(t)=\dfrac d{dx}+\phi_{\sigma_1}(t,.), \tag {1.22}$$ $$\phi_{\sigma_1}(t,x)=\partial_x\ln \psi_{0,\sigma_1}(t,x), \tag {1.23}$$ \setcounter{equation}{23} \begin{align} \begin{split} \psi_{0,\sigma_1}(t,x)=\dfrac 12(1-\sigma_1)\psi_{0,-1}(\lambda_1,t,x)+\dfrac 12 (1+\sigma_1)\psi_{0,+1}(\lambda_1,t,x),\\ \sigma_1\in (-1,1), \end{split} \end{align} $$\hf (H_0(t)-\lambda_1)\psi_{0,\pm 1}(\lambda_1,t)=0,\ (\partial_t-B_{V_0}(t))\psi_{0,\pm 1}(\lambda_1,t)=0, %\tag {1.25}$$ \begin{align} \begin{split} 0<\psi_{0,\pm 1}(\lambda_1,.,.)\in C^\infty(\R^2),\ \psi_{0,\pm 1}(\lambda_1, t,.)\in L^2((R,\pm \infty)),\\ (t,R)\in \R^2, %\tag {1.26} \end{split} \end{align} and $\lambda_1\in \R$ is chosen to be below the spectrum of $H_0(t)$ for some (and hence for all) $t\in \R$, $$\hf \lambda_1<\inf [\sigma (H_0(t))]. %\tag {1.27}$$ Then commuting $A_{\sigma_1}(t)$ and $A_{\sigma_1}(t)^*$ yields that $H_{0,\sigma_1}(t)$ defined as $$H_{0,\sigma_1}(t)=A_{\sigma_1}(t)^*A_{\sigma_1}(t)+\lambda_1=-\dfrac {d^2}{dx^2}+V_{0,\sigma_1}(t,.), \tag {1.28}$$ $$V_{0, \sigma_1}(t,x)=\phi_{\sigma_1}(t,x)^2-\phi_{\sigma_1,x}(t,x)+\lambda_1 \tag {1.29}$$ has an additional eigenvalue $\lambda_1$ in its spectrum $$\sigma (H_{0,\sigma_1}(t))=\sigma(H_0(t))\cup \{ \lambda_1\},\ \sigma_1\in (-1,1), \tag {1.30}$$ i.e., the eigenvalue $\lambda_1$ has been inserted into the lowest spectral gap $$(-\infty,\ \inf [\sigma (H_0(t))]) \tag {1.31}$$ of $H_0(t)$. In addition, (1.25) implies $$\text {KdV}(V_0)=\text { KdV}(V_{0,\sigma_1})=0,\ \text { mKdV}(\phi_{\sigma_1}; \lambda_1)=0. \tag {1.32}$$ $V_{0,\sigma_1}(t,x)$ is then called a one-soliton KdV solution relative to the background KdV solution $V_0(t,x)$. This procedure is now easily repeated to yield $N$-soliton KdV solutions relative to $V_0(t,x)$. Then mKdV solitons relative to an mKdV background solution $\phi_{0,\pm}(t,x)$ are constructed by introducing eigenvalues into the spectral gap $$(-|\inf [\sigma (H_0(t))]|^{1/2},\ |\inf [\sigma(H_0(t))]|^{1/2}) \tag {1.33}$$ of $Q_{0,\pm}(t)$, where $$H_0(t)=-\dfrac {d^2}{dx^2}+V_0(t,.), \tag {1.34}$$ $$V_0(t,x)=\phi_{0,\pm}(t,x)^2+\phi_{0,\pm,x}(t,x), \tag {1.35}$$ $$\phi_{0,\pm}(t,x)=\partial_x\ln \psi_{0,\pm 1}(0,t,x), \tag {1.36}$$ $$Q_{0,\pm}(t)=\pmatrix 0& -\dfrac d{dx}+\phi_{0,\pm}(t,.)\\ \dfrac d{dx}+\phi_{0,\pm}(t,.)&0\endpmatrix. \tag {1.37}$$ In order to avoid the main drawback of the single commutation method, which inserts eigenvalues only in selected spectral gaps (such as (1.31), (1.33)) of the background operator in question (in order to guarantee nonsingular (m)KdV solutions), we now turn to a short discussion of the double commutation method which inserts eigenvalues into {\bf any} spectral gap of the background operator. For this purpose we again start with a factorization of $H_0(t)$ like (1.20), but this time $\lambda_1$ need not satisfy the restriction (1.27), but merely satisfy $\lambda_1\in \R\backslash \sigma(H_0(t))$. Then \begin{align*} H_0(t)&=A_{1,\pm}(t)A_{1,\pm}(t)^++\lambda_1=-\dfrac {d^2}{dx^2}+V_0(t,.), \tag {1.38}\\ V_0(t,x)&=\phi_{1,\pm}(t,x)^2+\phi_{1,\pm,x}(t,x)+\lambda_1, \tag {1.39} \end{align*} where \setcounter{equation}{39} \begin{align} & A_{1,\pm}(t) = \partial_x+\phi_{1,\pm}(t,x),\; A_{1,\pm}(t)^+=-\partial_x+\phi_{1,\pm}(t,x), \lb{1.40}\\ & \phi_{1,\pm}(t,x) =\partial_x\ln \psi_{0,1,\pm}(t,x), \end{align} $$\psi_{0,1,\pm} \in C^\infty (\R^2),\ \psi_{0,1,\pm} (t,.)\in L^2((R, \pm \infty)),\ (t,R)\in \R^2, %\tag {1.42}$$ $$(H_0(t)-\lambda_1)\psi_{0,1,\pm}(t)=0,\ (\partial_t-B_{V_0}(t))\psi_{0,1,\pm}(t)=0. %\tag {1.43}$$ Performing the first commutation results in the (in general singular) differential expression \begin{align} \tilde H_{0,\pm}(t)&=A_{1,\pm}(t)^+A_{1,\pm}(t)+\lambda_1=-\partial^2_x+\tilde V_{0,\pm}(t,x), \\ %\tag {1.44}\\ \tilde V_{0,\pm}(t,x)&=\phi_{1,\pm}(t,x)^2-\phi_{1,\pm,x}(t,x)+\lambda_1. %\tag {1.45} \end{align} By inspection, $\tilde H_{0,\pm}(t)$ equals \begin{align} \tilde H_{0,\pm} (t)&=A_{1,\gamma_1, \pm}(t) A_{1,\gamma_1, \pm}(t)^++\lambda_1, \\ % \tag {1.46}\\ \tilde V_{0, \pm}(t,x)&=\phi_{1,\gamma_1, \pm}(t,x)^2+\phi_{1,\gamma_1, \pm,x}(t,x)+\lambda_1, \lb{1.47} \end{align} where \begin{gather} A_{1,\gamma_1,\pm}(t) =\partial_x+\phi_{1,\gamma_1,\pm}(t,x),\; A_{1,\gamma_1,\pm} (t)^+ =-\partial_x+\phi_{1,\gamma_1,\pm}(t,x), \lb{1.48}\\ \phi_{1,\gamma_1,\pm}(t,x) =\partial_x\ln \psi_{1,\gamma_{1,\pm}}(t,x), \hskip 2truein \lb{1.49} \end{gather} $$\psi_{1,\gamma_1, \pm}(t,x) =\psi_{0,1,\pm}(t,x)^{-1}[1\mp \gamma_{1,\pm}\int\limits^x_{\pm \infty} dx'\psi_{0,1,\pm} (t,x')^2],\; \gamma_{1,\pm} >0. \lb{1.50}$$ Finally, performing the second commutation, one obtains the well-defined Schr\"odinger operator \setcounter{equation}{50} \begin{gather} H_{\gamma_1,\pm} (t) =A_{1,\gamma_1,\pm}(t)^+A_{1,\gamma_1,\pm}(t)+\lambda_1 =-\dfrac {d^2}{dx^2}+V_{\gamma_1,\pm}(t,x), \lb{1.51} \\ V_{\gamma_1,\pm}(t,x) =V_0(t,x)-2\partial^2_x\ln [1\mp \gamma_{1,\pm} \int\limits^x_{\pm \infty}dx'\psi_{0,1,\pm}(t,x')^2] \lb{1.52} \end{gather} with spectrum $$\sigma (H_{\gamma_{1,\pm}}(t))=\sigma(H_0(t))\cup \{ \lambda_1\} \tag {1.53}$$ and eigenfunction $\psi_{1,\gamma_1,\pm}(t,.)^{-1}\in L^2(\R)$ associated with the eigenvalue $\lambda_1$ of $H_{\gamma_1,\pm} (t)$. In addition one has \setcounter{equation}{53} \begin{gather} \begin{split} \text {KdV}(V_0)=\text { KdV}(\tilde V_{0,\pm})=\text { KdV}(V_{\gamma_1,\pm})=0,\\ \text {mKdV}(\phi_{1,\pm};\lambda_1)=\text { mKdV}(\phi_{1,\gamma_1,\pm};\lambda_1)=0. \lb{1.54} \end{split} \end{gather} (Note however, that $\tilde V_{0,\pm}, \phi_{1,\pm}, \phi_{1,\gamma_1,\pm}$ are singular (m)KdV solutions unless (1.27) holds.) $V_{\gamma_1,\pm}(t,x)$ (like $V_{0,\sigma_1}(t,x))$ is called a one-soliton KdV solution relative to the background KdV solution $V_0(t,x)$. This procedure now can be iterated to yield $N$-soliton KdV solutions relative to $V_0(t,x)$, and an appropriate modification then also yields (nonsingular) mKdV solitons relative to a general mKdV background solution. (We should perhaps mention at this point that even if (1.27) holds, and hence both commutation methods apply, $V_{0,\sigma_1}$ differs from $V_{\gamma_1,\pm}$ unless the background solution $V_0$ was constant in $t$ and $x$.) After this informal discussion of some of the basic techniques used, we now turn to a more detailed description of the content of this paper. As mentioned in the beginning, we address two main topics:\ \ the construction of mKdV soliton solutions relative to general (m)KdV background solutions and, as its major application, a systematic treatment of $N$-soliton solutions relative to quasi-periodic finite-gap backgrounds. According to these two main themes we have split the paper into two parts as follows. Appendices A--D deal with solitons relative to general (m)KdV backgrounds, whereas Sections 2--3 (and Appendix E) deal with the major application to quasi-periodic finite-gap backgrounds. In Appendix A we give a complete treatment of the single commutation method which originated in the work of Jacobi \cite{48} and Darboux \cite{12} (see the historical account at the end of Appendix A). The basic formulas are summarized in (A.46)--(A.54), (A.57)--(A.61), and (A.62)--(A.69). Though special cases (involving constant backgrounds) of these formulas are well known, our systematic treatment appears to be the most general available in the literature. The double commutation technique is thoroughly presented in Appendix B. While again special cases (in connection with constant and spatially periodic backgrounds) are known (see the bibliographical comments at the end of Appendix B), our treatment of general backgrounds $V_0$ is new. The main formulas are summarized in (B.68)--(B.95). Whenever applicable (e.g., in the case of constant or spatially periodic backgrounds), the inverse spectral method of inserting eigenvalues into spectral gaps of the background operator $H_0$ based, e.g., on Marchenko's equation coincides with the double commutation method and produces reflectionless potentials relative to the background $V_0$. However, the double commutation method applies without any modifications to more general circumstances, such as cases where the background operator $H_0$ does not necessarily have uniform spectral multiplicity in its essential spectrum (e.g., $V_0(x)@>>{x\rightarrow \pm \infty}> V_\pm \in \R$ with $V_+\neq V_-$), and hence the notion of a reflectionless potential becomes meaningless without further refinements. While Appendices~A and B deal with stationary (i.e., $t$-independent) situations only and are of purely spectral theoretic nature, we turn to auto-B\"acklund transformations for the KdV equation and to connections between KdV and mKdV solutions related to each other by Miura-type transformations in Appendix~C. We also study the Lax-approach to (m)KdV systems and briefly sketch Hirota's $\tau$-function approach as far as it is needed in Sections~2--4. Moreover, since the double commutation method in general involves singular potentials $V,\phi$ in its intermediate step as described above, we decided to use a very general framework which applies to all known classes of singular (m)KdV solutions (see the references at the beginning of Appendix C). The main results of Appendix C are presented in formulas (C.19), (C.29), (C.30), and (C.36)--(C.40). In particular, the approach leading to (C.29) and (C.30) appears to be novel under these general conditions on $V_0$. In Appendix~D we finally combine Appendices~A--C and introduce $N$-soliton (m)KdV solutions relative to general (m)KdV background solutions. This appendix is naturally divided into two parts, the first devoted to solitons obtained by the single commutation method and the second to solitons constructed by the double commutation technique. As in the case of Appendices~A--C, the results for the special case of constant or spatially periodic backgrounds are known, but our general treatment, summarized in (D.13)--(D.36), (D.48)--(D.52), and (D.66)--(D.100), is new. In order to keep this paper reasonably self-contained, we have added Appendix~E on hyperelliptic curves and their theta functions which introduces all the basic algebro-geometric notions and facts employed in Sections~2--4. In Section~2 we review the well known construction of quasi-periodic finite-gap (m)KdV solutions (in terms of theta functions) which will serve as background solutions in Sections~3 and 4. We present a comprehensive yet fairly complete approach utilizing a combination of (single) commutation methods, algebro-geometric techniques, and Hirota's $\tau$-function approach. In Section~3, using the single-commutation method, we derive $N$-soliton (m)KdV solutions (in terms of Riemann theta functions) relative to the quasi-periodic finite-gap (m)KdV background solutions of Section~2 and study their asymptotic behavior as $x\rightarrow \pm \infty$. Our main new results are summarized in Theorem~3.1. Finally, in Section~4 we use the double commutation technique to derive $N$-soliton (m)KdV solutions relative to the quasi-periodic finite-gap (m)KdV background solutions of Section~2. We study their asymptotic behavior as $x\rightarrow \pm \infty$ summarizing our principal new results in Theorem~4.1. A brief historical account concludes Section~4. \bigskip\bigskip \noindent{\bf Acknowledgements.} F.G. would like to thank Percy Deift for discussions on Baker-Akhiezer functions and commutation methods. \chapter*{2. Quasi-Periodic Finite-Gap (m)KdV Solutions} \addtocounter{page}{-1} \renewcommand{\theequation}{2.\arabic{equation}} \renewcommand{\thelem}{2.\arabic{lem}} \setcounter{lem}{0} In this section we review the well known construction of quasi-periodic finite-gap KdV solutions in terms of Riemann theta functions \cite{13}, \cite{17}--\cite{20}, \cite{37g}, \cite{43}, \cite{47}, \cite{53}, \cite{54}, \cite{56}, \cite{60}, \cite{62}, \cite{65}, \cite{69}, \cite{73}, \cite{89} and, following \cite{35}, also indicate how one can transfer these solutions to the mKdV equation. We closely follow the treatment in \cite{35} which contains a detailed review of the corresponding KdV literature and also a constructive approach to the corresponding mKdV problem. We start by introducing the basic quantities associated with quasi-periodic finite-gap KdV solutions relying on the notation established in Appendix E. In order to avoid the trivial case $g=0$ we shall from now on assume $g\in \N$ throughout until the end of Section 4. \begin{defn} Let $\D_{P_1+\ldots+P_g}$ be a positive divisor of degree $g$ on the hyperelliptic curve $K_g$ of the type (E.3) with $\zeta_\alpha=z^{-\frac 12}_\alpha$ the local coordinate near $P_\infty=(\infty,\infty)\in K_g$ and $q:\ \C\rightarrow \C$ a polynomial. Then the Baker-Akhiezer (B-A) function $\psi_y:K_g\backslash \{P_\infty\}\rightarrow \C_\infty,\ y\in \C$ associated with $K_g,\ P_\infty,\ q,\ \zeta_\alpha,\ \D_{P_1+\ldots+P_g}$ is defined as follows:\\ (i) $\psi_y$ is meromorphic on $K_g\backslash \{P_\infty\}$ with poles at most at the points $P_1,\ldots,P_g$, i.e., the divisor $\D$ of $\psi_y$ on $K_g\backslash \{P_\infty\}$ satisfies $$\D\geq -\D_{P_1+\ldots+P_g}. \tag {2.1}$$ (ii) The function $$(\psi_y\circ z^{-1}_\alpha)(\zeta_\alpha)e^{-iyq(\zeta^{-1}_\alpha)} =c+0(\zeta_\alpha),\ c\in \C \tag {2.2}$$ is holomorphic near $\zeta_\alpha=0$. \end{defn} The next result is well known (see, e.g., \cite{18}). \begin{lem} %l2.1 For a given divisor $\D_{P_1+\ldots+P_g}\in \Div (K_g)$ and given $q, \zeta_\alpha,y$ as in Definition 2.1, the set of B-A-functions forms a linear space. Moreover, if $\D_{P_1+\ldots+P_g}$ is nonspecial and $q$ is such that the divisor $\D_0 = \D + \D_{P_1 + \cdots + P_g} \in \Div (K_g)$ of $\psi_y$ on $K_g\backslash \{P_\infty\}$ is nonspecial too, this linear space has (complex) dimension one. \end{lem} >From now on we shall normalize the B-A function $\psi_y$ such that $c=1$ in (2.2). In order to connect $\psi_y$ with Floquet theory and the KdV equation we introduce the system of differential equations \setcounter{equation}{2} \begin{align} \begin{split} \mu_{j,x}(t,x)&=2R_0(\mu_j(t,x))^{\frac 12}\prod\limits^g_{\Sb \ell=1\\ \ell \neq j\endSb}[\mu_\ell(t,x)-\mu_j(t,x)]^{-1},\\ \mu_{j,t}(t,x)&=2[\Lambda-2\sum\limits^g_{\Sb \ell=1\\ \ell\neq j\endSb} \mu_\ell(t,x)]\mu_{j,x}(t,x),\\ & \hspace*{.4in} (t,x)\in \R^2,\ 1\leq j\leq g, \lb{2.3} \end{split} \end{align} where $$\Lambda=\sum\limits^{2g}_{n=0}E_n. \lb{2.4}$$ Given the initial conditions \begin{align} \begin{split} \hat \mu_j(t_0, x_0)=&(\mu_j(t_0,x_0),\ R_0(\mu_j(t_0, x_0)^{1/2})\in K_g,\\ &\phantom{(}\mu_j(t_0, x_0)\in [E_{2j-1},E_{2j}],\ 1\leq j\leq g \lb{2.5} \end{split} \end{align} for some $(t_0, x_0)\in \R^2$, (2.3) yields a unique solution denoted by $$\underline {\hat \mu}(t,x)=(\hat \mu_1(t,x),\ldots,\ \hat \mu_g(t,x))\in K^g_g, \tag {2.6}$$ with $$\mu_j(t,x)\in [E_{2j-1},\ E_{2j}],\ 1\leq j\leq g,\ (t,x)\in \R^2. \tag {2.7}$$ In fact, (2.3) can be integrated with the help of the Abel map as shown below in Lemma~2.3. For notational simplicity we write $$\underline \alpha_{P_0}(\underline P)=\underline \alpha_{P_0}(\D_{P_1+\ldots+P_g}) \tag {2.8}$$ for $\underline P=(P_1\ldots, P_g)\in K^g_g$ with $P_j\neq P_\ell$ for $j\neq \ell,\ 1\leq j,\ell\leq g$ and $$\underline c_j=(c_{1,j},\ldots, c_{g,j}),\ 1\leq j\leq g, \tag {2.9}$$ where the $c_{j,\ell}$ are defined in (E.14). \begin{lem} Let $(t,x)\in \R^2$ then \begin{align*} \partial_x\underline \alpha_{P_0}(\underline {\hat \mu}(t,x))&=2(-1)^{g+1}\underline c_g, \tag {2.10}\\ \partial_t\underline \alpha_{P_0}(\underline {\hat \mu}(t,x))&=4(-1)^{g+1}[\Lambda \underline c_g+2\underline c_{g_{-1}}]. \tag {2.11} \end{align*} In particular, \begin{multline*} \underline \alpha_{P_0}(\underline {\hat \mu}(t,x))-\underline \alpha_{P_0}(\underline {\hat \mu}(t_0, x_0))=\Big\{2(-1)^{g+1}(x-x_0)\underline c_g\\ +4(-1)^{g+1}(t-t_0)[\Lambda \underline c_g+2\underline c_{g_{-1}}]\Big\} (\text {mod } L_g). \tag {2.12} \end{multline*} \end{lem} The $\mu_j(t,x_0),\ \mu_j(t_0,x)$ are known to be quasi-periodic functions of $t\in \R$ and $x\in \R$ with the same periods for $j=1,\ldots, g$ whose number is less or equal to $g$ (the periods and their number differ in general for $t$ and $x$) \cite{10}, \cite{17}, \cite{51}, \cite{58}, \cite{60}. Moreover, the $\mu_j(t_0,x)$ sweep the closure $\overline {\rho_j}=[E_{2j-1}, E_{2j}]$ of the spectral gap $\rho_j,\ 1\leq j\leq g$ of $H_0$ as $x$ varies in $\R$ (see (2.26) for the definition of $H_0$). Next, choosing $P_0=(E_0, 0)$ from now on, we define the normalized B-A-function \begin{multline*} \psi_0(P,t,x)=\exp[-i(x-x_0)\int\limits^P_{P_0} \omega^{(2)}_0-12i(t-t_0)\int\limits^P_{P_0}\omega^{(2)}_2] \left[\dfrac {\tau_0(P,t,x)}{\tau_0(t,x)}\dfrac {\tau_0(t_0,x_0)}{\tau_0(P,t_0,x_0)}\right],\\ \ P_0=(E_0,0),\ P\in K_g\backslash \{P_\infty\},\ (t,x)\in \R^2, \tag {2.13} \end{multline*} where \begin{multline*} \tau_0(P,t,x)=\theta(\underline \zeta_{P_0}-\underline A_{P_0}(P)+\underline \alpha_{P_0}(\underline {\hat \mu}(t_0, x_0))+((x-x_0)/2\pi)\underline U_0 +(6(t-t_0)/\pi)\underline U_2),\\ P\in K_g, \tag {2.14} \end{multline*} $$\tau_0(t,x)=\tau_0(P_\infty,t,x) \tag {2.15}$$ and $\omega^{(2)}_{2k},\ k=0,1$ are normalized DSK's with a single pole at $P_\infty$ and principal part $$\omega^{(2)}_{2k}=[\zeta^{-2-2k}_\alpha+0(1)]d\zeta_\alpha,\ k=0,1 \tag {2.16}$$ at $P_\infty$ whose $b$-periods are $\underline U_{2k}=(U_{2k,1},\ldots, U_{2k,g})$ $$\int\limits_{aj}\omega^{(2)}_{2k}=0,\ U_{2k,j}=\int\limits_{b_j}\omega^{(2)}_{2k},\ k=0,1. \tag {2.17}$$ (The integration paths from $P_0$ to $P$ in the exponent of (2.13) and in the Abel map $\underline A_{P_0}(P)$ in (2.14) and (2.15) are assumed to be identical.) By (E.39), (E.40), $\psi_0$ is single-valued on $K_g\backslash \{P_\infty\}$. From (E.13) and (E.19) one infers \seq{17} \begin{gather} U_{0,j} =\int\limits_{b_j}\omega^{(2)}_0=4\pi (-1)^{g+1}c_{j,g},\ 1\leq j\leq g, \lb{2.18}\\ U_{2,j} =\int\limits_{b_j}\omega^{(2)}_2=(2\pi/3)(-1)^{g+1}[\Lambda c_{j,g}+2c_{j,g_{-1}}],\ 1\leq j\leq g. \lb{2.19} \end{gather} In addition one can prove $$\int\limits^P_{P_0}\omega^{(2)}_{2k}\=\limits_{\zeta_\alpha\rightarrow 0}-(1+2k)^{-1}\zeta^{-1-2k}_\alpha+0(\zeta_\alpha),\ k=0,1 \tag {2.20}$$ and $$\omega^{(2)}_0=(-i/2)R_0(z)^{-1/2}\prod\limits^g_{j=1}(\lambda_j-z)dz, \tag {2.21}$$ where $$\lambda_j\in (E_{2j-1}, E_{2j}),\ 1\leq j\leq g. \tag {2.22}$$ Moreover, \begin{multline*} \psi_0(P,t,x)\psi_0(P^*,t,x)=\prod\limits^g_{j=1} \left[\dfrac {\mu_j(t,x)-z}{\mu_j(t_0,x_0)-z}\right],\\ P=(z,R_0(z)^{1/2}),\ P^*=(z, -R_0(z)^{1/2}),\ z\in \C. \tag {2.23} \end{multline*} Next, defining $$V_0(t,x)=E_0+\sum\limits^g_{j=1}[E_{2j-1}+E_{2j}-2\mu_j(t,x)],\ (t,x)\in \R^2, \tag {2.24}$$ we recall the celebrated Its-Matveev formula \cite{47}. \begin{thm} \mbox{\rm (\cite{47}, {\rm see also} \cite{65}).} $$V_0(t,x)=E_0+\sum\limits^g_{j=1}(E_{2j-1} +E_{2j}-2\lambda_j)-2\partial^2_x\ln \tau_0(t,x). \tag {2.25}$$ \end{thm} Introducing in $L^2(\R)$ the Schr\"odinger-type operator $$H_0(t)=-\dfrac {d^2}{dx^2}+V_0(t,.),\ \D (H_0(t))=H^2(\R),\ t\in \R \tag {2.26}$$ and for $\partial^m_x V\in L^\infty (\R^2),\ m=0,1$ the operator $$\hspace*{2mm} B_V(t)=-4\dfrac {d^3}{dx^3}+6V(t,.)\dfrac d{dx}+3V_x(t,.), \ \D(B_V(t))=H^3(\R),\ t\in \R \tag {2.27}$$ one verifies the Lax representation $$\dfrac d{dt}H_0-[B_{V_0}, H_0]=\text { KdV }(V_0) \tag {2.28}$$ either as an operator relation on $H^5(\R)$ or in the sense of differential expressions. \begin{thm} One has $$(H_0(t)-z)\psi_0(P^{(*)},t)=0, \tag {2.29}$$ $$(\partial_t-B_{V_0}(t))\psi_0(P^{(*)}, t)=0 \tag {2.30}$$ in the weak sense and hence $$\text {KdV }(V_0)=0. \tag {2.31}$$ (Here $P^{(*)}$ denotes $P$ or $P^*$.) Moreover, the spectrum of $H_0(t)$ is $t$-independent, purely absolutely continuous, of multiplicity two, and consists of finitely many gaps $$\sigma (H_0(t))=\bigcup\limits^g_{j=1}[E_{2(j-1)}, E_{2j-1}]\cup [E_{2g}, \infty),\ t\in \R. \tag {2.32}$$ In addition, $V_0\in C^\infty(\R^2)$ is a (real-valued) quasi-periodic function of $t\in \R$ and $x\in \R$. $V_0$ is periodic in $x$ with fundamental period $a>0$ iff \setcounter{equation}{32} \begin{align} \begin{split} a & =\min \left\{\omega>0|2\omega|c_{j,g}|\in \N,\ 1\leq j\leq g\right\}\\ & =\min \left\{\omega>0|(\omega|U_{0,j}|\,/2\pi)\in \N,\ 1\leq j\leq g\right\}. \lb{2.33} \end{split} \end{align} Finally, (2.25) describes all KdV solutions that yield the finite gap spectrum (2.32) (parametrized by the initial conditions (2.5)). \end{thm} As can be read off from (2.23), the divisor $\D_{\hat \mu_1(t,y)+\ldots +\hat \mu_g(t,y)}\in \sigma^gK_g,\ y\in \R$ is a Dirichlet divisor associated with $H_0(t)$, i.e., the projections $\mu_j(t,y)=\ti \pi(\hat \mu_j(t,y)),\ 1\leq j\leq g$ are the $g$ eigenvalues of $H_0(t)$ restricted to the interval $[y,\infty)$ and $(-\infty,y]$ with a Dirichlet boundary condition at $y$. The above results can be found in \cite{13}, \cite{17}, \cite{20}, \cite{47}, \cite{69} and are reviewed in \cite{18}, \cite{19}, \cite{35}, \cite{43}, \cite{53}, \cite{54}, \cite{56}, \cite{60}, \cite{62}, \cite{65}, \cite{73}, \cite{89}. In the remainder of this section we shall recall the corresponding construction of quasi-periodic finite-gap mKdV solutions following \cite{35}. In order to get nonsingular mKdV solutions we assume $H_0(t)$ to be a finite-gap operator of the type (2.26) $$H_0(t)=-\dfrac {d^2}{dx^2}+V_0(t,.),\ \D (H_0(t))=H^2(\R),\ t\in \R \lb {2.34}$$ such that H_0(t)=H_2(t)+\lambda_0,\ \lambda_00$iff (2.33) holds. \end{thm} \begin{rem} %2.8 Although we assume$E_j0$is determined by $$\kappa_{0,j}=i\int\limits^{Q_{\ell_{0,j}}}_{P_0}\omega^{(2)}_0,\ \ell_{0,j}=\max \left\{\ell\in \{j,\ldots, N\}|\sigma_\ell\in (-1,1)\right\} \tag {3.42}$$ if$\left\{\ell\in \{j,\ldots, N\}\big|\sigma_\ell\in (-1,1)\right\}$is nonempty. (In the trivial case where\hfill\break$\left\{\ell\in \{j,\ldots, N\}\big|\sigma_\ell \in (-1,1)\right\}=\emptyset$, i.e., where$\sigma_\ell\in \{-1,1\}$for all$j\leq \ell\leq N$, equality in (3.40) and (3.41) holds for all$(t,x)\in \R^2$with$0(e^{-2\kappa_{0,j}|x|})replaced by zero.) Finally, a comparison of the leading order terms in (3.40) and (3.41), taking into account (3.33) and (3.39), yields that \setcounter{equation}{42} \begin{align} \begin{split} & V_{0,\epsilon_{j,+},\ldots,\epsilon_{N,+}}(t,x) =V_{0,\epsilon_{j,-},\ldots,\epsilon_{N,-}}(t,x), \; \phi_{\epsilon_{j,+},\ldots,\epsilon_{N,+}}(t,x) =\phi_{\epsilon_{j,-},\ldots,\epsilon_{N,-}}(t,x)\\ & \text { iff }2\sum\limits^N_{\ell=j}\beta_\ell \underline A_{P_0} (Q_\ell)=\underline 0(\text {mod } L_g), \lb{3.43} \end{split} \end{align} where $$\beta_\ell=\cases 1,\ \sigma_\ell\in (-1,1)\\ 0,\ \sigma_\ell\in \{-1,1\}.\endcases \tag {3.44}$$ \end{thm} \begin{pf} It remains to verify (3.40)--(3.42). This follows from $$V_{0,\sigma_j,\ldots,\sigma_N}(t,x)=\phi_{\sigma_j,\ldots, \sigma_N}(t,x)^2-\phi_{\sigma_j,\ldots,\sigma_N,x}(t,x)+\lambda_j, \tag {3.45}$$ $$\phi_{\sigma_j,\ldots, \sigma_N}(t,x)=\partial_x\ln \psi_{0,\sigma_j,\ldots,\sigma_N}(t,x) \tag {3.46}$$ and the fact that for fixedt\in \R$$$\psi_{0,\sigma_j,\ldots, \sigma_N}(t,x)\=\limits_{x\rightarrow \pm \infty}C_{j,\ldots,N}\psi_{0,\epsilon_{j,\pm},\ldots, \epsilon_{N,\pm}}(t,x)[1+0(e^{-2\kappa_{0,j}|x|})] \tag {3.47}$$ for some constant$C_{j,\ldots,N}$and with$\kappa_{0,j}$defined in (3.42) as can be verified from (D.13), (D.14), (D.26), (D.28), and (D.29). \end{pf} The Galilei-type transformation (D.38) then produces KdV and mKdV solu\-tions as illustrated in (D.37)--(D.46) and their interpretation as soliton solutions relative to quasi-periodic background (m)KdV solutions is identical to the one in the paragraphs following (D.46) and (D.52) respectively. As briefly mentioned after (3.25), Theorem 3.1 (especially (3.30)--(3.33)) illustrates the effects of single commutation as translations by$[\underline A_{P_0}(P_\infty)-\underline A_{P_0}(Q_j)]$on the Jacobi variety pioneered in \cite{7}, put into the context of B\"acklund transformations for the KdV equation in \cite{24}--\cite{26}, and discussed in connection with spectral theory of Hill's equation in \cite{35}, \cite{37g}, \cite{66}--\cite{68}. For notational convenience only we have chosen$Q_j=(\lambda_j, \lim\limits_{\epsilon\downarrow 0}R_0(\lambda_j+i\epsilon)^{1/2}),\\ 1\leq j\leq N$in this section (see (3.29)). Our results trivially extend to the case where$Q_j=(\lambda_j,\ \lim\limits_{\epsilon \downarrow 0}R_0(\lambda_j+i\epsilon s_j)^{1/2}),\ s_j\in \{-1,1\},\ 1\leq j\leq N$. The obvious limitations of the single commutation technique concerning the insertion of eigenvalues only into the lowest spectral gap$(-\infty, E_0)$of$H_0$will be overcome in the next Section 4 where we shall use the double commutation method instead. \chapter*{4. (m)KdV Soliton Solutions on Quasi-Periodic Finite-Gap Backgrounds. II.\ The Double Commutation Method} %\addtocounter{page}{-1} \renewcommand{\theequation}{4.\arabic{equation}} \renewcommand{\thelem}{4.\arabic{lem}} \setcounter{lem}{0} In this section we construct (m)KdV soliton solutions relative to quasi-periodic finite-gap (m)KdV background solutions using the double commutation technique described in Appendix B and in the second part of Appendix D. In particular, we shall avoid the shortcomings of the single commutation method which distinguishes the lowest spectral gap$(-\infty, E_0)$of the background operator$H_0(t)$. Employing the notation of Section 2 and Appendices B and D we assume that$H_0(t),\ V_0(t,x),\ \Lambda_0,\ \psi_{0,\pm}(\lambda,t,x)=\psi_{0,\pm 1}(\lambda, t,x),\ \tau_0(t,x),\ \tau_{\pm 1}(\lambda,t,x)=\tau_{0,\pm 1}(\lambda,t,x)$, and$\underline \alpha_{P_0}(\underline {\hat \mu}_0(t,x))$are defined as in (3.1)--(3.3), (3.5)--(3.9) and hence$V_0$is the quasi-periodic finite-gap background KdV solution (2.25). Following the strategy of Section 3 we shall first relate the leading asymptotic terms$V_{0,\ldots,0,\pm}, V_{\infty,\ldots,\infty,\pm}$and$\phi_{0,\ldots,0,\pm},\ \phi_{\infty,\ldots,\infty,\pm}$in (D.70) and (D.89) to the theta function associated with the underlying hyperelliptic curve$K_g$and the$\tau-functions introduced in (C.36)--(C.40) and (2.49)--(2.54). Writing $$V_0(t,x)=\phi_{1,\pm}(t,x)^2+\phi_{1,\pm,x}(t,x)+\lambda_1, \tag {4.1}$$ (2.54)--(2.59) (or (D.56)--(D.58)) and (C.36)--(C.40) yield \begin{align*} &\text {KdV }(\tilde V_{1,0,\pm}+\lambda_1)=\text { KdV}(V_{2,1,\pm}+\lambda_1)=0,\tag {4.2}\\ &\text {mKdV }(\phi_{1,\pm}; \lambda_1)=0, \tag {4.3} \end{align*} where \setcounter{equation}{3} \begin{align} \begin{split} \phi_{1,\pm}(t,x) =\mp i\int\limits^{Q_1}_{P_0}\omega^{(2)}_0 +\partial_x\ln [\tau_{\pm 1}(t,x)/\tau_0(t,x)],&\\ Q_1=(\lambda_1, \lim\limits_{\epsilon \downarrow 0} R_0(\lambda_1+i\epsilon)^{1/2}),& \lb{4.4} \end{split} \end{align} \begin{align} \begin{split} &\tilde V_{1,0,\pm}(t,x)+\lambda_1=\phi_{1,\pm}(t,x)^2-\phi_{1,\pm,x}(t,x) +\lambda_1\\ &=\phi_{1,\infty,\pm}(t,x)^2+\phi_{1,\infty,\pm,x}(t,x)+\lambda_1\\ & =\phi_{1,0,\pm}(t,x)^2+\phi_{1,0,\pm,x}(t,x)+\lambda_1\\ &=V_{2,1,\pm}(t,x)+\lambda_1 =\Lambda_0-2\partial^2_x\ln \tau_{\pm 1}(t,x), \lb{4.5} \end{split} \end{align} with {\allowdisplaybreaks \begin{align*} &\tau_{\pm 1}(t,x)=\tau_{\pm 1}(\lambda_1, t,x), \tag {4.6}\\ &\phi_{1,0,\pm}(t,x)=\partial_x\ln \psi_{1,0,\pm}(t,x) =-\phi_{1,\pm}(t,x), \tag {4.7} \\ &\psi_{1,0,\pm}(t,x)=\psi_{0,1,\pm}(t,x)^{-1}, \tag {4.8} \\ &\phi_{1,\infty,\pm}(t,x)=\partial_x\ln \psi_{1,\infty,\pm}(t,x), \tag {4.9}\\ &\psi_{1,\infty,\pm}(t,x)=\mp \psi_{0,1,\pm}(t,x)^{-1} \int\limits^x_{\pm \infty}dx'\psi_{0,1,\pm}(t,x')^2. \tag {4.10} \end{align*} } We emphasize that unless\lambda_1\leq E_0$, both$\tilde V_{1,0,\pm}=V_{2,1,\pm}$and$\phi_{1,\pm}$are singular solutions since for$\lambda_1>E_0,\ \lambda_1\in \rho(H_0)\ \tau_{\pm 1}$will have infinitely many zeros by Sturm oscillation arguments. However, since Appendix C (and especially (C.36)--(C.40)) applies to such singular solutions we simply iterate the procedure and perform the second commutation in order to complete the first double commutation step. We rewrite$\tau_{\pm 1}$in (4.6) as $$\tau_{\pm 1}(t,x)=\theta(\underline \zeta_{P_0}\mp \underline A_{P_0}(P_\infty)+\underline \alpha_{P_0}(\underline {\hat \mu}_{\pm 1}(t,x))), \tag {4.11}$$ with $$\underline \alpha_{P_0}(\underline {\hat \mu}_{\pm 1}(t,x))=\pm [\underline A_{P_0}(P_\infty)-\underline A_{P_0}(Q_1)]+\underline \alpha_{P_0}(\underline {\hat \mu}_0(t,x)). \tag {4.12}$$ Then (C.36)--(C.40) and (D.59)--(D.65) with$\psi_{1,\gamma_1,\pm}$replaced by$\psi_{1,0,\pm}=\psi^{-1}_{0,1,\pm}$respectively by$\psi_{1,\infty,\pm}$(substitute$\psi_{1,\gamma_1,\pm}$by$\gamma^{-1}_{1,\pm}\psi_{1,\gamma_1,\pm}$and let$\gamma_{1,\pm}\rightarrow \infty) yield $$\text {KdV }(V_{0,\pm})=\text {KdV }(V_{\infty,\pm})=0, \tag {4.13}$$ $$\text {mKdV }(\phi_{1,0,\pm}; \lambda_1)=\text {mKdV }(\phi_{1,\infty,\pm}; \lambda_1)=0, \tag {4.14}$$ where \setcounter{equation}{14} \begin{align} \begin{split} & \phi_{1,0,\pm}(t,x) =\pm i\int\limits^{Q_1}_{P_0}\omega^{(2)}_0+\partial_x\ln [\tau_{\pm 1,\mp 1}(t,x)/\tau_{\pm 1}(t,x)]\\ &=-\phi_{1,\pm}(t,x)=\pm i\int\limits^{Q_1}_{P_0}\omega^{(2)}_0+\partial_x[\ln \tau_0(t,x)/\tau_{\pm 1}(t,x)], \lb{4.15} \end{split} \end{align} $$\phi_{1,\infty,\pm}(t,x)=\mp i\int\limits^{Q_1}_{P_0}\omega^{(2)}_0+\partial_x\ln [\tau_{\pm 1, \pm 1}(t,x)/\tau_{\pm 1}(t,x)], \lb{4.16}$$ \begin{align} \begin{split} V_{0,\pm}(t,x)& =\phi_{1,0,\pm}(t,x)^2-\phi_{1,0,\pm,x}(t,x)+\lambda_1\\ &=\Lambda_0-2\partial^2_x\ln \tau_{\pm 1, \mp 1}(t,x)\\ &=\phi_{1,\pm}(t,x)^2+\phi_{1,\pm, x}(t,x)+\lambda_1\\ & =V_0(t,x)=\Lambda_0-2\partial^2_x\ln \tau_0(t,x), \lb{4.17} \end{split} \end{align} \begin{align} \begin{split} & V_{\infty,\pm}(t,x) =\phi_{1,\infty,\pm}(t,x)^2-\phi_{1,\infty,\pm,x}(t,x)+\lambda_1\\ &=\Lambda_0-2\partial^2_x\ln \tau_{\pm 1, \pm 1}(t,x). \lb{4.18} \end{split} \end{align} Here $$\tau_{\epsilon_1,\epsilon_2}(\lambda,t,x)=\theta(\underline \zeta_{P_0}-\epsilon_2\underline A_{P_0}(P)+\underline \alpha_{P_0}(\underline {\hat \mu}_{\epsilon_1}(t,x))), \ \epsilon_1, \epsilon_2\in \{-1,1\}, \lb{4.19}$$ \begin{align} \begin{split} \tau_{\epsilon_1,\epsilon_2}(t,x) & =\tau_{\epsilon_1,\epsilon_2}(\lambda_1, t,x)\\ & =\theta(\underline \zeta_{P_0}-\epsilon_2\underline A_{P_0}(Q_1)+\underline \alpha_{P_0}(\underline {\hat \mu}_{\epsilon_1}(t,x)))\\ & =\theta(\underline \zeta_{P_0}-\epsilon_2\underline A_{P_0}(P_\infty)+\underline \alpha_{P_0}(\underline {\hat \mu}_{\epsilon_1,\epsilon_2}(t,x))), \lb{4.20} \end{split} \end{align} with $$\underline \alpha_{P_0}(\underline {\hat \mu}_{\epsilon_1,\epsilon_2}(t,x))=(\epsilon_1+\epsilon_2)[\underline A_{P_0}(P_\infty)-\underline A_{P_0}(Q_1)]+\underline \alpha_{P_0}(\underline {\hat \mu}_0(t,x)) \tag {4.21}$$ andQ_1$and$P$defined in (4.4) and (3.10). In particular, $$\tau_{\pm 1,\mp 1}(t,x)=\tau_0(t,x), \tag {4.22}$$ $$\tau_{\pm 1, \pm 1}(t,x)=\theta (\underline \zeta_{P_0}\mp\underline A_{P_0}(P_\infty)\pm 2[\underline A_{P_0}(P_\infty)-\underline A_{P_0}(Q_1)]+\underline \alpha_{P_0}(\underline {\hat \mu}_0(t,x))). \tag {4.23}$$ The fact that the leading asymptotic terms$V_{0,\pm}$and$V_{\infty,\pm}$in (D.70) for$V_{\gamma_1,\pm}$in general differ as$x\rightarrow +\infty$and$x\rightarrow -\infty$has been originally observed in \cite{31}, \cite{57}, \cite{86} in the context of spatially periodic backgrounds$V_0$. Since $$\theta(\underline \zeta_{P_0}-\underline A_{P_0}(P)+\underline \xi_{\pm 1, \pm 1}(t,x))\not\equiv 0 \tag {4.24}$$ with respect to$P\in K_g$, where $$\underline \xi_{\pm 1,\pm 1}(t,x)=\pm 2[\underline A_{P_0}(P_\infty)-\underline A_{P_0}(Q_1)]+\underline \alpha_{P_0}(\underline {\hat \mu}_0(t,x)), \tag {4.25}$$ there exists a unique divisor$\D_{\pm 1,\pm 1}(t,x)\in \sigma^gK_g$such that $$\xi_{\pm 1,\pm 1}(t,x)=[\underline \alpha_{P_0}(\D_{\pm 1,\pm 1}(t,x))](\text {mod }L_g) \tag {4.26}$$ by Theorem E.8 (with$\underline \xi$replaced by$\underline \xi+\underline \zeta_{P_0}$) and hence we may identify\hfill\break$\underline {\hat \mu}_{\pm 1,\pm 1}(t,x)$and$\D_{\pm 1,\pm 1}(t,x)$(according to our simplified notation (2.8)). The divisor$\D_{\pm 1,\pm 1}(t,x)$is nonspecial by Theorem E.12 (iii) (with$\underline \xi$replaced by$\underline \xi+\underline \zeta_{P_0}$) and determined by the$gzeros of (4.24). This completes the first double commutation step. Iterating the above procedure then yields \seq{26} {\allowdisplaybreaks \begin{gather} \text {KdV }(V_{\Sb 0,\ldots,0,\pm\\ N\text { times }\endSb}) =\text { KdV }(V_{\Sb \infty,\ldots, \infty,\pm\\N\text { times }\endSb})=0, \lb{4.27}\\ \text {mKdV }(\phi_{\Sb N,0,\ldots,0,\pm\\ N\text { times }\endSb};\lambda_n)=\text { mKdV }(\phi_{\Sb N,\infty,\ldots,\infty,\pm\\ N\text { times}\endSb};\lambda_N)=0, \lb{4.28} \end{gather} } where \setcounter{equation}{28} {\allowdisplaybreaks \begin{gather} \phi_{\Sb N,0,\ldots,0,\pm\\ N\text { times }\endSb}(t,x) =-\phi_{1,\pm}(t,x)=\pm i\int\limits^{Q_N}_{P_0}\omega^{(2)}_0+\partial_x\ln [\tau_0(t,x)/\tau_{\pm 1}(t,x)], \lb{4.29}\\ \phi_{\Sb N,\infty,\ldots,\infty,\pm\\ N\text { times }\endSb}(t,x) =\mp i\int\limits^{Q_N}_{P_0}\omega^{(2)}_0+\partial_x\ln [\tau_{\Sb \pm 1,\ldots,\pm 1\\ 2N\text { times }\endSb}(t,x) /\tau_{\Sb \pm 1,\ldots,\pm 1 \\ (2N-1)\text { times }\endSb}(t,x)], \lb{4.30} \end{gather} } {\allowdisplaybreaks \begin{align} \begin{split} V_{\Sb 0,\ldots,0,\pm\\ N \text { times }\endSb}(t,x) & =\phi_{\Sb N,0,\ldots,0,\pm\\ N\text { times }\endSb}(t,x)^2-\phi_{\Sb N,0,\ldots,0,\pm,x\\ N\text { times }\endSb}(t,x)+\lambda_N\\ & =V_0(t,x)=\Lambda_0-2\partial^2_x\ln \tau_0(t,x), \lb{4.31} \end{split} \end{align} } \begin{align} \begin{split} V_{\Sb \infty,\ldots,\infty,\pm\\ N\text { times }\endSb}(t,x) & =\phi_{\Sb N,\infty,\ldots,\infty,\pm\\ N\text { times }\endSb} (t,x)^2-\phi_{\Sb N,\infty,\ldots,\infty,\pm,x\\ N \text { times }\endSb} (t,x)+\lambda_N\\ & =\Lambda_0-2\partial^2_x\ln \tau_{\Sb \pm 1,\ldots,\pm 1\\ 2N\text{ times }\endSb}(t,x).\lb{4.32} \end{split} \end{align} Here \seq{32} \begin{align} \begin{split} &\tau_{\Sb \pm 1,\ldots,\pm 1\\ 2N\text { times }\endSb}(t,x)\\ & =\theta(\underline \zeta_{P_0}\mp \underline A_{P_0}(P_\infty)\pm 2\sum\limits^N_{j=1}[\underline A_{P_0}(P_\infty)-\underline A_{P_0}(Q_j)]+\underline \alpha_{P_0}(\underline {\hat \mu}_0(t,x))), \lb{4.33} \end{split} \end{align} \begin{align} \begin{split} &\tau_{\Sb \pm 1,\ldots,\pm 1\\ (2N-1)\text { times }\endSb}(t,x)\\ & =\theta(\underline \zeta_{P_0}\mp \underline A_{P_0}(Q_N)\pm 2\sum\limits^{N-1}_{j=1}[\underline A_{P_0}(P_\infty)-\underline A_{P_0}(Q_j)]+\underline \alpha_{P_0}(\underline {\hat \mu}_0(t,x))), \lb{4.34} \end{split} \end{align} with $$\hspace*{2mm} \underline \alpha_{P_0}(\underline {\hat \mu}_0(t,x))=\underline \alpha_{P_0}(\underline {\hat \mu}(t_0,x_0))+((x-x_0)/2\pi)\underline U_0+(6(t-t_0)/\pi)\underline U_2, \tag{4.35}$$ $$Q_j=(\lambda_j,\lim\limits_{\epsilon \downarrow 0}R_0(\lambda_j+i\epsilon)^{1/2}),\ 1\leq j\leq N. \tag {4.36}$$ The divisor\underline {\hat \mu}_{\Sb \pm 1,\ldots,\pm 1\\ 2N\text { times }\endSb}(t,x)\in \sigma^gK_g$defined by $$\hspace*{2mm} \underline \alpha_{P_0}(\underline {\hat \mu}_{\Sb \pm 1,\ldots,\pm 1\\ 2N\text { times }\endSb}(t,x))=\pm 2\sum\limits^N_{j=1}[\underline A_{P_0}(P_\infty)-\underline A_{P_0}(Q_j)]+\underline \alpha_{P_0}(\underline {\hat \mu}_0(t,x))\tag {4.37}$$ is nonspecial and uniquely determined by the$g$zeros$\{\underline {\hat \mu}_{\Sb \pm 1,\ldots,\pm 1,m\\ 2N \text { times }\endSb}\}^g_{m=1}$of $$\theta(\underline \zeta_{P_0}-\underline A_{P_0}(.)\pm 2\sum\limits^N_{j=1}[\underline A_{P_0}(P_\infty)-\underline A_{P_0}(Q_j)]+\underline \alpha_{P_0}(\underline {\hat \mu}_0(t,x))). \tag {4.38}$$ As in Section 3 (see (3.36)--(3.38)) these zeros satisfy the system of differential equations (2.3), (2.4) with initial conditions at$(t_0, x_0)$of the type (2.5). They are located in the closure of the corresponding spectral gaps of$\sigma (H_{\infty,\ldots,\infty,\pm}(t))=\sigma (H_0(t))$as in (2.7), i.e., the projections $$\ti\pi (\hat \mu_{\Sb \pm 1,\ldots,\pm 1,m\\ 2N\text { times }\endSb}(t,x))\in [E_{2m-1}, E_{2m}],\ 1\leq m\leq g,\ (t,x)\in \R^2\tag {4.39}$$ sweep$\overline {\rho_m}=[E_{2m-1}, E_{2m}]$as$x$varies in$\R$. In particular, $$\underline {\hat \mu}_{\Sb \pm 1,\ldots,\pm 1\\ 2N\text { times }\endSb}(t,x)=\D_{\hat \mu_{\Sb \pm 1,\ldots, \pm 1,1\\ 2N\text { times }\endSb}(t,x)+\ldots+\hat \mu_{\Sb\pm 1,\ldots,\pm 1,g\\ 2N\text { times }\endSb}(t,x)} \tag {4.40}$$ is the Dirichlet divisor associated with$V_{\Sb \infty,\ldots, \infty,\pm\\ N\text { times }\endSb}(t,x)$(cf. the paragraph following Theorem~2.5). While (4.31)--(4.33) describe the leading asymptotic terms$V_{0,\ldots,0,\pm}, V_{\infty,\ldots, \infty,\pm}$of$V_{\gamma_1,\ldots,\gamma_N,\pm}$in (D.70) in terms of the Riemann theta function of$K_g$, the solutions$\phi_{N,0,\ldots,0,\pm}, \phi_{N,\infty,\ldots,\infty,\pm}$in (4.29) and (4.30) are singular in general. In order to relate the nonsingular leading asymptotic terms$\phi_{0,\ldots,0, \epsilon,\epsilon'}, \phi_{\infty,\ldots,\infty,\epsilon, \epsilon'}$in (D.89) to the theta function of$K_g$we now assume that $$H_{\gamma_1,\ldots,\gamma_N,\epsilon}(t)\geq 0,\ \epsilon=\pm, \gamma_j>0,\ 1\leq j\leq N \tag {4.41}$$ for some (and hence for all)$t\!\in\!\R$and define$\phi_{\gamma_1,\ldots,\gamma_N,\epsilon,\epsilon'},\ \phi_{0,\ldots,0,\epsilon, \epsilon'}=\phi_{0,\epsilon'}$, and$\phi_{\infty,\ldots,\infty,\epsilon, \epsilon'}$,$\epsilon, \epsilon'\in \{\pm\}$as in (D.86), (D.90), and (D.91). Then (D.87)--(D.89) and (D.92)--(D.99) hold. Writing $$V_0(t,x)=\phi_{0,\epsilon}(t,x)^2+\phi_{0,\epsilon,x}(t,x) =\Lambda_0-2\partial^2 _x\ln \tau_0(t,x), \tag {4.42}$$ equation (2.54) (with$\lambda_0=0) implies $$\phi_{0,\epsilon}(t,x)=-\epsilon i\int\limits^{Q_0}_{P_0}\omega^{(2)}_0+\partial_x\ln [\tau_\epsilon (t,x)/\tau_0(t,x)],\ Q_0=(0,\lim\limits_{\epsilon \downarrow 0}R_0(i\epsilon)^{1/2}), \tag {4.43}$$ where $$\tau_\epsilon (\lambda,t,x)=\theta(\underline \zeta_{P_0}-\epsilon \underline A_{P_0}(P)+\underline \alpha_{P_0}(\underline {\hat \mu}_0(t,x))),\ P=(\lambda, \lim\limits_{\epsilon \downarrow 0}R_0(\lambda +i\epsilon)^{1/2}), \tag {4.44}$$ \begin{align*} \tau_\epsilon(t,x)& =\tau_\epsilon(0,t,x),\tag {4.45}\\ \tau_0(t,x) & =\tau_\epsilon(-\infty,t,x),\ \epsilon=\pm. \tag {4.46}\end{align*} Similarly, writing \setcounter{equation}{46} \begin{align} \begin{split} V_{\infty,\epsilon}(t,x)&=\phi_{\infty,\epsilon,\epsilon'}(t,x)^2+ \phi_{\infty,\epsilon,\epsilon',x} (t,x)\\ &=\Lambda_0-2\partial^2_x\ln \tau_{\epsilon,\epsilon}(t,x), \end{split} \end{align} (D.91) and (D.93) yield $$\phi_{\infty,\epsilon,\epsilon'}(t,x)=\partial_x\ln \Psi_{\infty,\epsilon,\epsilon'}(0,t,x). \tag {4.48}$$ A comparison with (2.50) then implies \begin{multline*} \Psi_{\infty,\epsilon,\epsilon'}(\lambda,t,x)=\exp[-\epsilon'i (x-x_0)\int\limits^P_{P_0}\omega^{(2)}_0-\epsilon'12i(t-t_0) \int\limits^P_{P_0}\omega^{(2)}_2]\bullet\\ \bullet \left[\dfrac {\tau_{\epsilon,\epsilon,\epsilon'}(\lambda,t,x)} {\tau_{\epsilon,\epsilon}(t,x)}\dfrac {\tau_{\epsilon,\epsilon} (t_0,x_0)} {\tau_{\epsilon,\epsilon,\epsilon'}(\lambda,t_0,x_0)}\right],\ P=(\lambda,\lim\limits_{\epsilon \downarrow 0}R_0(\lambda+i\epsilon)^{1/2}) \tag {4.49} \end{multline*} and hence $$\phi_{\infty,\epsilon,\epsilon'}(t,x)=-\epsilon'i\int\limits^{Q_0}_{P_0} \omega^{(2)}_0+\partial_x\ln [\tau_{\epsilon,\epsilon,\epsilon'}(t,x)/\tau_{\epsilon,\epsilon}(t,x)], \tag {4.50}$$ where \setcounter{equation}{50} \begin{align} \begin{split} & \tau_{\epsilon,\epsilon,\epsilon'}(\lambda,t,x) =\theta(\underline \zeta_{P_0}-\epsilon'\underline A_{P_0}(P)+\underline \alpha_{P_0}(\underline {\hat \mu}_{\epsilon,\epsilon}(t,x)))\\ &=\theta(\underline \zeta_{P_0}-\epsilon'\underline A_{P_0}(P)+2\epsilon[\underline A_{P_0}(P_\infty)-\underline A_{P_0}(Q_1)]+\underline \alpha_{P_0}(\underline {\hat \mu}_0(t,x))), \lb{4.51} \end{split} \end{align} \begin{align} &\tau_{\epsilon,\epsilon,\epsilon'}(t,x)=\tau_{\epsilon,\epsilon, \epsilon'} (0,t,x), \lb{4.52} \\ &\tau_{\epsilon,\epsilon}(t,x) =\tau_{\epsilon,\epsilon,\epsilon'}(-\infty,t,x).\lb{4.53} \end{align} Iterating this procedure then yields $$V_{\Sb 0,\ldots,0,\epsilon\\ N\text { times }\endSb}(t,x) =\phi_{\Sb 0,\ldots,0,\epsilon,\epsilon'\\ N\text { times }\endSb} (t,x)^2+\phi_{\Sb 0,\ldots,0,\epsilon,\epsilon',x\\ N\text { times }\endSb}(t,x)=V_0(t,x),\lb{4.54}$$ $$\phi_{\Sb 0,\ldots,0,\epsilon,\epsilon'\\ N\text { times }\endSb}(t,x) =\phi_{0,\epsilon'}(t,x)\lb{4.55}$$ (cf. (4.42)--(4.46)) and \begin{align} \begin{split} V_{\Sb \infty,\ldots,\infty,\epsilon\\ N\text { times }\endSb}(t,x) & =\phi_{\Sb \infty,\ldots,\infty,\epsilon,\epsilon'\\ N\text { times}\endSb}(t,x)^2+\phi_{\Sb \infty,\ldots,\infty,\epsilon, \epsilon',x\\ N\text {times }\endSb}(t,x)\\ \phantom{V_{\Sb \infty,\ldots,\infty,\epsilon\\ N\text { times }\endSb}(t,x)} & =\Lambda_0-2\partial^2_x\ln \tau_{\Sb \epsilon,\ldots,\epsilon\\ 2N\text { times }\endSb}(t,x), \lb{4.56} \end{split} \end{align} \begin{align} \begin{split} & \phi_{\Sb \infty,\ldots,\infty,\epsilon,\epsilon'\\ N \text { times }\endSb} (t,x) =\partial_x\ln \Psi_{\Sb \infty,\ldots,\infty,\epsilon,\epsilon'\\N\text { times }\endSb} (t,x)\\ &=-\epsilon'\int\limits^{Q_0}_{P_0}\omega^{(2)}_0+\partial_x\ln [\tau_{\Sb \epsilon,\ldots,\epsilon,\epsilon'\\ 2N\text { times }\endSb}(t,x)/\tau_{\Sb \epsilon,\ldots, \epsilon\\ 2N\text { times}\endSb}(t,x)], \lb{4.57} \end{split} \end{align} where \begin{align} \begin{split} & \tau_{\Sb \epsilon,\ldots,\epsilon,\epsilon'\\ 2N\text { times }\endSb}(t,x)\\ &=\theta(\underline \zeta_{P_0}-\epsilon'\underline A_{P_0}(Q_0)+2\epsilon \sum\limits^N_{j=1}[\underline A_{P_0}(P_\infty)-\underline A_{P_0}(Q_j)]+\underline \alpha_{P_0}(\underline {\hat \mu}_0(t,x))), \lb{4.58} \end{split} \end{align} \begin{align} \begin{split} & \tau_{\Sb \epsilon,\ldots,\epsilon\\ 2N \text { times }\endSb}(t,x)\\ & =\theta(\underline \zeta_{P_0}-\epsilon \underline A_{P_0}(P_\infty)+2\epsilon\sum\limits^N_{j=1}[\underline A_{P_0}(P_\infty)-\underline A_{P_0}(Q_j)]+\underline \alpha_{P_0}(\underline {\hat \mu}_0(t,x))), \; N\in \N. \lb{4.59} \end{split} \end{align} We also note that \begin{multline*} \Psi_{\Sb \infty,\ldots,\infty,\epsilon,\epsilon'\\ N\text { times }\endSb}(\lambda,t,x)=\exp [-\epsilon'i(x-x_0)\int\limits^P_{P_0}\omega^{(2)}_0- \epsilon'12i(t-t_0)\int\limits^P_{P_0}\omega^{(2)}_2]\bullet\\ \bullet \left[\dfrac {\tau_{\Sb \epsilon,\ldots,\epsilon,\epsilon'\\ 2N \text { times }\endSb}(\lambda,t,x)}{\tau_{\Sb \epsilon,\ldots,\epsilon\\2N \text { times }\endSb}(t,x)}\dfrac {\tau_{\Sb \epsilon,\ldots,\epsilon\\ 2N\text { times }\endSb}(t_0,x_0)}{\tau_{\Sb \epsilon,\ldots,\epsilon,\epsilon'\\ 2N\text { times }\endSb}(\lambda,t_0,x_0)}\right],\tag {4.60} \end{multline*} with \setcounter{equation}{60} \begin{align} \begin{split} & \tau_{\Sb \epsilon,\ldots,\epsilon,\epsilon'\\ 2N\text { times }\endSb}(\lambda,t,x)\\ & =\theta(\underline \zeta_{P_0}-\epsilon'\underline A_{P_0}(P)+2\epsilon \sum\limits^N_{j=1}[\underline A_{P_0}(P_\infty)-\underline A_{P_0}(Q_j)]+\underline \alpha_{P_0}(\underline {\hat \mu}_0(t,x))),\\ & \hspace*{2.25in} \ P=(\lambda,\ \lim\limits_{\epsilon\downarrow 0}R_0 (\lambda+i\epsilon)^{1/2}), \lb{4.61} \end{split} \end{align} \begin{align} \tau_{\Sb \epsilon,\ldots,\epsilon,\epsilon'\\ 2N\text { times }\endSb}(t,x) & =\tau_{\Sb \epsilon,\ldots,\epsilon,\epsilon'\\ 2N \text { times }\endSb}(0,t,x), \\ \tau_{\Sb \epsilon,\ldots,\epsilon\\ 2N\text { times }\endSb}(t,x)&=\tau_{\Sb \epsilon,\ldots,\epsilon,\epsilon'\\ 2N\text { times }\endSb}(-\infty,t,x),\ \epsilon,\epsilon'\in \{\pm\}. \lb{4.63} \end{align} We summarize these findings in \begin{thm} %4.1 Assume the background KdV solutionV_0$to be given by (3.1)--(3.4) and define (D.68), (D.69), (D.75)--(D.82). Then (D.66), (D.70)--(D.74), (D.83), and (D.84) hold. Assume in addition that$H_{\gamma_1,\ldots,\gamma_N,\pm}(t)\geq 0,\ \gamma_{j,\pm}>0, 1\leq j\leq N$for some (and hence for all)$t\in \R$, and define (D.86). Then (D.87)--(D.99) hold. In particular, the leading asymptotic terms$V_{\Sb 0,\ldots,0,\epsilon\\ \infty,\ldots,\infty,\epsilon\endSb}$and$\phi_{\Sb 0,\ldots,0,\epsilon,\epsilon'\\ \infty ,\ldots,\infty,\epsilon,\epsilon'\endSb}$in (D.70) and (D.89) are explicitly expressed in terms of Riemann's theta function associated with the hyperelliptic curve$K_g$by (4.31)--(4.33) and (4.43)--(4.46), (4.55), (4.57)--(4.59). Moreover, the asymptotic relations (D.70) and (D.89) for fixed$t\in \Rcan be improved as follows: \seq{63} \begin{align} \begin{split} V_{\gamma_1,\ldots,\gamma_{N,+}}(t,x)&\=\limits_{x\rightarrow \pm \infty}V_{\Sb 0,\ldots,0,+\\ \infty,\ldots,\infty,+\\ N\text { times }\endSb}(t,x)+0(e^{-2\kappa_0|x|}),\\ V_{\gamma_1,\ldots,\gamma_{N,-}}(t,x)&\=\limits_{x\rightarrow \pm \infty}V_{\Sb \infty,\ldots,\infty,-\\0,\ldots,0,-\\ N\text { times }\endSb}(t,x)+0(e^{-2\kappa_0|x|}), \lb{4.64} \end{split}\\ \begin{split} \phi_{\gamma_1,\ldots,\gamma_{N,+,\epsilon'}}(t,x) &\=\limits_{x\rightarrow \pm \infty}\phi_{\Sb 0,\ldots,0,+,\epsilon'\\ \infty,\ldots,\infty,+,\epsilon'\\N\text { times }\endSb} (t,x)+0(e^{-2\kappa_0|x|}),\\ \phi_{\gamma_1,\ldots,\gamma_{N,-,\epsilon'}}(t,x) &\=\limits_{x\rightarrow \pm \infty} \phi_{\Sb \infty,\ldots,\infty,-,\epsilon'\\ 0,\ldots,0,-,\epsilon'\endSb}(t,x)+0(e^{-2\kappa_0|x|}),\\ &\qquad \epsilon'=\pm,\ \gamma_{j,\pm} >0,\ 1\leq j\leq N, \lb{4.65} \end{split} \end{align} where the Floquet exponent\kappa_0>0is determined by $$\kappa_0=\min\limits_{1\leq j\leq N} [\kappa_{0,j}],\ \kappa_{0,j}=|\Im \int\limits^{Q_j}_{P_0}\omega^{(2)}_0|,\ 1\leq j\leq N. \tag {4.66}$$ Finally, a comparison of the leading order terms in (4.64) and (4.65), taking into account (4.31)--(4.33), (4.43)--(4.46), (4.55), (4.56)--(4.59) yields that \begin{align*} &V_{\Sb \infty,\ldots,\infty,\epsilon\\ N\text { times }\endSb}(t,x)=V_0(t,x),\qquad \phi_{\Sb \infty,\ldots,\infty,\epsilon,\epsilon'\\ N\text { times }\endSb} (t,x)=\phi_{0,\epsilon'}(t,x)\\ &\text{iff } 2\sum\limits^N_{j=1}\underline A_{P_0}(Q_j)=\underline 0(\text {mod } L_g).\tag {4.67} \end{align*} \end{thm} \begin{pf} It remains to verify (4.64)--(4.66). This follows from \seq{67} \begin{gather} V_{\gamma_1,\ldots,\gamma_{N,\epsilon}}(t,x)= \phi_{\gamma_1,\ldots,\gamma_{N,\epsilon,\epsilon'}}(t,x)^2+ \phi_{\gamma_1,\ldots,\gamma_{N,\epsilon,\epsilon',x}}(t,x), \lb{4.68}\\ \phi_{\gamma_1,\ldots,\gamma_{N,\epsilon,\epsilon'}}(t,x)=\partial_x\ln \Psi_{\gamma_1,\ldots,\gamma_{N,\epsilon,\epsilon'}}(0,t,x) \lb{4.69} \end{gather} and the fact that for fixedt\in \R\begin{align*} \Psi_{\gamma_1,\ldots,\gamma_{N,+,\epsilon'}}(0,t,x) &\=\limits_{x\rightarrow \pm \infty}\Psi_{\Sb 0,\ldots ,0,+,\epsilon'\\ \infty,\ldots,\infty,+,\epsilon'\\ N\text { times }\endSb}(0,t,x)[1+0(e^{-2\kappa_0|x|})],\\ \Psi_{\gamma_1\ldots,\gamma_{N,-,\epsilon'}}(0,t,x) &\=\limits_{x\rightarrow \pm \infty}\Psi_{\Sb \infty,\ldots,\infty,-,\epsilon'\\0,\ldots,0,-, \epsilon'\\ N\text { times }\endSb}(0,t,x)[1+0(e^{-2\kappa_0|x|})] \tag {4.70} \end{align*} with\kappa_0defined in (4.66). Relations (4.70) can be verified from (D.75)--(D.82), (2.50) and \begin{align*} \int\limits^\infty_xdx'\psi_{0,j,+}(t,x')\psi_{0,\ell,+}(t,x') &\=\limits_{x\rightarrow \pm \infty}0(e^{-(\kappa_{0,j}+\kappa_{0,\ell})x}),\\ \int\limits^x_{-\infty}dx'\psi_{0,j,-} (t,x')\psi_{0,\ell,-}(t,x')&\=\limits_{x\rightarrow \pm \infty}0(e^{(\kappa_{0,j}+\kappa_{0,\ell})x}),\ 1\leq j,\ell\leq N. \tag {4.71} \end{align*} While (4.71) is obvious forj\neq \ell$from (B.54) and the Floquet-type structure of$\psi_{0,j,\pm}(t,x)$in (2.50), the case$j=\ell$seems to require a few arguments. For simplicity we only consider the case$g=2$since nothing new enters in the general situation$g\geq 3$. From $$\int\limits^x_{\pm \infty}dx'\psi_{0,j,\pm}(t,x')^2=W(\partial_\lambda\psi_{0,\pm} (t,\lambda) |_{\lambda=\lambda_j},\psi_{0,j,\pm}(t))(x) \tag {4.72}$$ one obtains by (2.50) that $$\int\limits^x_{\pm \infty}dx'\psi_{0,j,\pm}(t,x')^2=e^{\mp 2\kappa_{0,j}(x-x_0)}f_{j,\pm}(t,x), \tag {4.73}$$ where$f_{j,\pm}(t,x)$is quasi-periodic in$x$and hence of the type $$f_{j,\pm}(t,x)=F_{j,\pm}(t,\omega_1x, \omega_2x) \tag {4.74}$$ with$F_{j,\pm}(t,x_1, x_2)$periodic in$x_1$and$x_2$of period one and$\omega_1$and$\omega_2$rationally independent such that $$U_{0,\ell}/2\pi=n_{\ell,1}\omega_1+n_{\ell,2}\omega_2,\ (n_{\ell,1},n_{\ell,2})\in \Z^2,\ 1\leq \ell\leq 2. \tag {4.75}$$ It suffices to discuss$\psi_{0,j,+}$and the limit$x\rightarrow -\infty$in (4.73). Assume that$f_{j,+}(t,x)$does not stay bounded away from zero as$x\rightarrow -\infty$. Then there exists a sequence$\{(x_{1,n}, x_{2,n})\}_{n\in \N}\subset [0, \omega^{-1}_1]\times [0, \omega^{-1}_2]$such that$F_{j,+}(t, x_{1,n}, x_{2,n})@>>{n\rightarrow \infty}>0$and by the continuity of$F_{j,+}$and the compactness of$[0,\omega^{-1}_1]\times [0,\omega^{-1}_2]$one infers the existence of an$(\hat x_1, \hat x_2)\in [0,\omega^{-1}_1]\times [0, \omega^{-1}_2]$such that $$F_{j,+}(t,\hat x_1, \hat x_2)=0. \tag {4.76}$$ Since (4.76) contradicts (4.73) if$\hat x_1=\hat x_2$we may assume that$\hat x_1\neq \hat x_2$. By Theorem 10.3.1 of \cite{58} the Jacobi inversion problem $$\underline \alpha_{P_0}(\underline {\hat \mu}(t,x_1,x_2))=\underline \alpha_{P_0}(\underline {\hat \mu}(t,x_0))+\pmatrix n_{1,1} \omega_1(x_1-x_0)+n_{1,2}\omega_2(x_2-x_0)\\ n_{2,1}\omega_1(x_1-x_0)+n_{2,2}\omega_2(x_2-x_0)\endpmatrix \tag {4.77}$$ has a unique solution$\underline {\hat \mu}(t,x_1,x_2)$continuous in$(x_1, x_2)$with the projections satisfying$\ti\pi (\hat \mu_\ell(t,x_1,x_2))\in [E_{2\ell-1},\ E_{2\ell}],\ \ell=1,2$. In particular, $$\underline {\hat \mu}(t,x)=\underline {\hat \mu}(t,x,x),\ (t,x)\in \R^2 \tag {4.78}$$ and since$\ti\pi [\hat \mu_\ell(t,x)]$sweeps$[E_{2\ell-1},\ E_{2\ell}]$if$x$varies in$\R$, there exists$\tilde x_1, \tilde x_2\in \R$such that $$\hat \mu_1(t,\tilde x_1)=\hat \mu_1(t,\hat x_1, \hat x_2),\ \hat \mu_2(t,\tilde x_2)=\hat \mu_2(t,\hat x_1, \hat x_2). \tag {4.79}$$ If$\tilde x_1=\tilde x_2$, then$f_{j,+}(t,\tilde x_1)=F_{j,+}(t,\hat x_1, \hat x_2)=0$contradicts (4.73). If$\tilde x_1\neq \tilde x_2define $$\underline {\hat {\tilde \mu}}(t,x_0)=\underline {\hat \mu}(t,\hat x_1, \hat x_2) \tag {4.80}$$ and $$\tilde \psi_{0,+}(\lambda_j, t,x)=e^{-\kappa_{0,j}(x-x_0)} \dfrac {\tilde \tau_{+1}(\lambda_j,t,x)\tilde \tau_0(t,x_0)}{\tilde \tau_{+1}(\lambda_j,t,x_0)\tilde \tau_0(t,x)},\tag {4.81}$$ where \seq{81} \begin{align} \begin{split} \tilde \tau_{+1}(\lambda_j, t,x) =\theta(\underline \zeta_{P_0}-\underline A_{P_0}(Q_j)+\underline \alpha_{P_0}(\underline {\hat {\tilde \mu}}(t,x_0))+((x-x_0)/2\pi)\underline U_0),&\\ Q_j=(\lambda_j, \lim\limits_{\epsilon\downarrow 0}R_0(\lambda_j+i\epsilon)^{1/2})\in \Pi_+,& \lb{4.82} \end{split} \end{align} $$\tilde \tau_0(t,x) =\theta(\underline \zeta_{P_0}-\underline A_{P_0}(P_\infty)+\underline \alpha_{P_0}(\underline {\hat {\tilde \mu}}(t,x_0))+((x-x_0)/2\pi)\underline U_0). \lb{4.83}$$ As in (4.73) one computes $$\int^\infty_xdx'\tilde \psi_{0,+}(\lambda_j,t,x')^2=-e^{-2\kappa_{0,j}(x-x_0)}\tilde f_{j,+}(t,x) \tag {4.84}$$ with $$\tilde f_{j,+}(t,x_0)=F_{j,+}(t,\hat x_1, \hat x_2)=0. \tag {4.85}$$ This contradiction proves thatf_{j,+}(t,x)$in (4.73) stays bounded away from zero as$x\rightarrow -\infty$. \end{pf} The interpretation of$V_{\gamma_1,\ldots,\gamma_{N,\epsilon}}$and$\phi_{\gamma_1,\ldots,\gamma_{N,\epsilon,\epsilon'}},\ \gamma_{j,\epsilon}>0,\ 1\leq j\leq N$as (m)KdV soliton solutions relative to the background solutions$V_0$and$\phi_{0,\epsilon'}$is then identical to the one following (D.69) and (D.99). As noted in a similar context at the end of Section~3,$Q_j$in (4.36) can be replaced by$Q_j=(\lambda_j, \lim\limits_{\epsilon\downarrow 0}R_0(\lambda_j+i\epsilon s_j)^{1/2}),\ s_j\in \{-1,1\},\ 1\leq j\leq N$. We omit the obvious details. In the elliptic case, where$g=1$, the KdV results of this section were first derived in \cite{57} (see also \cite{86}). A generalization to$g\geq 1$, based on an algebro-geometric approach was soon after sketched in \cite{52} and further discussed in \cite{19}. The latter reference briefly mentions the connections with Marchenko's approach to inverse spectral theory. An extension of \cite{52} is shortly presented in \cite{82}. In \cite{6} the authors consider the Cauchy problem for the KdV equation in the class of solutions (not necessarily of soliton-type) that tend to isospectral finite-gap solutions as$x\rightarrow \pm \infty$employing inverse spectral techniques. A different method to produce solitons relative to a finite-gap background solution based on singularizations of the hyperelliptic curve$K_g$appeared in \cite{64}, \cite{68}, \cite{88} (see also \cite{89}, \cite{90}). The corresponding mKdV results for$g=1$expressed in terms of elliptic functions first appeared in \cite{39}. Their generalization to$g\geq 1$in this paper is new. We would like to emphasize that all results in this paper extend in a straightforward manner to the entire (m)KdV hierarchy. However, in order to keep this exposition within a reasonable length we omit the details. \appendix \chapter{Single Commutation Methods} \addtocounter{page}{-1} \renewcommand{\theequation}{A.\arabic{equation}} \renewcommand{\thelem}{A.\arabic{lem}} \setcounter{lem}{0} In this appendix we describe the method of inserting additional eigenvalues into the lowest spectral gap$(-\infty,E_0)$of a given background Schr\"odinger operator$H_0=-\dfrac {d^2}{dx^2}+V_0$in$L^2(\R)$using the single commutation method (also called the Crum-Darboux method in the literature) \cite{11}, \cite{14}, \cite{80}. Our basic hypothesis in this appendix reads \medskip {\bf (H.A.1).} Suppose$\dfrac {d^m}{dx^m}V_0\in C^\infty(\R)\cap L^\infty(\R),\ m\in \N_0$to be real-valued. \medskip Given (H.A.1) we define in$L^2(\R)$the self-adjoint operator $$H_0=-\dfrac {d^2}{dx^2}+V_0,\ \D(H_0)=H^2(\R), \tag {A.1}$$ with$H^m(.), m\in \N$the standard Sobolev space. Next let E_0=\inf [\sigma (H_0)],\ \lambda_N0$ near $\pm \infty$. (Here $W(f,g)(x)=f(x)g'(x)-f'(x)g(x)$ denotes the Wronskian of $f$ and $g$.) Given any positive solutions $\psi_{\pm 1}(\lambda,x)$ of (A.13) near $\pm \infty$ we note that \hat \psi_{0,\pm 1}(\lambda,x)=\pm \psi_{0,\pm 1}(\lambda,x)\int\limits^x_{x_\pm}dx'\psi_{\pm 1}(\lambda,x')^{-2},\ \lambda0$near$\pm \infty$. We briefly outline the argument for (A.32). (A.33)--(A.35) then follow as (A.14)--(A.16) in the case of$H_0. Since $$\dfrac {d^m}{dx^m}\phi_{\sigma_N}\in C^\infty (\R)\cap L^\infty (\R),\ m\in \N_0 \tag {A.36}$$ (this follows from Corollary XI.6.5 of \cite{44}, see also Lemma C.3 for a generalization) we infer from (A.3) that \begin{align*} &A^*_{\sigma_N}\psi_{0,\pm 1}(\lambda,.)\in L^2((R,\pm \infty)),\ \sigma_N\in [-1,1],\ \lambda<\lambda_N,\ R\in \R\\ &\text {iff }\psi'_{0,\pm 1}(\lambda,.)\in L^2((R,\pm \infty)), \ R\in \R. \tag {A.37} \end{align*} But the latter is clearly satisfied by (A.4), (A.36) and the fact that $$\dfrac {d^m}{dx^m}V_0,\ \dfrac {d^m}{dx^m}V_{0,\sigma_N}\in C^\infty(\R)\cap L^\infty (\R),\ m\in \N_0. \tag {A.38}$$ This identifiesA^*_{\sigma_N}\psi_{0,\pm 1}(\lambda)$(up to normalization) as principal solutions of (A.31) near$\pm \infty$. Since $$\lambda<\lambda_N\leq \inf [\sigma(H_{0,\sigma_N})] \tag {A.39}$$ by (A.22), these principal solutions are necessarily positive on$\R$(see, e.g., Lemma 3.2 in \cite{40}). This also shows that the normalization of$\psi_{0,\pm 1,\sigma_N}in (A.29) is meaningful and completes the proof of (A.32). As in (A.19)--(A.21) one infers \setcounter{equation}{39} \begin{align} \begin{split} &\psi^{-1}_{0,\sigma_{N-1},\sigma_N}\in H^2(\R),\ \sigma_{N-1} \in (-1,1),\ \sigma_N\in [-1,1],\\ &\psi^{-1}_{0,\pm 1,\sigma_N}\notin L^2(\R),\ \sigma_N\in [-1,1]. \lb{a.40} \end{split} \end{align} Defining $$H_{0,\sigma_{N-1},\sigma_N}=A^*_{\sigma_{N-1},\sigma_N}A_{\sigma_{N-1}, \sigma_N} +\lambda_{N-1},\ \D (H_{0,\sigma_{N-1},\sigma_N})=H^2(\R) \tag {A.41}$$ we have \seq{41} \begin{align} \begin{split} H_{0,\sigma_N} & =-\dfrac {d^2}{dx^2}+V_{0,\sigma_N},\\ V_{0,\sigma_N}(x) & =\phi_{\sigma_N}(x)^2-\phi'_{\sigma_N}(x)+\lambda_N\\ & =\phi_{\sigma_{N-1},\sigma_N}(x)^2+\phi'_{\sigma_{N-1},\sigma_N}(x) +\lambda_{N-1 }, \lb{A.42} \end{split}\\ \begin{split} H_{0,\sigma_{N-1},\sigma_N} & =-\dfrac {d^2}{dx^2}+V_{0,\sigma_{N-1},\sigma_N},\\ V_{0,\sigma_{N-1},\sigma_N}(x)& =\phi_{\sigma_{N-1},\sigma_N}(x)^2-\phi'_{\sigma_{N-1},\sigma_N}(x) +\lambda_{N-1 } \lb{A.43} \end{split} \end{align} and hence $$\sigma(H_{0,\sigma_{N-1},\sigma_N})=\cases \sigma (H_{0,\sigma_N})\cup \{\lambda_{N-1}\},\ \sigma_{N-1}\in (-1,1)\\ \sigma (H_{0,\sigma_N}),\ \sigma_{N-1}=\pm 1 \endcases \tag {A.44}$$ in analogy to (A.22). We also have \setcounter{equation}{44} \begin{align} \begin{split} & V_{0,\sigma_{N-1},\sigma_N}(x) =V_{0,\sigma_N}(x)-2\phi'_{\sigma_{N-1},\sigma_N}(x)\\ &=V_{0,\sigma_N}(x)-2\dfrac {d^2}{dx^2}\ln \psi_{0,\sigma_{N-1},\sigma_N}(x)\\ & = V_0(x)-2\dfrac {d^2}{dx^2}\ln [\psi_{0,\sigma_{N-1},\sigma_N}(x)\psi_{0,\sigma_N}(x)] \lb{a.45} \end{split} \end{align} by (A.23), (A.42) and (A.43). At this point it is clear how to proceed inductively. Assume \lambda_1<\lambda_2<\ldots<\lambda_{N-1}<\lambda_N0,\ 1\leq j\leq N. \tag {B.3} \end{align*} Similar to Appendix A we pick \seq{3} \begin{align} \begin{split} &\psi_{0,\pm} (\lambda,.)\in C^\infty (\R),\\ &\psi_{0,\pm} (\lambda,x)\text { real-valued for }\lambda\in \rho(H_0) \cap \R, \lb{B.4} \end{split} \end{align} $$\psi_{0,\pm}(\lambda,.)\in L^2((R,\pm \infty)),\ \lambda\in \rho(H_0),\ R\in \R \tag {B.5}$$ satisfying $$H_0\psi_{0,\pm}(\lambda)=\lambda\psi_{0,\pm}(\lambda),\ \lambda\in \rho(H_0). \tag {B.6}$$ We abbreviate $$\psi_{0,j,\pm}(x)=\psi_{0,\pm}(\lambda_j,x),\ 1\leq j\leq N. \tag {B.7}$$ In the first step, aiming at inserting the additional eigenvalue\lambda_1$into$\rho(H_0), we introduce the following (singular) differential expressions \setcounter{equation}{7} \begin{align} \begin{split} & A_{1,\pm} =\psi^{-1}_{0,1,\pm}\dfrac d{dx}\psi_{0,1,\pm}=\dfrac d{dx}+\phi_{1,\pm},\\ & A^+_{1,\pm} =-\psi_{0,1,\pm}\dfrac d{dx}\psi^{-1}_{0,1,\pm}=-\dfrac d{dx}+\phi_{1,\pm}, \lb{B.8} \end{split} \end{align} \begin{align} \begin{split} & H_{2,0} =A_{1,\pm}A^+_{1,\pm}=H_0-\lambda_1=-\dfrac {d^2}{dx^2}+V_{2,0},\\ & V_{2,0} =\phi^2_{1,\pm}+\phi'_{1,\pm},\ \phi_{1,\pm}=\dfrac d{dx}\ln \psi_{0,1,\pm}, \lb{B.9} \end{split} \end{align} \begin{align} & H_{2,0}\psi_{0,1,\pm} =0. \lb{B.10}\\ \begin{split} & \tilde H_{1,0,\pm} =A^+_{1,\pm}A_{1,\pm}=-\dfrac {d^2}{dx^2}+\tilde V_{1,0,\pm},\\ & \tilde V_{1,0,\pm} =\phi^2_{1,\pm}-\phi'_{1,\pm}, \lb{B.11} \end{split} \end{align} \begin{align} & \tilde H_{1,0,\pm}(\psi^{-1}_{0,1,\pm}) =0. \lb{B.12}\\ \begin{split} & A_{1,\gamma_1,\pm} =\psi^{-1}_{1,\gamma_1,\pm}\dfrac d{dx}\psi_{1,\gamma_1,\pm}=\dfrac d{dx}+\phi_{1,\gamma_1,\pm},\\ & A^+_{1,\gamma_1,\pm} =-\psi_{1,\gamma_1,\pm}\dfrac d{dx}\psi^{-1}_{1,\gamma_1,\pm}=-\dfrac d{dx}+\phi_{1,\gamma_1,\pm}, \lb{B.13} \end{split} \end{align} \begin{align} \begin{split} & H_{2,1,\pm} =A_{1,\gamma_1,\pm}A^+_{1,\gamma_1,\pm}=-\dfrac {d^2}{dx^2}+V_{2,1,\pm}, \\ & V_{2,1,\pm} =\phi^2_{1,\gamma_1,\pm}+\phi'_{1,\gamma_1,\pm},\ \phi_{1,\gamma_1,\pm}=\dfrac d{dx}\ln \psi_{1,\gamma_1,\pm}, \lb{B.14} \end{split} \end{align} $$\psi_{1,\gamma_1,\pm}(x) =\psi_{0,1,\pm }(x)^{-1}[1\mp \gamma_{1,\pm}\int\limits^x_{\pm \infty}dx'\psi_{0,1,\pm}(x')^2],\ \gamma_{1,\pm}>0, \lb{B.15}$$ $$H_{2,1,\pm}\psi_{1,\gamma_1,\pm} =0. \lb{B.16}$$ By inspection, $$\tilde H_{1,0,\pm}=H_{2,1,\pm},\ \tilde V_{1,0,\pm}=V_{2,1,\pm}. \tag {B.17}$$ Finally we need to introduce \setcounter{equation}{17} \begin{align} \begin{split} H_{1,1,\gamma_1,\pm}&=A^+_{1,\gamma_1,\pm}A_{1,\gamma_1,\pm}=-\dfrac {d^2}{dx^2}+V_{1,1,\gamma_1,\pm},\\ V_{1,1,\gamma_1,\pm}&=\phi^2_{1,\gamma_1,\pm}-\phi'_{1,\gamma_1,\pm}, \lb{B.18} \end{split} \end{align} then $$H_{1,1,\gamma_1,\pm}(\psi^{-1}_{1,\gamma_1,\pm})=0. \tag {B.19}$$ Combining (B.9), (B.11), (B.14), (B.15), (B.17), and (B.18) yields \setcounter{equation}{19} \begin{align} \begin{split} & V_{1,1,\gamma_1,\pm}(x) =V_{2,1,\pm}(x)-2\dfrac {d^2}{dx^2}\ln \psi_{1,\gamma_1,\pm}(x)\\ &=V_{2,0}(x)-2\dfrac {d^2}{dx^2}\ln [1\mp \gamma_{1,\pm} \int\limits^x_{\pm \infty}dx'\psi_{0,1,\pm}(x')^2]. \lb{B.20} \end{split} \end{align} Completing step one we define \begin{align} \begin{split} & V_{\gamma_{1,\pm}}(x) =V_{1,1,\gamma_1,\pm}(x)+\lambda_1\\ &=V_0(x)-2\dfrac {d^2}{dx^2}\ln [1\mp \gamma_{1,\pm}\int\limits^x_{\pm \infty}dx'\psi_{0,1,\pm}(x')^2]. \lb{B.21} \end{split} \end{align} In order not to interrupt the flow in this appendix we make the simplifying assumption $$\dfrac {d^m}{dx^m}V_{\gamma_1,\pm}\in L^\infty(\R),\ m\in \N_0 \tag {B.22}$$ and define the self-adjoint operator $$H_{\gamma_{1,\pm}}=-\dfrac {d^2}{dx^2}+V_{\gamma_{1,\pm}},\ \D (H_{\gamma_{1,\pm}})=H^2(\R) \tag {B.23}$$ inL^2(\R)$. (For a short discussion of the general case, avoiding (B.22), see the comments at the end of this appendix.) In order to prove that indeed $$\lambda_1\in \sigma_p(H_{\gamma_{1,\pm}}) \tag {B.24}$$ as desired by step one (here$\sigma_p(.)$denotes the point spectrum, i.e., the set of eigenvalues), we argue as follows. By hypothesis $$\lambda_1\notin \sigma (H_0) \tag {B.25}$$ and $$\psi^{-1}_{0,\pm 1}\notin L^2 ((R, \pm \infty)),\ R\in \R \tag {B.26}$$ due to (B.5). But then also $$\psi^{-1}_{0,1,\pm}\int\limits^{\pm \infty}_x dx' \psi^2_{0,1,\pm}\notin L^2(\R) \tag {B.27}$$ by the following reasoning. If$\lambda_1>\inf [\sigma (H_0)],\ \psi_{0,1,\pm}$has at least one zero which proves (B.27). If$\lambda_1<\inf [\sigma (H_0)],\ \psi_{0,1,\pm}\neq 0$on$\R$are principal solutions of$H_0\psi=\lambda_1\psi$near$\pm \infty$(see the discussion following (A.13)) and$\tilde H_{1,0,\pm}$defined on$H^2(\R)$is self-adjoint in$L^2(\R)$. Thus$\psi^{-1}_{0,1,\pm}$is a principal solution of$\tilde H_{1,0,\pm}\psi=0$near$\mp \infty$. Consequently$\psi^{-1}_{0,1,\pm}\int\limits^x_{\pm\infty}dx'\psi^2_{0,1,\pm}$is a nonprincipal solution of$\tilde H_{1,0,\pm}\psi=0$proving (B.27). The purpose of (B.26) and (B.27) was to prove that$(\tilde H_{1,0,\pm}+\lambda_1)\psi=\lambda \psi$has no solution$\psi \in L^2(\R)$for$\lambda=\lambda_1$. Therefore the second commutation in (B.18) is crucial. Since $$\| \psi^{-1}_{1,\gamma_1,\pm}\|^2_2=\gamma^{-1}_{1,\pm}, \tag {B.28}$$ (B.19) together with $$V_{\gamma_{1,\pm}}\in L^\infty(\R) \tag {B.29}$$ proves $$\psi^{-1}_{1,\gamma_1,\pm}\in H^2(\R) \tag {B.30}$$ and hence (B.24). In addition one can prove that $$\sigma (H_{\gamma_1,\pm})=\sigma (H_0)\cup \{\lambda_1\} \tag {B.31}$$ as shown at the end of this appendix. In the second step we now insert the eigenvalue$\lambda_2. Define the (singular) differential expressions \setcounter{equation}{31} \begin{align} \begin{split} A_{2,\gamma_1,\pm} &=\psi^{-1}_{1,2,\gamma_1,\pm}\dfrac d{dx}\psi_{1,2,\gamma_1,\pm}=\dfrac d{dx}+\phi_{1,2,\gamma_1,\pm},\\ A^+_{2,\gamma_1,\pm}& =-\psi_{1,2,\gamma_1,\pm}\dfrac d{dx}\psi^{-1}_{1,2,\gamma_1,\pm}=-\dfrac d{dx}+\phi_{1,2,\gamma_1,\pm}, \lb{B.32} \end{split} \end{align} \begin{align} \begin{split} H_{2,1,\gamma_1,\pm} & =A_{2,\gamma_1,\pm}A^+_{2,\gamma_1,\pm}=H_{1,1,\gamma_1,\pm} +\lambda_1-\lambda_2 =-\dfrac {d^2}{dx^2}+V_{2,1,\gamma_1,\pm},\\ V_{2,1,\gamma_1,\pm} & =\phi^2_{1,2,\gamma_1,\pm}+\phi'_{1,2,\gamma_1,\pm}, \ \phi_{1,2,\gamma_1,\pm}=\dfrac d{dx}\ln \psi_{1,2,\gamma_1,\pm}, \lb{B.33} \end{split}\end{align} where\psi_{1,2,\gamma_1,\pm}is a distributional solution of $$H_{2,1,\gamma_1,\pm}\psi_{1,2,\gamma_1,\pm}=0,\ \psi_{1,2,\gamma_1,\pm}\in C^\infty(\R)\cap L^2 ((R,\pm \infty)),\ R\in \R \tag {B.34}$$ to be determined later. In addition we introduce \setcounter{equation}{34} \begin{align} \begin{split} &\tilde H_{1,1,\gamma_1,\pm}=A^+_{2,\gamma_1,\pm}A_{2,\gamma_1,\pm}=-\dfrac {d^2}{dx^2}+\tilde V_{1,1,\gamma_1,\pm},\\ &\tilde V_{1,1,\gamma_1,\pm}=\phi^2_{1,2,\gamma_1,\pm}-\phi'_{1,2,\gamma_1,\pm}, \lb{B.35} \end{split} \end{align} $$\tilde H_{1,1,\gamma_1,\pm}(\psi^{-1}_{1,2,\gamma_{1,\pm}})=0, \lb{B.36}$$ \begin{align} \begin{split} &A_{2,\gamma_1,\gamma_2,\pm}=\psi^{-1}_{2,\gamma_1,\gamma_2,\pm}\dfrac d{dx}\psi_{2,\gamma_1,\gamma_2,\pm}=\dfrac d{dx}+\phi_{2,\gamma_1,\gamma_2,\pm},\\ &A^+_{2,\gamma_1,\gamma_2,\pm}=-\psi_{2,\gamma_1,\gamma_2,\pm}\dfrac d{dx}\psi^{-1}_{2,\gamma_1,\gamma_2,\pm}=-\dfrac d{dx}+\phi_{2,\gamma_1,\gamma_2,\pm}, \lb{B.37} \end{split} \end{align} \begin{align} \begin{split} &H_{2,2,\gamma_1,\pm}=A_{2,\gamma_1,\gamma_2,\pm} A^+_{2,\gamma_1,\gamma_2,\pm}=- \dfrac {d^2}{dx^2}+V_{2,2,\gamma_1,\pm},\\ &V_{2,2,\gamma_1,\pm}=\phi^2_{2,\gamma_1,\gamma_2,\pm} +\phi'_{2,\gamma_1,\gamma_ 2,\pm}, \ \phi_{2,\gamma_1,\gamma_2,\pm}=\dfrac d{dx}\ln \psi_{2,\gamma_1,\gamma_2,\pm}, \lb{B.38} \end{split} \end{align} $$\psi_{2,\gamma_1,\gamma_2,\pm}(x)=\psi_{1,2,\gamma_1,\pm}(x)^{-1}[1\mp \gamma_{2,\pm}\int\limits^x_{\pm \infty}dx'\psi_{1,2,\gamma_1,\pm} (x')^2], \lb{B.39}$$ $$H_{2,2,\gamma_1,\pm}\psi_{2,\gamma_1,\gamma_2,\pm}=0. \lb{B.40}$$ By inspection, $$\tilde H_{1,1,\gamma_1,\pm}=H_{2,2,\gamma_1,\pm},\ \tilde V_{1,1,\gamma_1,\pm}=V_{2,2,\gamma_1,\pm}. \tag {B.41}$$ Moreover, we introduce \setcounter{equation}{41} \begin{align} \begin{split} & H_{1,2,\gamma_1,\gamma_2,\pm}=A^+_{2,\gamma_1,\gamma_2,\pm} A_{2,\gamma_1,\gamma_2,\pm} =-\dfrac {d^2}{dx^2}+V_{1,2,\gamma_1,\gamma_2,\pm}, \\ & V_{1,2,\gamma_1,\gamma_2,\pm}=\phi^2_{2,\gamma_1,\gamma_2,\pm} -\phi'_{2,\gamma_1,\gamma_2,\pm}, \lb{B.42} \end{split}\\ & H_{1,2,\gamma_1,\gamma_2,\pm}(\psi^{-1}_{2,\gamma_1,\gamma_2,\pm})=0. \lb{B.43} \end{align} In analogy to (B.20) one obtains $$V_{1,2,\gamma_1,\gamma_2,\pm}(x)=V_{1,1,\gamma_1,\pm}(x)+\lambda_1- \lambda_2-2\dfrac {d^2}{dx^2}\ln [1\mp \gamma_{2,\pm} \int\limits^x_{\pm \infty}dx' \psi_{1,2,\gamma_1,\pm}(x')^2] \tag {B.44}$$ and similarly to (B.21), (B.23) one then introduces \begin{align*} & V_{\gamma_1,\gamma_2,\pm}(x)=V_{1,2,\gamma_1,\gamma_2,\pm}(x) +\lambda_2=\phi_{2,\gamma_1,\gamma_2,\pm} (x)^2-\phi'_{2,\gamma_1,\gamma_2,\pm}(x)+\lambda_2\\ &=V_0(x)-2\dfrac {d^2}{dx^2}\ln \{[1\mp \gamma_{1,\pm}\int\limits^x_{\pm \infty}dx'\psi_{0,1,\pm}(x')^2][ 1\mp \gamma_{2,\pm}\int\limits^x_{\pm \infty}dx'\psi_{1,2,\gamma_1,\pm}(x')^2]\} \tag {B.45} \end{align*} and $$H_{\gamma_1,\gamma_2,\pm}=-\dfrac {d^2}{dx^2} +V_{\gamma_1,\gamma_2,\pm},\ \D (H_{\gamma_1,\gamma_2,\pm})=H^2(\R) \tag {B.46}$$ inL^2(\R)$assuming again $$\dfrac {d^m}{dx^m}V_{\gamma_1,\gamma_2,\pm}\in L^\infty(\R),\ m\in \N_0. \tag {B.47}$$ It remains to identify$\psi_{1,2,\gamma_1,\pm}$. For that purpose we recall that for differential expressions$A,A^+$of the type introduced above $$AA^+\psi=\lambda \psi\text { implies }A^+A(A^+\psi)=\lambda(A^+\psi) \tag {B.48}$$ within the algebra of such differential expressions. Repeated use of (B.48) shows that $$\tilde \psi_{1,2,\gamma_1,\pm}(x)=N_\pm (A^+_{1,\gamma_1,\pm}A^+_{1,\pm}\psi_{0,2,\pm})(x) \tag {B.49}$$ (with$N_\pman appropriate normalization constant to be determined later) satisfies $$H_{2,1,\gamma_1,\pm}\tilde \psi_{1,2,\gamma_1,\pm}=0. \tag {B.50}$$ Explicit computations then yield \setcounter{equation}{50} \begin{align} \begin{split} & \tilde \psi_{1,2,\gamma_1,\pm}(x) =N_\pm (\lambda_1-\lambda_2)\psi_{0,2,\pm} (x)\\ & \pm N_\pm \gamma_{1,\pm}\psi_{0,1,\pm}(x)[1\mp \gamma_{1,\pm}\int\limits^x_{\pm \infty}dx'\psi_{0,1,\pm}(x')^2]^{-1}W(\psi_{0,1,\pm}, \psi_{0,2,\pm})(x)\\ & =N_\pm (\lambda_1-\lambda_2)\Big\{\psi_{0,2,\pm}(x)\pm \dfrac {\gamma_{1,\pm} \psi_{0,1,\pm}(x)\int\limits^x_{\pm \infty}dx'\psi_{0,1,\pm}(x')\psi_{0,2,\pm}(x')}{1\mp \gamma_{1,\pm}\int\limits^x_{\pm \infty}dx'\psi_{0,1,\pm}(x')^2}\Big\}, \lb{B.51} \end{split}\\ \begin{split} & \tilde \psi_{1,2,\gamma_1,\pm}(x)^2 \\ & =N^2_\pm (\lambda_1-\lambda_2)^2\left\{\psi_{0,2,\pm}(x)^2\pm \dfrac d{dx}\left[\dfrac {\gamma_{1,\pm}[\int\limits^x_{\pm \infty}dx'\psi_{0,1,\pm} (x')\psi_{0,2,\pm}(x')]^2}{1\mp \gamma_{1,\pm}\int\limits^x_{\pm \infty}dx'\psi_{0,1,\pm}(x')^2}\right]\right\}, \lb{B.52} \end{split} \end{align} where we used $$\int\limits^x_{\pm \infty}dx'\psi_{0,1,\pm}(x')\psi_{0,2,\pm}(x')=(\lambda_1-\lambda_2)^{-1} W(\psi_{0,1,\pm}, \psi_{0,2,\pm})(x) \tag {B.53}$$ and $$\lim\limits_{x\rightarrow \pm \infty} W(\psi_{0,1,\pm}, \psi_{0,2,\pm})(x)=0 \tag {B.54}$$ since\psi_{0,j,\pm} \in H^2 ((R,\pm \infty)),\ 1\leq j\leq N, \ R\in \R$due to (B.5) and (B.6). In particular, $$\tilde \psi_{1,2,\gamma_1,\pm}\in C^\infty(\R)\cap L^2((R,\pm \infty)),\ R\in \R \tag {B.55}$$ as required in (B.34). Hence we define $$\psi_{1,2,\gamma_1,\pm}(x)=\tilde \psi_{1,2,\gamma_1,\pm}(x). \tag {B.56}$$ Repeating the analysis that led to (B.30) then results in $$\psi^{-1}_{2,\gamma_1,\gamma_2,\pm}\in H^2(\R) \tag {B.57}$$ and hence $$\lambda_2\in \sigma_p(H_{\gamma_1,\gamma_2,\pm}). \tag {B.58}$$ Moreover, (B.39) yields $$\|\psi^{-1}_{2,\gamma_1,\gamma_2,\pm}\|^2_2=\gamma^{-1}_{2,\pm} \tag {B.59}$$ since$\psi_{1,2,\gamma_1,\pm}$are not square integrable near$\mp \infty$. (Otherwise$0\in \sigma_p(H_{2,1,\gamma_1,\pm})$which contradicts$\lambda_2\in \rho(H_0)\cap \rho (H_{\gamma_1,\pm})$.) Similarly to (B.31) one can show $$\sigma(H_{\gamma_1,\gamma_2,\pm})=\sigma(H_0)\cup \{\lambda_1,\lambda_2\}. \tag {B.60}$$ This completes the second step except for one point. We left open the choice of$N_\pm$in (B.49).$N_\pm$can be fixed by the following simple observation. Repeat the entire two-step procedure (B.8)--(B.60) but interchange the role of$\lambda_1$and$\lambda_2$, i.e., first introduce the eigenvalue$\lambda_2$and then$\lambda_1$into$\rho(H_0)$. Denote the resulting self-adjoint operator in$L^2(\R)by $$H_{\gamma_2,\gamma_1,\pm}=-\dfrac {d^2}{dx^2}+V_{\gamma_2,\gamma_1,\pm}. \tag {B.61}$$ If one requires $$H_{\gamma_2,\gamma_1,\pm}=H_{\gamma_1,\gamma_2,\pm},\text { i.e., }V_{\gamma_2,\gamma_1,\pm}=V_{\gamma_1,\gamma_2,\pm}, \tag {B.62}$$ a straightforward computation yields $$N_\pm=(\lambda_1-\lambda_2)^{-1} \tag {B.63}$$ and hence \seq{63} \begin{align} \begin{split} V_{\gamma_1,\gamma_2,\pm}(x)&=V_0(x)-2\dfrac {d^2}{dx^2}\ln \det [1+C_{\gamma_1,\gamma_2,\pm}(x)]\\ &=V_{\gamma_2,\gamma_1,\pm}(x), \lb{B.64} \end{split} \end{align} where \begin{align} \begin{split} C_{\gamma_1,\gamma_2,\pm}(x)&=[c_{\gamma_1,\gamma_2,\pm, \ell, m}(x)]^2_{\ell, m=1},\\ c_{\gamma_1,\gamma_2,\pm, \ell,m}(x)&=\mp \gamma_{\ell,\pm}\int\limits^x_{\pm \infty}dx'\psi_{0,\ell,\pm}(x')\psi_{0,m,\pm}(x'),\\ & \hspace*{1in} 1\leq \ell,m\leq 2. \lb{B.65} \end{split} \end{align} The eigenfunctions\Psi_{\gamma_1,\gamma_2,j,\pm},j=1,2$of$H_{\gamma_1,\gamma_2, \pm}are then determined by \begin{align*} \Psi_{\gamma_1,\gamma_2,\pm}(x)&=\left[1+C_{\gamma_1,\gamma_2,\pm}(x) \right]^{-1 }\Psi^0_{\gamma_1,\gamma_2, \pm}(x),\\ \Psi_{\gamma_1,\gamma_2,\pm}(x)&=\pmatrix \Psi_{\gamma_1,\gamma_2,1,\pm}(x)\\ \Psi_{\gamma_1,\gamma_2,2,\pm}(x)\endpmatrix, \Psi^0_{\gamma_1,\gamma_2,\pm}(x)=\pmatrix \gamma_{1,\pm}\psi_{0,1,\pm}(x)\\ \gamma_{2,\pm}\psi_{0,2,\pm}(x)\endpmatrix \tag {B.66} \end{align*} and one infers \seq{66} \begin{align} \begin{split} & H_{\gamma_1,\gamma_2,\pm}\Psi_{\gamma_1,\gamma_2,j,\pm} =\lambda_j\Psi_{\gamma_1 ,\gamma_2,j,\pm},\\ & \Psi_{\gamma_1,\gamma_2,j,\pm}\in H^2(\R),\ \|\Psi_{\gamma_1,\gamma_2,j,\pm}\|^2_2 =\gamma_{j,\pm},\ j=1,2. \lb{B.67} \end{split} \end{align} \medskip At this point we immediately jump to the final result after completing theN$-th step. The corresponding self-adjoint operator in$L^2(\R)is then given by \begin{align*} H_{\gamma_1,\ldots,\gamma_N,\pm}&=-\dfrac {d^2}{dx^2}+V_{\gamma_1,\ldots,\gamma_N,\pm},\ \D (H_{\gamma_1,\ldots,\gamma_N,\pm})=H^2(\R),\\ V_{\gamma_1,\ldots, \gamma_N,\pm}(x)&=V_0(x)-2\dfrac {d^2}{dx^2}\ln \det \left[1+C_{\gamma_1,\ldots, \gamma_N,\pm}(x)\right], \tag {B.68} \end{align*} where \begin{align*} C_{\gamma_1,\ldots, \gamma_N,\pm}(x)=&\left[c_{\gamma_1,\ldots, \gamma_N,\pm, \ell, m}(x)\right]^N_{\ell,m=1},\ N\in \N,\\ c_{\gamma_1,\ldots, \gamma_N,\pm,\ell,m}(x)&=\mp \gamma_{\ell,\pm}\int\limits^x_{\pm \infty}dx'\psi_{0,\ell,\pm}(x')\psi_{0,m,\pm}(x'),\ 1\leq \ell,m\leq N. \tag {B.69} \end{align*} It is well known that a matrix of the type (B.69) is positive definite, i.e., $$\det \left[1+C_{\gamma_1,\ldots,\gamma_N,\pm}(x)\right]>1,\ x\in \R. \tag {B.70}$$ Here we again assumed implicitly that $$\dfrac {d^m}{dx^m}V_{\gamma_1,\ldots,\gamma_j,\pm}\in L^\infty(\R), \ m\in \N_0,\ 1\leq j\leq N \tag {B.71}$$ (see, however, the comments at the end of this appendix). The eigenfunctions\Psi_{\gamma_1,\ldots,\gamma_N,j,\pm},1\leq j\leq N$of$H_{\gamma_1,\ldots, \gamma_N,\pm}are then determined by \setcounter{equation}{71} $$\Psi_{\gamma_1,\ldots, \gamma_N,\pm}(x) =\left[1+C_{\gamma_1,\ldots,\gamma_N,\pm}(x)\right]^{-1}\Psi^0_{\gamma_1 ,\ldots,\gamma_N,\pm}(x), \lb{B.72}$$ \begin{align} \begin{split} \Psi_{\gamma_1,\ldots, \gamma_N,\pm}(x) & =(\Psi_{\gamma_1,\ldots,\gamma_N,1,\pm}(x),\ldots, \Psi_{\gamma_1,\ldots, \gamma_N,N,\pm}(x))^T,\\ \Psi^0_{\gamma_1,\ldots, \gamma_N,\pm}(x) & =(\gamma_{1,\pm}\psi_{0,1,\pm}(x),\ldots, \gamma_{N,\pm}\psi_{0,N,\pm}(x))^T \lb{B.73} \end{split} \end{align} and one obtains \begin{align} \begin{split} & H_{\gamma_1,\ldots, \gamma_N,\pm}\Psi_{\gamma_1,\ldots,\gamma_N,j,\pm} =\lambda_j\Psi_{\gamma_1,\ldots, \gamma_N,j,\pm,}\\ & \Psi_{\gamma_1,\ldots, \gamma_N,j,\pm}\in H^2(\R),\; \|\Psi_{\gamma_1,\ldots,\gamma_N,j,\pm} \|^2_2=\gamma_{j,\pm},\; 1\leq j\leq N. \lb{b.74} \end{split} \end{align} Moreover, as indicated at the end of this appendix, one has $$\sigma (H_{\gamma_1,\ldots,\gamma_N,\pm})=\sigma(H_0)\cup \{\lambda_1,\ldots,\lambda_N\}. \tag {B.75}$$ Next we study the asymptotic behavior ofV_{\gamma_1,\ldots,\gamma_N,\pm}$as$|x|\rightarrow \infty$. For this purpose we now lift the restriction$\gamma_{j,\pm}>0$in (B.3) and admit the values 0 and$\infty$for$\gamma_{j,\pm}. This then yields \setcounter{equation}{75} \begin{align} \begin{split} V_{\gamma_1,\ldots,\gamma_N,+}(x)&\=\limits_{x\rightarrow \pm \infty} V_{\Sb 0,\ldots,0,+\\ \infty,\ldots,\infty,+\\N\text { times }\endSb}(x) +o(V_{\Sb 0,\ldots,0,+\\ \infty,\ldots,\infty,+\\ N\text { times }\endSb}(x)),\\ V_{\gamma_1,\ldots,\gamma_N,-}(x)&\=\limits_{x\rightarrow \pm \infty} V_{\Sb \infty,\ldots,\infty,-\\ 0,\ldots,0,-\\ N\text { times }\endSb}(x) +o(V_{\Sb \infty,\ldots,\infty,-\\ 0,\ldots,0,-\\ N\text { times }\endSb}(x)),\\ &\hskip 1truein \gamma_{j,\pm}>0,\ 1\leq j\leq N, \lb{B.76} \end{split} \end{align} where \begin{align*} &V_{\Sb 0,\ldots,0,\pm\\ N\text { times }\endSb}(x)=V_0(x), \tag {B.77}\\ &V_{\Sb \infty,\ldots,\infty,\pm\\ N\text { times }\endSb}(x)=V_0(x)-2\dfrac {d^2}{dx^2}\ln \det [C_{N,\infty,\pm}(x)], \tag {B.78} \end{align*} with $$C_{N,\infty,\pm}(x)=\left[\mp \int\limits^x_{\pm \infty}dx'\psi_{0,\ell,\pm}(x') \psi_{0,m,\pm}(x')\right]^N_{\ell,m=1},\ N\in \N. \tag {B.79}$$ In the following we shall assume thatV_{\Sb \infty,\ldots,\infty,\pm\\ N\text { times }\endSb}$satisfies (B.71). Then$H_{\Sb \infty,\ldots,\infty,\pm\\ N\text { times }\endSb}$and$H_0$are isospectral (i.e.,$\sigma (H_{\Sb \infty,\ldots,\infty,\pm\\ N\text { times }\endSb})=\sigma(H_0)). Next we introduce \seq{79} \begin{align} \begin{split} \Psi_{\gamma_1,\ldots,\gamma_N,\epsilon,\epsilon'}(\lambda,x) & =\psi_{0,\epsilon'}(\lambda,x)\\ &\qquad +\epsilon\sum\limits^N_{j=1} \Psi_{\gamma_1,\ldots,\gamma_{N,j,\epsilon}}(x) (\lambda_j-\lambda)^{-1}W(\psi_{0,j,\epsilon}, \psi_{0,\epsilon'}(\lambda))(x),\\ & \qquad \epsilon,\epsilon'\in \{\pm\},\gamma_{j,\epsilon}>0,\ 1\leq j\leq N,\ \lambda\in \rho(H_0)\backslash \{\lambda_j\}^N_{j=1} \lb{B.80} \end{split} \end{align} and \begin{align} \begin{split} \Psi_{\Sb \infty,\ldots,\infty,\epsilon,\epsilon'\\ N\text { times }\endSb}(\lambda,x) & =\psi_{0,\epsilon'} (\lambda,x)\\ &\qquad +\epsilon\sum\limits^N_{j=1} \Psi_{\Sb \infty,\ldots,\infty,j,\epsilon\\ N\text { times }\endSb}(x) (\lambda_j-\lambda)^{-1}W(\psi_{0,j,\epsilon},\psi_{0,\epsilon'} (\lambda))(x),\\ & \hspace*{1in} \epsilon,\epsilon'\in \{\pm\},\ \lambda\in \rho (H_0)\backslash \{\lambda_j\}^N_{j=1}, \lb{B.81} \end{split} \end{align} where \setcounter{equation}{81} \begin{align} \begin{split} \Psi_{\Sb \infty,\ldots,\infty,\epsilon\\ N\text { times }\endSb}(x) & = [C_{N,\infty,\epsilon}(x)]^{-1}\Psi^0_{\infty,\epsilon}(x),\\ \Psi_{\Sb \infty,\ldots,\infty,\epsilon\\ N\text { times }\endSb} (x) & =(\Psi_{\Sb \infty,\ldots,\infty,1,\epsilon\\ N\text { times }\endSb}(x),\ldots,\Psi_{\Sb \infty,\ldots,\infty,N,\epsilon\\ N \text { times }\endSb}(x))^T, \lb{b.82} \end{split} \end{align} $$\Psi^0_{\infty,\epsilon}(x) =(\psi_{0,1,\epsilon}(x),\ldots, \psi_{0,N,\epsilon}(x))^T. \lb{B.83}$$ Then \begin{multline*} H_{\gamma_1,\ldots,\gamma_N,\epsilon} \Psi_{\gamma_1,\ldots,\gamma_N,\epsilon,\epsilon'}(\lambda)=\lambda \Psi_{\gamma_1,\ldots,\gamma_N,\epsilon,\epsilon'}(\lambda),\\ \gamma_{j,\epsilon}>0,\ 1\leq j\leq N,\ \lambda\in \rho(H_0)\backslash \{\lambda_j\}^N_{j=1} \tag {B.84} \end{multline*} and $$H_{\Sb \infty,\ldots,\infty,\epsilon\\ N\text { times }\endSb}\Psi_{\Sb \infty,\ldots,\infty,\epsilon,\epsilon'\\ N\text { times }\endSb}(\lambda)=\lambda \Psi_{\Sb \infty,\ldots,\infty,\epsilon,\epsilon'\\ N\text { times }\endSb}(\lambda),\ \lambda\in \rho(H_0)\backslash \{\lambda_j\}^N_{j=1} \tag {B.85}$$ in the distributional sense, where $$H_{\Sb \infty,\ldots,\infty,\epsilon\\ N\text { times }\endSb}=-\dfrac {d^2}{dx^2}+V_{\Sb \infty,\ldots,\infty,\epsilon\\N\text { times }\endSb}, \ \D(H_{\Sb \infty,\ldots,\infty,\epsilon\\ N\text { times }\endSb})=H^2(\R). \tag {B.86}$$ Moreover, one verifies for\epsilon=\epsilon'$that \seq{86} \begin{gather} \Psi_{\gamma_1,\ldots,\gamma_N,\epsilon,\epsilon}(\lambda_\ell,x) =\gamma^{-1}_{\ell,\epsilon}\Psi_{\gamma_1,\ldots,\gamma_N,\ell,\epsilon}(x),\ \gamma_{\ell, \epsilon}>0,\ 1\leq \ell \leq N, \lb{B.87}\\ \Psi_{\infty,\ldots,\infty,\epsilon,\epsilon}(\lambda_\ell,x) =\Psi_{\infty,\ldots,\infty,\ell,\epsilon}(x),\ 1\leq \ell \leq N. \lb{B.88} \end{gather} Assuming in addition that $$H_{\gamma_1,\ldots,\gamma_N,\epsilon}\geq 0, \epsilon=\pm, \gamma_{j,\epsilon}>0,\ 1\leq j\leq N \tag {B.89}$$ (implying$H_{\Sb \infty,\ldots,\infty,\pm\\ N\text { times }\endSb}\geq 0,\,H_0\geq 0), we may write $$V_{\gamma_1,\ldots,\gamma_N,\epsilon}(x)=\phi_{\gamma_1,\ldots, \gamma_N,\epsilon , \epsilon'}(x)^2+\phi'_{\gamma_1,\ldots,\gamma_N,\epsilon,\epsilon'} (x),\ \gamma_{j,\epsilon}>0,\ 1\leq j\leq N, \tag {B.90}$$ where $$\phi_{\gamma_1,\ldots,\gamma_N,\epsilon,\epsilon'}(x)=\dfrac d{dx}\ln \Psi_{\gamma_1,\ldots,\gamma_N,\epsilon,\epsilon'}(0,x),\ \gamma_{j,\epsilon}>0,\ 1\leq j\leq N \tag {B.91}$$ and hence $$\quad \dfrac {d^m}{dx^m}\phi_{\gamma_1,\ldots,\gamma_N,\epsilon,\epsilon'}\in C^\infty(\R)\cap L^\infty(\R),\ m\in \N_0, \gamma_{j,\epsilon}>0,\ 1\leq j\leq N \tag {B.92}$$ by Lemma C.3. This finally yields the asymptotic relations \setcounter{equation}{92} \begin{align} \begin{split} \phi_{\gamma_1,\ldots,\gamma_N,+,\epsilon'}(x)&\=\limits_{x\rightarrow \pm \infty}\phi_{\Sb 0,\ldots,0,+,\epsilon'\\ \infty,\ldots,\infty,+,\epsilon'\\ N\text { times }\endSb}(x)+o(\phi_{\Sb 0,\ldots,0,+,\epsilon'\\ \infty,\ldots,\infty,+,\epsilon'\\ N\text { times }\endSb}(x)),\\ \phi_{\gamma_1,\ldots,\gamma_N,-,\epsilon'}(x)&\=\limits_{x\rightarrow \pm \infty}\phi_{\Sb \infty,\ldots,\infty,-,\epsilon'\\ 0,\ldots,0,-,\epsilon'\\ N\text { times }\endSb}(x)+o(\phi_{\Sb \infty,\ldots,\infty,-,\epsilon'\\ 0,\ldots,0,-,\epsilon'\\ N\text { times }\endSb}(x)),\\ &\hskip 1.5truein \gamma_{j,\epsilon}>0,\ 1\leq j\leq N, \lb{B.93} \end{split} \end{align} where \begin{align} \phi_{\Sb 0,\ldots,0,\epsilon,\epsilon'\\ N\text { times }\endSb}(x) & =\phi_{0,\epsilon'}(x)=\dfrac d{dx}\ln \psi_{0,\epsilon'}(0,x), \lb{B.94}\\ \phi_{\Sb \infty,\ldots,\infty,\epsilon,\epsilon'\\ N\text { times } \endSb} (x) & =\dfrac d{dx}\ln \Psi_{\Sb \infty,\ldots,\infty,\epsilon,\epsilon'\\ N\text { times }\endSb} (0,x) \lb{B.95} \end{align} and consequently, $$\dfrac {d^m}{dx^m}\phi_{\Sb 0,\ldots,0,\epsilon,\epsilon'\\ N \text { times }\endSb},\ \dfrac {d^m}{dx^m}\phi_{\Sb \infty,\ldots,\infty,\epsilon,\epsilon'\\ N\text { times }\endSb}\in C^\infty(\R)\cap L^\infty(\R),\ m\in \N_0 \tag {B.96}$$ and $$V_{\Sb \infty,\ldots,\infty,\epsilon\\ N\text { times }\endSb}(x) =\phi_{\Sb \infty,\ldots,\infty,\epsilon,\epsilon'\\ N\text { times }\endSb} (x)^2+\phi'_{\Sb \infty,\ldots,\infty,\epsilon,\epsilon'\\ N\text { times }\endSb}(x). \tag {B.97}$$ At this point we return toH_{\gamma_{1,\epsilon}}$and provide a short proof of (B.31). For this purpose we first construct the resolvent of$H_{\gamma_{1,\epsilon}}$. One computes \begin{multline*} ((H_{\gamma_{1,\epsilon}}-z)^{-1}f)(x)=\int\limits_{\R} dx'G_{\gamma_{1,\epsilon} }(z,x,x')f(x'),\\ f\in L^2(\R),\ \epsilon=\pm, z\in \rho(H_0)\backslash \{\lambda_1\},\ 0\leq \gamma_{1,\epsilon}\leq \infty, \tag {B.98} \end{multline*} where \begin{multline*} G_{\gamma_{1,\epsilon}}(z,x,x')=W(\psi_{0,+}(z), \psi_{0,-}(z))^{-1}\cases \Psi_{\gamma_{1,\epsilon,-}}(z,x')\Psi_{\gamma_{1,\epsilon,+}}(z,x),\ x'\leq x\\ \Psi_{\gamma_{1,\epsilon,+}}(z,x')\Psi_{\gamma_{1,\epsilon,-}}(z,x),\ x\leq x', \endcases\\ \epsilon=\pm,\ z\in \rho (H_0)\backslash \{\lambda_1\},\ 0\leq \gamma_{1,\epsilon}\leq \infty \tag {B.99} \end{multline*} since $$W(\Psi_{\gamma_{1,\epsilon,+}}(z), \Psi_{\gamma_{1,\epsilon,-}}(z))=W(\psi_{0,+}(z), \psi_{0,-}(z)),\ z\in \rho(H_0)\backslash \{\lambda_1\} \tag {B.100}$$ (see (B.80), (B.81)). This together with $$\Psi_{\gamma_{1,\epsilon,\epsilon}}(\lambda_1,.)=\psi^{-1}_{1,\gamma_1, \epsilon} \cases \in H^2(\R),&\gamma_{1,\epsilon}\in (0, \infty)\\ \notin L^2(\R),&\gamma_{1,\epsilon}=0, \infty\endcases \tag {B.101}$$ proves $$\sigma(H_{\gamma_{1,\epsilon}})=\cases \sigma(H_0)\cup \{\lambda_1\}, \ \gamma_{1,\epsilon}\in (0, \infty)\\ \sigma (H_0),\ \gamma_{1,\epsilon}=0,\infty \endcases \tag {B.102}$$ and hence (B.31). We may add that as$z\rightarrow \lambda_1$, $$G_{\gamma_{1,\epsilon}} (z,x,x')\=\limits_{z\rightarrow \lambda_1}\gamma_{1,\epsilon}(\lambda_1-z)^{-1} \psi_{1,\gamma_1,\epsilon}(x)^{-1}\psi_{1,\gamma_1,\epsilon} (x')^{-1}+0(1),\ \gamma_{1,\epsilon}\in [0, \infty) \tag {B.103}$$ as expected. The rank-one residue in (B.103) then proves that$\lambda_1$is a simple eigenvalue of$H_{\gamma_{1,\epsilon}}$iff$\gamma_{1,\epsilon}\in (0,\infty)$. (Simplicity of$\lambda_1$of course also follows from the fact that$H_{\gamma_{1,\epsilon}}$is in the limit point case at$\pm \infty$by hypothesis.) The general case (B.75) is proved analogously. Since$V_{\gamma_{1,+}}\neq V_{\gamma_{1,-}},\ \gamma_{1,\pm}\in (0,\infty)$as long as$V_0$is nonconstant, (B.102) exhibits an interesting nonuniqueness of general inverse spectral problems:\ Even though$V_{\gamma_{1,+}}$and$V_{\gamma_{1,-}}$are isospectral and the associated norming constants$\gamma_{1,+}$and$\gamma_{1,-}$(see (B.28)) can be chosen to be equal,$V_{\gamma_{1,+}}$and$V_{\gamma_{1,-}}$do not coincide unless the background potential$V_0$is trivial, i.e., constant. In order to keep this appendix within a reasonable length we adopted the simplifying hypothesis (B.71). However, a detailed study of the double commutation method reveals the fact that$\tau_{\gamma_{1,\pm}}=-\dfrac {d^2}{dx^2}+V_{\gamma_{1,\pm}},\ \gamma_{1,\pm}\in (0,\infty)$is in the limit point case at$\pm \infty$whenever$\tau_0=-\dfrac {d^2}{dx^2}+V_0$is (assuming$V_0\in L^1_{\loc}(\R)$to be real-valued). In addition, if$\tau_0$is in the limits point case at$+\infty$and$-\infty$and if$\sigma_{\ess}(H_0)\neq \emptyset$or$\sigma_{\ess}(H_0)=\emptyset$and$-\infty$is not an accumulation point of the discrete spectrum of$H_0$, then$\tau_{\gamma_{1,\pm}}$is also in the limit point case at$\mp \infty$and (B.75) is valid. Moreover, in analogy to the comment following (A.58) in the context of single commutation, one can prove that$H_{\gamma_1,\ldots,\gamma_N,\pm}$, restricted to the orthogonal complement of the eigenspace corresponding to the eigenvalues$\{\lambda_1,\ldots,\lambda_N\}$, is unitarily equivalent to$H_0$. Since (in contrast to the single commutation method) the proof of this fact is based on the Kodaira-Titchmarsh-Weyl theory of the associated matrix spectral function and requires additional preparations, this argument is separately presented in \cite{37}. Historically the results (B.68)--(B.75) have first been derived in \cite{50} in the case$V_0=0$. Extensions of these results (also for$V_0=0$) and their application to soliton solutions of the KdV equation appeared in the seminal paper \cite{34} and later in \cite{14}. The case of periodic backgrounds$V_0$is briefly treated in \cite{30}, \cite{31}, \cite{33}, \cite{57}. The case$V_0\in L^1(\R;\ (1+|x|)dx)$is considered in detail in \cite{58}. The close connection between double commutation and inverse spectral methods is exploited, e.g., in \cite{6}, \cite{20}, \cite{39}, \cite{57}. General backgrounds$V_0$are briefly discussed in Chapter~4 of \cite{23}. \chapter{Lax Pairs,$\tau$-Functions and B\"acklund Transformations} \renewcommand{\theequation}{C.\arabic{equation}} \addtocounter{page}{-1} %\setcounter{page}{59} \setcounter{lem}{0} \renewcommand{\thelem}{C.\arabic{lem}} In this appendix we recall auto-B\"acklund transformations for the KdV equation and connections between KdV and mKdV solutions related to each other by Miura's transformation \cite{70}. The methods employed involve the study of Lax pairs for the KdV equation, Hirota's$\tau$-function approach \cite{45} (see also \cite{46}, \cite{81}, \cite{87}) and commutation methods \cite{11}, \cite{14}, \cite{34a}--\cite{35}, \cite{37}--\cite{37e}, \cite{37g}, \cite{39}. These commutation methods are by no means restricted to (m)KdV equations. In fact, they have been applied to the Gelfand-Dickey hierarchy and its modified analog, the Drinfeld-Sokolov hierarchy \cite{37c}, \cite{37k}, the (modified) Kadomtsev-Petviashvili hierarchy \cite{37f}, \cite{37a} as well as to the Toda hierarchy and its modified version, the Kac-van Moerbeke hierarchy \cite{6a}, \cite{37b} (see also the references cited therein). Although we are mainly interested to study the class of (smooth) soliton solutions on quasi-periodic finite-gap background solutions in the main body of this paper, our hypotheses in this appendix will be chosen general enough to include the classes of singular (m)KdV solutions discussed, e.g., in \cite{1}--\cite{4}, \cite{9}, \cite{22}, \cite{32}, \cite{38}, \cite{49}, \cite{55}, \cite{63}, \cite{74} and the references therein. This is necessary since we encounter various singular (m)KdV solutions as a byproduct in intermediate steps in connection with double commutation techniques. We shall use the following general hypothesis. \medskip {\bf (H.C.1).}\ \ Let$V$be of the type $$V(t,x)=C-2\partial^2_x\ln \tau(t,x),\ C\in \C, \tag {C.1}$$ where$\tau:\C^2\rightarrow \C$is entire such that near any pole$x_j(t)$of$V(t,.)$one has a Laurent expansion of the form $$V(t,x)=2[x-x_j(t)]^{-2}+c_0(t)+\sum\limits^\infty_{n=2} c_n(t) [x-x_j(t)]^n,\ t\in \R, \tag {C.2}$$ with$c_n,\ x_j\in C^0(\R),\ n\in \N\backslash \{1\},\ j\in J$(some index set,$J\subseteq \Z$). Moreover, we assume that there exists a solution$\psi$of $$-\psi_{xx}(t,x)+V(t,x)\psi(t,x)=0,\ (t,x)\in \R^2 \tag {C.3}$$ satisfying $$\psi(t,x)=e^{Dx+Ft}\dfrac {\tilde \tau(t,x)}{\tau (t,x)},\ D,F\in \C \tag {C.4}$$ with$\tilde \tau:\C^2\rightarrow \C$entire and the Laurent expansion $$\psi(t,x)=[x-x_j(t)]^{-1}d_{-1}+\sum\limits^\infty_{n=2} d_n(t)[x-x_j(t)]^n,\ t\in \R, \tag {C.5}$$ with$d_{-1}\in \C,\ d_n\in C^0(\R),\ n\geq 2\ (d_2\neq 0$if$d_{-1}=0),\ j\in J$. \medskip Assuming$V_2to satisfy (H.C.1) we define the (possibly singular) differential expressions \begin{align*} A(t)&=\partial_x+\phi(t,x),\ A(t)^+=-\partial_x+\phi(t,x),\tag {C.6}\\ H_2(t)&=A(t)A(t)^+=-\partial^2_x+V_2(t,x),\\ H_1(t)&=A(t)^+A(t)=-\partial^2_x+V_1(t,x),\tag {C.7}\\ V_j(t,x)&=\phi(t,x)^2+(-1)^j\phi_x(t,x),\ j=1,2, \tag {C.8} \end{align*} where $$\phi(t,x)=\partial_x\ln \psi_2(t,x) \tag {C.9}$$ for a fixed solution\psi_2$of $$H_2(t)\psi(t)=0 \tag {C.10}$$ satisfying (C.5). Due to (H.C.1), the Laurent expansion for$\phi=\partial_x\ln \psi$near poles$x_j$of$V$looks like $$\phi(t,x)=\cases -[x-x_j(t)]^{-1}+\sum\limits^\infty_{n=2} e_n(t)[x-x_j(t)]^n,& e_2(t)=\dfrac {3d_2(t)}{d_{-1}},\ d_{-1}\neq 0\\ 2[x-x_j(t)]^{-1}+\sum\limits^\infty_{n=0} f_n(t)[x-x_j(t)]^n,& f_0(t)=\dfrac {d_3(t)}{d_2(t)},\ d_{-1}=0,\\ & \hspace*{.3in} d_2(t)\neq 0,\; t\in \R. \endcases \tag{C.11}$$ Near (necessarily first-order) zeros$y_{j'}(t),\ j'\in J'$(another index set,$J'\subseteq \Z$) of$\psi (t,x)$which are not poles of$V(t,x),\ \phi$behaves as $$\phi(t,x)=[x-y_{j'}(t)]^{-1}+\sum\limits^\infty_{n=0}g_n(t) [x-y_{j'}(t)]^n,\ t\in \R. \tag {C.12}$$ Here$e_n, f_n, g_n\in C^0(\R)$are appropriate coefficients determined by$\psi. In addition we introduce \setcounter{equation}{12} \begin{align} \begin{split} \psi_1(t,x)&=\psi_2(t,x)^{-1},\\ \Psi_1(t,x)&=\alpha_1(t)\psi_1(t,x)+\beta_1(t)\psi_1(t,x) \int\limits^x_{y_0}dx'\psi_1(t,x')^{-2}, \\ &\hskip 1.2truein \alpha_1,\beta_1\in C^\infty(\R),\ y_0\in \R, \lb{C.13} \end{split} \end{align} wherey_0$is independent of$t\in \R$. Then clearly $$H_1(t)\psi_1(t)=0,\ H_1(t)\Psi_1(t)=0. \tag {C.14}$$ We also define the (singular) differential expression $$B_V(t)=-4\partial^3_x+6V(t,x)\partial_x+3V_x(t,x), \tag {C.15}$$ where$Vsatisfies (H.C.1) and introduce \seq{15} \begin{align} \begin{split} H_0(t)&=H_2(t)+\lambda_0=-\partial^2_x+V_0(t,x),\\ V_0(t,x)&=V_2 (t,x)+\lambda_0=\phi (t,x)^2+\phi_x(t,x)+\lambda_0, \lb{C.16} \end{split} \end{align} \begin{align} \begin{split} \tilde H_0(t)&=H_1(t)+\lambda_0=-\partial^2_x+\tilde V_0(t,x),\\ \tilde V_0(t,x)&=V_1(t,x)+\lambda_0=\phi(t,x)^2-\phi_x(t,x)+\lambda_0 \lb{C.17} \end{split} \end{align} for some\lambda_0\in \R$. Then$\dfrac d{dt}H_0-[B_{V_0}, H_0] =\text { KdV}(V_0)$applied to$\psi_2yields $$\psi_2\text { KdV}(V_0)=-(H_0-\lambda_0)(\partial_t-B_{V_0})\psi_2 \tag {C.18}$$ and hence $$\text { KdV}(V_0)=0\text { iff }(H_0-\lambda_0)(\partial_t-B_{V_0})\psi_2=0. \tag {C.19}$$ We also note the identities \setcounter{equation}{19} \begin{align} &\text {KdV}(V_0)=\text { KdV}(V_2)-6\lambda_0V_{2,x}, \lb{C.20}\\ &\text {KdV}(\tilde V_0)=\text { KdV}(V_1)-6\lambda_0V_{1,x}, \lb {C.21}\\ &B_{V_0}=B_{V_2}+6\lambda_0\partial_x,\ B_{\tilde V_0}=B_{V_1}+6\lambda_0\partial_x, \lb {C.22}\\ &(\partial_t-B_{\tilde V_0})\psi_1=-\psi^2_1(\partial_t-B_{V_0})\psi_2, \lb{C.23}\\ \begin{split} &\text {KdV}(V_0)=(2\phi+\partial_x)[\phi_t-6(\phi^2+\lambda_0) \phi_x+\phi_{xxx}],\\ &\text {KdV}(\tilde V_0)=(2\phi-\partial_x)[\phi_t-6(\phi^2+\lambda_0)\phi_x+\phi_{xxx}]. \lb{C.24} \end{split}\end{align} In addition, assuming $$H_2(\partial_t-B_{V_0})\psi_2=(H_0-\lambda_0)(\partial_t-B_{V_0})\psi_2=0 \tag {C.25}$$ we may write \seq{25} \begin{align} \begin{split} [\partial_t-B_{V_0}(t)]\psi_2(t,x) & =\alpha_2(t)\psi_2(t,x)+\beta_2(t) \psi_2(t,x)\int\limits^x_{y_0}dx'\psi_2(t,x')^{-2},\\ & \hspace*{1.6in} \alpha_2,\beta_2\in C^\infty(\R) \lb{C.26} \end{split} \end{align} implying\text {KdV}(V_0)=0by (C.19). Combining (C.23) and (C.26) yields \setcounter{equation}{26} \begin{align} \begin{split} & (\tilde H_0-\lambda_0)(\partial_t-B_{\tilde V_0})\psi_1\\ &=(\tilde H_0-\lambda_0)[-\alpha_2\psi_1-\beta_2\psi_1\int\limits^x_{y_0} dx'\psi^2_1] =4\beta_2\psi^2_1\psi_{1,x}. \lb{C.27} \end{split} \end{align} Since (as in (C.19)) $$\text {KdV}(\tilde V_0)=0\text { iff }(\tilde H_0-\lambda_0)(\partial_t-B_{\tilde V_0})\psi_1=0, \tag {C.28}$$ we obtain $$\text {KdV}(\tilde V_0)=0\text { iff }\beta_2=0\text { in (C.26) or }V_0=\tilde V_0=\lambda_0. \tag {C.29}$$ Due to the identities (C.24) we obtain in addition $$\phi_t-6(\phi^2+\lambda_0)\phi_x+\phi_{xxx}=0\text { iff }\beta_2=0\text { in (C.26) or }V_0=\tilde V_0=\lambda_0=\phi^2. \tag {C.30}$$ In this context we also note that \setcounter{equation}{30} \begin{align} \begin{split} & A^+(\partial_t-B_{V_0}) =(\partial_t-B_{\tilde V_0})A^+,\ (\partial_t-B_{V_0})A=A(\partial_t-B_{\tilde V_0})\\ & \text {iff }\phi_t-6(\phi^2+\lambda_0)\phi_x+\phi_{xxx}=0, \lb{C.31} \end{split} \end{align} or equivalently, \begin{gather} \begin{split} \dfrac d{dt}Q-[B,Q]=\{\phi_t-6(\phi^2+\lambda_0)\phi_x+\phi_{xxx}\} \pmatrix 0&1\\ 1&0\endpmatrix,\\ Q(t)=\pmatrix 0&A(t)^+\\ A(t)&0\endpmatrix,\ B(t)=\pmatrix B_{\tilde V_0}(t)&0\\ 0&B_{V_0}(t)\endpmatrix.\lb{C.32} \end{split} \end{gather} These formulas may be compared toA^+H_0=\tilde H_0A^+,\ H_0A=A\tilde H_0$. Relations such as (C.29) and (C.30) are not restricted to (m)KdV systems. In fact, generalizing the methods of this appendix we have obtained the analog of (C.29) and (C.30) for the entire Gelfand-Dickey hierarchy and its modified version, the Drinfeld-Sokolov hierarchy, in \cite{37c}, \cite{37k} as well as for the Toda hierarchy and its modified analog, the Kac-van Moerbeke hierarchy, in \cite{6a}, \cite {37b}. Returning to$\Psi_1in (C.13) we note the identities \setcounter{equation}{32} \begin{align} \begin{split} & (\partial_t-B_{\tilde V_0})\Psi_1 =\ {\DOT \alpha}_1\psi_1+\ {\DOT \beta}_1\psi_1\int\limits^x_{y_0}dx'\psi^{-2}_1-\alpha_1\psi^2_1 (\partial_t-B_{V _0})\psi_2\\ & -\beta_1\psi^2_1(\int\limits^x_{y_0}dx'\psi^{-2}_1)(\partial_t-B_{V_0}) \psi_2+2\ beta_1\psi_1\int\limits^x_{y_0}dx'[\psi^{-1}_1(\partial_t-B_{V_0} )\psi_2]\\ & -2\beta_1[\tilde V_0(t,y_0)+2\lambda_0]\psi_2(t,y_0)^2\psi_1 \lb{C.33} \end{split}\\ \begin{split} &=\ {\DOT \alpha}_1\psi_1+\ {\DOT \beta}_1\psi_1\int\limits^x_{y_0}dx'\psi^{-2}_1+\alpha_1 (\partial_t-B_{\tilde V_0})\psi_1+\beta_1(\int\limits^x_{y_0}dx'\psi^{-2}_1)(\partial_t -B_{\tilde V_0})\psi_1 \hskip 1truein\\ & -2\beta_1\psi_1\int\limits^x_{y_0}dx'[\psi^{-3}_1(\partial_t-B_{\tilde V_0})\psi_1]-2\beta_1[\tilde V_0(t,y_0)+2\lambda_0]\psi_1 (t,y_0)^{-2}\psi_1 \lb{C.34} \end{split} \end{align} (here\bullet$denotes$\dfrac d{dt}$) and $$H_1(\partial_t-B_{V_1})\Psi_1=4\beta_2\psi_1\psi_{1,x}\Psi_1 \lb{C.35}$$ (with$\beta_2introduced in (C.26)). Next, in accordance with (H.C.1), we make the ansatz \begin{align*} \psi_2(t,x)&=e^{Dx+Et}\tau_1(t,x)/\tau_2(t,x), \tag {C.36}\\ V_2(t,x)&=C-2\partial^2_x\ln \tau_2(t,x), \tag {C.37}\end{align*} with\tau_j:\C^2\rightarrow \C,\ j=1,2entire. Then a straightforward calculation yields \setcounter{equation}{37} \begin{align} V_1(t,x)&=C-2\partial^2_x\ln \tau_1(t,x),\lb{C.38}\\ \phi(t,x)&=D+\partial_x\ln [\tau_1(t,x)/\tau_2(t,x)], \lb{C.39} \end{align} $$C-D^2 = 2D\tau^{-1}_1\tau_{1,x}-2D\tau^{-1}_2\tau_{2,x}-2 \tau^{-1}_1\tau^{-1}_2\tau_{1,x}\tau_{2,x} +\tau^{-1}_1\tau_{1,xx} + \tau^{-1}_2\tau_{2,xx} \lb{C.40}$$ (see also Appendix D in \cite{35} for further details). So far our considerations have been purely algebraic, i.e., (C.6)--(C.10),\ (C.13)--(C.40) hold away from singularities of\psi_1,\psi_2, V_1,V_2,\phi$etc. In order to obtain a functional analytic treatment we finally strengthen (H.C.1) and introduce {\bf (H.C.2).}\ \ Suppose$V$to be real-valued, $$\partial^m_xV\in C^\infty(\R^2)\cap L^\infty(\R^2),\ m\in \N_0 \tag {C.41}$$ and$0<\psi\in C^\infty(\R^2)$satisfying $$-\psi_{xx}(t,x)+V(t,x)\psi(t,x)=0\tag {C.42}$$ in the distributional sense. \setcounter{lem}{2} \begin{lem} %c.3 {(\cite{44},\ Corollary XI.6.5, \cite{34b}, \cite{37d}, \cite{39})} Suppose$V_2, \psi_2$satisfy (H.C.2).\hfill\break Then$V_1$and$\phi$also satisfy (C.41), i.e.,$V_1$and$\phiare real-valued and $$\partial^m_xV_1,\ \partial^m_x\phi\in C^\infty(\R^2) \cap L^\infty(\R^2),\ m\in \N_0.\tag {C.43}$$ \end{lem} Lemma C.3 then guarantees that \setcounter{equation}{43} \begin{align} \begin{split} A(t)&=\dfrac d{dx}+\phi(t,.),\ \D(A(t))=H^1(\R),\ t\in \R,\\ A(t)^*&=-\dfrac d{dx}+\phi(t,.),\ \D(A(t)^*)=H^1(\R),\ t\in \R \lb{C.44} \end{split} \end{align} are closed linear operators inL^2(\R)$and that \begin{multline*} H_2(t)=A(t)A(t)^*\geq 0,\ H_1(t)=A(t)^*A(t)\geq 0,\ \D(H_j(t))=H^2(\R),\ j=1,2,\\ t\in \R \tag {C.45} \end{multline*} are nonnegative and self-adjoint in$L^2(\R)$. We may also add that in connection with (H.C.2) it is often convenient to choose$y_0=\pm \infty$in (C.13), (C.26) (see, e.g., Appendices A,B, and D) in which case expressions such as$[\tilde V_0(t,y_0)+2\lambda_0]\psi_2(t,y_0)^2$in (C.33) are interpreted as$\lim\limits_{y_0\rightarrow \pm \infty}[\tilde V_0(t,y_0)+2\lambda_0]\psi_2(t,y_0)^2$and hence equal zero whenever$\tilde V_0\in L^\infty(\R^2)$and$\psi_2(t,.) \in H^2((R,\pm \infty)),\ R\in \R$. \chapter{(m)KdV Soliton Solutions Relative to General Backgrounds} \addtocounter{page}{-1} \renewcommand{\theequation}{D.\arabic{equation}} \setcounter{lem}{0} \renewcommand{\thelem}{D.\arabic{lem}} In this appendix we combine Appendices A,B, and C and study soliton solutions relative to general KdV background solutions$V_0(t,x)$. In addition we use Miura-type transformations to accomplish the analogous goal for the mKdV equation. We start with single commutation and assume hypothesis {\bf (H.D.1).}\ \ Let$\partial^m_xV_0\in C^\infty(\R^2)\cap L^\infty(\R^2),\ m\in \N_0$be real-valued. In the following we freely use the notation introduced in Appendix A adding the variable$t\in \R$whenever appropriate. We also assume$\sigma_j\in [-1,1],\ 1\leq j\leq N$in (A.46) to be$t$-independent. If$\psi_{0,\pm 1}(\lambda,t,x),\ \lambda0$we call$\phi_{N,+}$(resp.$\phi_{N,-}$) a$2M$-soliton mKdV solution relative to the background mKdV solution$\phi_{0,+}$(resp.$\phi_{0,-}$) since then$Q_{N,\pm}(t)$has$2M$eigenvalues in the spectral gap$(-|E_0|^{1/2},\ |E_0|^{1/2})$of$Q_{0,\pm}(t)$. (Here$M$coincides with$M$in the paragraph following (D.46).) If$\lambda_1=0$and$\sigma_1\in \{-1,1\}$then$\phi_{N,+}$(resp.$\phi_{N,-}$) is also called a$2M$-soliton solution, whereas if$\lambda_1=0$and$\sigma_1 \in (-1,1)$then$\phi_{N,+}=\phi_{N,-}$is called a$(2M-1)$-soliton solution relative to$\phi_{0,\pm}$since in the former case$0\notin \sigma (Q_{N,\pm}(t))$whereas$0\in \sigma (Q_{N,\pm} (t))$in the latter case. (Since all eigenvalues of$Q_{N,\pm}(t)$in$(-|E_0|^{1/2},\ |E_0|^{1/2})$are simple, these cases just distinguish whether$Q_{N,\pm}(t)$has$2M$or$(2M-1)$eigenvalues in$(-|E^{1/2}_0,\ |E_1|^{1/2})$. We also recall that the spectra of$Q_{0,\pm},\ Q_{N,\pm}$are symmetric with respect to the origin and easily determined from$Q^2_{0,\pm},\ Q^2_{N,\pm}$\cite{34a}--\cite{35}, \cite{39}. If$\lambda_1=0$and$\sigma_1\in (-1,1)$then$\psi_{0,+1,\sigma_1,\ldots,\sigma_N}=\psi_{0,-1,\sigma_1,\ldots, \sigma_N}$implying$\phi_{N,+}=\phi_{N,-}$in this case.) We stress that (D.24), (D.25), (D.41)--(D.43) remain valid for complex-valued$V_0$and general$\sigma_j\in \C$. In this case, however, the corresponding (m)KdV solutions in general will be singular (with singularities as described in Appendix C). The same remark applies to the case where$\lambda_j\in \rho(H_0)\cap \R$but$\lambda_j>E_0$for some$j\in \{1,\ldots, N\}$. Finally we turn to double commutation. In addition to the basic hypothesis (H.D.1) we shall assume (see (B.71)) \medskip {\bf (H.D.2).}\ \$\partial^m_xV_{\gamma_1,\ldots,\gamma_j,\pm}\in L^\infty(\R^2),\ m\in \N_0,\ 1\leq j\leq N$\medskip \noindent and freely use the notation of Appendix B adding the variable$t\in \R$whenever appropriate. The constants$\gamma_{j,\pm}>0, \ 1\leq j\leq N$in (B.3) are assumed to be$t$-independent. We pick$\psi_{0,\pm}(\lambda,t,x),\ \lambda\in \rho(H_0)satisfying \setcounter{equation}{52} \begin{align} \begin{split} \psi_{0,\pm}(\lambda,.,.)&\in C^\infty(\R^2),\ \lambda\in \rho (H_0),\\ \psi_{0,\pm}(\lambda,t,.)&\in L^2((R,\pm \infty)),\ \lambda\in \rho(H_0),\ R\in \R,\ t\in \R \lb{D.53} \end{split} \end{align} and $$(H_0(t)-\lambda)\psi_{0,\pm}(\lambda,t)=0,\ \lambda\in \rho(H_0), \ t\in \R, \lb{D.54}$$ $$(\partial_t-B_{V_0}(t))\psi_{0,\pm}(\lambda,t)=0,\ \lambda\in \rho(H_0),\ t\in \R. \lb{D.55}$$ Thus $$\text {KdV}(V_0)=0,\ V_0=\phi^2_{1,\pm}+\phi_{1,\pm,x}+\lambda_1 \lb{D.56}$$ (and hence\sigma(H_0(t))$is$t$-independent) by (C.19). Next, combining (C.17), (C.29), and (C.30) (with$\lambda_0=\lambda_1) one gets (cf. (B.14)--(B.18)) \setcounter{equation}{56} \begin{align} \begin{split} \text {KdV}(\tilde V_{1,0,\pm}+\lambda_1)&=\text { KdV}(V_{2,1,\pm}+\lambda_1)=0,\\ \tilde V_{1,0,\pm}=\phi^2_{1,\pm}-\phi_{1,\pm,x} &=\phi^2_{1,\gamma_1,\pm}+\phi_{1,\gamma_1,\pm,x}=V_{2,1,\pm} \lb{D.57} \end{split} \end{align} and $$\text {mKdV}(\phi_{1,\pm};\lambda_1)=0. \lb{D.58}$$ By (C.33) one concludes \begin{multline} (\partial_t-B_{\tilde V_{1,0,\pm}+\lambda_1})\psi_{1,\gamma_1,\pm}\\ =\lim\limits_{y_0\rightarrow \pm \infty}\{-2\gamma_{1,\pm}[\tilde V_{1,0,\pm}(t,y_0)+3\lambda_1]\psi_{0,1,\pm}(t,y_0)^2\} \psi^{-1}_{0,1,\pm}=0 \lb{D.59} \end{multline} since $$\tilde V_{1,0,\pm}\psi^2_{0,1,\pm}=2\psi^2_{0,1,\pm,x}-\psi^2_{0,1,\pm} [V_0-\lambda_1] \lb{D.60}$$ and $$\psi_{0,1,\pm}(t,.)\in H^2((R,\pm \infty)),\ R\in \R,\ t\in \R \lb {D.61}$$ because ofV_0\in L^\infty((R, \pm \infty)). (Note that we do not need the full strength of (H.D.1).) Thus $$(\partial_t-B_{V_{2,1,\pm}+\lambda_1})\psi_{1,\gamma_1,\pm}=0 \lb{D.62}$$ implying \begin{align} &(\partial_t-B_{V_{\gamma_{1,\pm}}})\psi^{-1}_{1,\gamma_1,\pm}=0, \lb {D.63}\\ &\text {KdV}(V_{\gamma_{1,\pm}})=0,\ V_{\gamma_1,\pm}=\phi^2_{1,\gamma_1,\pm}-\phi_{1,\gamma_1,\pm,x} +\lambda_1, \lb{D.64}\\ &\text {mKdV}(\phi_{1,\gamma_1,\pm};\lambda_1)=0 \lb{D.65} \end{align} using (C.23), (C.29), and (C.30). Iterating this procedure yields \begin{align} \begin{split} &\text {KdV}(V_{\gamma_1,\ldots,\gamma_j,\pm})=0,\\ & V_{\gamma_1,\ldots,\gamma_j,\pm} =\phi^2_{j,\gamma_1,\ldots,\gamma_j,\pm}-\phi_{j ,\gamma_1,\ldots,\gamma_j,\pm,x}+\lambda_j,\ 1\leq j\leq N, \lb{D.66} \end{split} \end{align} $$\text {mKdV}(\phi_{j,\gamma_1,\ldots,\gamma_j,\pm};\lambda_j)=0,\ 1\leq j\leq N, \lb{D.67}$$ where $$V_{\gamma_1,\ldots,\gamma_j,\pm}(t,x)=V_0(t,x)-2\partial^2_x\ln \det [1+C_{\gamma_1,\ldots,\gamma_j,\pm}(t,x)], \lb{D.68}$$ \begin{align} \begin{split} & C_{\gamma_1,\ldots,\gamma_j,\pm}(t,x) =[c_{\gamma_1,\ldots,\gamma_j,\pm,\ell,m}(t ,x)]^j_{\ell,m=1},\\ & c_{\gamma_1,\ldots,\gamma_j,\pm,\ell,m}(t,x)=\mp\gamma_{\ell,\pm} \int\limits^x_{ \pm \infty} dx'\psi_{0,\ell,\pm}(t,x')\psi_{0,m,\pm}(t,x'),\\ &\hskip 2truein 1\leq \ell,m\leq j,\ 1\leq j\leq N. \lb{D.69} \end{split} \end{align} In analogy to the discussion following (D.46),V_{\gamma_1,\ldots,\gamma_N,\pm},\ \gamma_{j,\pm}>0,\ 1\leq j\leq N$are$N$-soliton KdV solutions relative to the background KdV solution$V_0$. The asymptotic relations (B.76) for fixed$t\in \Rthen read in the present context \begin{align} \begin{split} V_{\gamma_1,\ldots,\gamma_N,+}(t,x)&\=\limits_{x\rightarrow \pm \infty} V_{\Sb 0,\ldots,0,+\\ \infty,\ldots,\infty,+\\ N\text { times }\endSb}(t,x)+o(V_{\Sb 0,\ldots,0,+\\ \infty,\ldots,\infty,+\\ N \text { times }\endSb} (t,x)),\\ V_{\gamma_1,\ldots,\gamma_N,-}(t,x)&\=\limits_{x\rightarrow \pm \infty}V_{\Sb \infty,\ldots,\infty,-\\ 0,\ldots,0,-\\ N\text { times}\endSb} (t,x)+o (V_{\Sb \infty,\ldots,\infty,-\\ 0,\ldots,0,-\\ N\text { times }\endSb}(t,x)),\\ &\hskip 1truein \gamma_{j,\pm}>0,\ 1\leq j\leq N, \lb{D.70} \end{split} \end{align} where \begin{align} &V_{\Sb 0,\ldots,0,\pm\\ N\text { times }\endSb}(t,x)=V_0(t,x), \lb {D.71}\\ &V_{\Sb \infty,\ldots,\infty,\pm\\ N\text { times }\endSb}(t,x)=V_0(t,x)-2\partial^2_x\ln \det [C_{N,\infty,\pm}(t,x)], \lb{D.72} \end{align} with \begin{align} \begin{split} C_{N,\infty,\pm}(t,x)=\left[\mp \int\limits^x_{\pm \infty}dx'\psi_{0,\ell,\pm}(t,x') \psi_{0,m,\pm}(t,x')\right]^N_{\ell,m=1},\\ \hspace*{2in} N\in \N. \lb{D.73} \end{split} \end{align} As in Appendix B we assume thatV_{\Sb \infty,\ldots,\infty,\pm\\ N-\text {times }\endSb}$satisfies (H.D.2) in the following. In particular, the leading asymptotic terms$V_{0,\ldots,0,\pm}=V_0,\ V_{\infty,\ldots, \infty,\pm}$in (D.70) satisfy the KdV equation, i.e., $$\text {KdV}(V_{\Sb 0,\ldots,0,\pm\\ N\text { times }\endSb})=\text { KdV}(V_{\Sb \infty,\ldots,\infty,\pm\\ N\text { times }\endSb})=0,\ N\in \N. \lb{D.74}$$ (The result for$V_{\infty,\pm}$in the case$N=1$follows from (D.59)--(D.64) by replacing$\psi_{1,\gamma_1,\pm}$by$\gamma^{-1}_{1,\pm}\psi_{1,\gamma_1,\pm}$and taking$\gamma_{1,\pm}\rightarrow \infty$. The case$N\geq 2can be treated analogously.) Next we introduce (see Appendix B) \begin{align} \begin{split} \Psi_{\gamma_1,\ldots,\gamma_N,\epsilon,\epsilon'} (\lambda,t,x)&=\psi_{0,\epsilon'}(\lambda,t,x)\\ &+\epsilon\sum\limits^N_{j=1}\Psi_{\gamma_1,\ldots, \gamma_{N,j,\epsilon}}(t,x) (\lambda_j - \lambda)^{-1}W(\psi_{0,j,\epsilon}(t), \psi_{0,\epsilon'}(\lambda,t))(x),\\ &\qquad \epsilon,\epsilon'\in \{\pm\},\ \gamma_{j,\epsilon}>0,\ 1\leq j\leq N,\ \lambda\in \rho (H_0)\backslash \{\lambda_j\}^N_{j=1}, \lb{D.75} \end{split}\\ \begin{split} \Psi_{\Sb \infty,\ldots,\infty,\epsilon,\epsilon'\\ N\text { times }\endSb}(\lambda,t,x)&=\psi_{0,\epsilon'}(\lambda,t,x)\\ &+\epsilon\sum\limits^N_{j=1}\Psi_{\Sb \infty,\ldots,\infty,j,\epsilon\\ N\text { times }\endSb}(t,x)(\lambda_j-\lambda)^{-1} W(\psi_{0,j,\epsilon}(t),\psi_{0,\epsilon'} (\lambda,t))(x),\\ &\hspace*{1.75in} \epsilon,\epsilon'\in \{\pm \},\ \lambda\in \rho(H_0)\backslash \{\lambda_j\}^N_{j=1}, \lb{D.76} \end{split} \end{align} where \begin{gather} \Psi_{\gamma_1,\ldots,\gamma_N,\epsilon}(t,x) =[1+C_{\gamma_1,\ldots, \gamma_N,\epsilon}(t,x)]^{-1} \Psi^0_{\gamma_1,\ldots,\gamma_N,\epsilon}(t,x), \lb{D.77}\\ \Psi_{\Sb \infty,\ldots,\infty,\epsilon\\ N\text { times }\endSb}(t,x) =[C_{N,\infty,\epsilon}(t,x)]^{-1}\Psi^0_{\Sb \infty,\ldots,\infty,\epsilon\\ N\text { times }\endSb}(t,x), \lb{D.78}\\ \Psi_{\gamma_1,\ldots,\gamma_N,\epsilon}(t,x) =(\Psi_{\gamma_1,\ldots, \gamma_N,1,\epsilon}(t,x),\ldots,\Psi_{\gamma_1,\ldots,\gamma_N,N, \epsilon} (t,x))^T, \lb{D.79}\\ \hspace*{1.2in} \gamma_{j,\epsilon}>0,\ 1\leq j\leq N,\notag \\ \Psi_{\Sb \infty,\ldots,\infty,\epsilon\\ N\text { times }\endSb}(t,x) =(\Psi_{\Sb \infty,\ldots,\infty,1,\epsilon\\ N\text { times }\endSb}(t,x),\ldots, \Psi_{\Sb \infty,\ldots,\infty,N,\epsilon\\ N \text { times }\endSb}(t,x))^T, \lb{D.80}\\ \Psi^0_{\gamma_1,\ldots,\gamma_N,\epsilon}(t,x) =(\gamma_{1,\epsilon} \psi_{0,1,\epsilon}(t,x),\ldots, \gamma_{N,\epsilon}\psi_{0,N,\epsilon}(t,x))^T, \lb{D.81}\\ \hspace*{2in} \gamma_{j,\epsilon}>0,\ 1\leq j\leq N, \notag \end{gather} $$\Psi^0_{\Sb \infty,\ldots,\infty,\epsilon\\ N\text { times }\endSb}(t,x) =(\psi_{0,1,\epsilon}(t,x),\ldots,\psi_{0,N,\epsilon}(t,x))^T. \lb{D.82}$$ Then \begin{align} (H_{\gamma_1,\ldots,\gamma_N,\epsilon}(t)-\lambda) &\Psi_{\gamma_1,\ldots,\gamma_ N,\epsilon,\epsilon'}(\lambda,t)=0, \lb{D.82-1}\\ (H_{\Sb \infty,\ldots,\infty,\epsilon\\ N\text { times }\endSb} (t)-\lambda)&\Psi_{\Sb \infty,\ldots,\infty,\epsilon,\epsilon'\\ N\text {times }\endSb} (\lambda,t)=0. \lb{D.83} \end{align} Assuming in addition that $$H_{\gamma_1,\ldots,\gamma_N,\epsilon}(t)\geq 0,\ \epsilon=\pm,\ \gamma_{j,\epsilon}>0,\ 1\leq j\leq N \lb{D.84}$$ for some (and hence for all)t\in \Rwe introduce $$\quad\phi_{\gamma_1,\ldots,\gamma_N,\epsilon,\epsilon'} (t,x)=\partial_x \ln \Psi_{\gamma_1,\ldots,\gamma_N,\epsilon,\epsilon'}(0,t,x),\ \gamma_{j,\epsilon}>0,\ 1\leq j\leq N. \lb{D.85}$$ Then $$\partial^m_x\phi_{\gamma_1,\ldots,\gamma_N,\epsilon,\epsilon'}\in C^\infty(\R^2)\cap L^\infty(\R^2),\ m\in \N_0,\ \gamma_{j,\epsilon}>0,\ 1\leq j\leq N \lb{D.86}$$ and \begin{multline} V_{\gamma_1,\ldots,\gamma_N,\epsilon}(t,x)= \phi_{\gamma_1,\ldots,\gamma_N,\epsilon,\epsilon'}(t,x)^2+ \phi_{\gamma_1,\ldots, \gamma_N,\epsilon,\epsilon',x}(t,x),\\ \gamma_{j,\epsilon}>0,\ 1\leq j\leq N. \lb{D.87} \end{multline} The asymptotic relations (B.93) then read \begin{align} \begin{split} \phi_{\gamma_1,\ldots,\gamma_N,+,\epsilon'}(t,x)&\=\limits_{x\rightarrow \pm \infty}\phi_{\Sb 0,\ldots,0,+,\epsilon'\\ \infty,\ldots,\infty,+,\epsilon'\\ N\text { times }\endSb}(t,x)+o(\phi_{\Sb 0,\ldots,0,+,\epsilon'\\ \infty,\ldots,\infty,+,\epsilon'\\ N\text { times }\endSb}(t,x)),\\ \phi_{\gamma_1,\ldots,\gamma_N,-,\epsilon'}(t,x)&\=\limits_{x\rightarrow \pm \infty}\phi_{\Sb \infty,\ldots,\infty,-,\epsilon'\\ 0,\ldots,0,-,\epsilon'\\ N\text { times }\endSb}(t,x)+o(\phi_{\Sb \infty,\ldots,\infty,-,\epsilon'\\ 0,\ldots,0,-,\epsilon'\\ N\text { times }\endSb}(t,x),\\ &\hskip 1.2truein \gamma_{j,\epsilon}>0,\ 1\leq j\leq N, \lb{D.88} \end{split} \end{align} where, according to (B.94)--(B.97), \begin{align} &\phi_{\Sb 0,\ldots,0,\epsilon,\epsilon'\\ N\text { times }\endSb} (t,x)=\phi_{0,\epsilon'}(t,x)=\partial_x \ln \psi_{0,\epsilon'}(0,t,x), \lb{D.89}\\ &\phi_{\Sb \infty,\ldots,\infty,\epsilon,\epsilon'\\ N\text { times }\endSb}(t,x)=\partial_x\ln \Psi_{\Sb \infty,\ldots,\infty,\epsilon,\epsilon'\\ N\text { times }\endSb}(0,t,x)\lb{D.90} \end{align} and hence $$\partial^m_x\phi_{\Sb 0,\ldots,0,\epsilon,\epsilon'\\ N\text { times }\endSb}, \partial^m_x\phi_{\Sb \infty,\ldots,\infty,\epsilon,\epsilon'\\ N\text { times }\endSb}\in C^\infty(\R^2)\cap L^\infty(\R^2),\ m\in \N_0 \lb{D.91}$$ and $$V_{\Sb \infty,\ldots,\infty,\epsilon\\ N\text { times }\endSb} (t,x)=\phi_{\Sb \infty,\ldots,\infty,\epsilon,\epsilon'\\ N\text { times }\endSb}(t,x)^2+\phi_{\Sb \infty,\ldots,\infty,\epsilon,\epsilon',x\\ N\text { times }\endSb} (t,x). \lb{D.92}$$ Since \begin{gather} (H_{\gamma_1,\ldots,\gamma_N,\epsilon}(t)-\lambda)\Psi_{\gamma_1,\ldots,\gamma_N ,\epsilon,\epsilon'}(\lambda,t)=0, \lb{D.93}\\ (\partial_t-B_{V_{\gamma_1,\ldots,\gamma_N,\epsilon}}(t)) \Psi_{\gamma_1,\ldots, \gamma_N,\epsilon,\epsilon'}(\lambda,t)=0, \lb{D.94}\\ (H_{\Sb \infty,\ldots,\infty\\ N\text { times }\endSb} (t)-\lambda)\Psi_{\Sb \infty,\ldots,\infty,\epsilon,\epsilon'\\ N\text {times }\endSb}(\lambda,t)=0, \lb{D.95}\\ (\partial_t-B_{V_{\Sb \infty,\ldots,\infty,\epsilon\\ N \text { times }\endSb}}(t))\Psi_{\Sb \infty,\ldots,\infty,\epsilon,\epsilon'\\ N\text { times }\endSb} (\lambda,t)=0,\ \lambda \in \rho (H_0)\backslash \{\lambda_j\}^N_{j=1} \lb{D.96} \end{gather} as indicated below, (C.30) (with\lambda_0=0) yields \begin{align} &\text {mKdV}(\phi_{\gamma_1,\ldots,\gamma_N,\epsilon,\epsilon'})=0, \lb{D.97}\\ \begin{split} &\text {mKdV}(\phi_{\Sb 0,\ldots,0,\epsilon,\epsilon'\\ N\text { times }\endSb})=\text { mKdV}(\phi_{\Sb \infty,\ldots,\infty,\epsilon,\epsilon'\\ N\text { times }\endSb})=0,\\ & \hspace*{2.4in} N\in \N, \lb{D.98} \end{split} \end{align} i.e., the leading order terms in (D.89) satisfy the mKdV equation. Introducing in analogy to (D.49) the Dirac-type operators \begin{align} \begin{split} & Q_{0,\epsilon}(t)=\pmatrix 0&-\dfrac d{dx}+\phi_{0,\epsilon}(t,.)\\ \dfrac d{dx}+\phi_{0,\epsilon}(t,.)&0\endpmatrix, \D(Q_{0,\epsilon}(t))=[H^1(\R)]^2,\\ & Q_{N,\epsilon,\epsilon'}(t) =\pmatrix 0&-\dfrac d{dx}+\phi_{\gamma_1,\ldots,\gamma_N,\epsilon,\epsilon'}(t,.)\\ \dfrac d{dx}+\phi_{\gamma_1,\ldots,\gamma_{N,\epsilon,\epsilon'}}(t,.)&0 \endpmatrix,\\ & \D (Q_{N,\epsilon,\epsilon'}(t)) =[H^1(\R)]^2,\ t\in \R,\ \epsilon,\epsilon'\in \{\pm\},\ N\in \N \lb{D.99} \end{split} \end{align} in[L^2(\R^2)]^2$then, as in the discussion following (D.52),$\phi_{\gamma_1,\ldots,\gamma_N,\epsilon,\epsilon'},\ \gamma_{j,\epsilon}>0,\ 1\leq j\leq N$are$2N$respectively$(2N-1)$-soliton mKdV solutions relative to the background mKdV solution$\phi_{0,\epsilon'}$depending on whether$0\notin \sigma (Q_{N,\epsilon,\epsilon'}(t))$respectively$0\in \sigma (Q_{N,\epsilon,\epsilon'}(t))$. (As in (D.49) and (D.50),$\sigma (Q_{0,\epsilon}(t))$and$\sigma (Q_{N,\epsilon,\epsilon'}(t))$are$t$-independent due to (D.98), (D.99).) It remains to sketch a proof of (D.94)--(D.97). We consider$N=1first. Then (D.94) and (D.96) are easily verified. In order to prove (D.95) one computes from (C.31) \begin{align} \begin{split} & (\lambda_1-\lambda)(\partial_t-B_{V_{\gamma_1,\epsilon}}(t)) \Psi_{\gamma_1,\epsilon,\epsilon'}(\lambda,t) = (\partial_t-B_{V_{\gamma_1,\epsilon}}(t))(A^+_{1,\gamma_1,\epsilon} (t) A^+_{1,\epsilon}(t)\psi_{0,\epsilon'})(\lambda,t)\\ &=A^+_{1,\gamma_1,\epsilon}(t)(\partial_t-B_{V_{2,1,\epsilon}+\lambda_1} (t))(A^+_{1,\epsilon}(t)\psi_{0,\epsilon'})(\lambda,t)\\ &=A^+_{1,\gamma_1,\epsilon}(t)(\partial_t-B_{\tilde V_{1,0,\epsilon}+\lambda_1} (t))(A^+_{1,\epsilon}(t)\psi_{0,\epsilon'})(\lambda,t)\\ &=A^+_{1,\gamma_1,\epsilon}(t)A^+_{1,\epsilon}(t) (\partial_t-B_{V_0}(t))\psi_{0, \epsilon'}(\lambda,t)=0. \lb{D.100} \end{split} \end{align} The caseN\geq 2$then follows by iterating this argument. (D.97) is proved analogously. \medskip In addition to the bounded$C^\infty(\R^2)$solutions$V_{\gamma_1,\ldots,\gamma_N,\epsilon}$and$\phi_{\gamma_1,\ldots,\gamma_N, \epsilon, \epsilon'}$of the KdV and mKdV equation this procedure also leads to an abundance of singular solutions such as (D.57) and$\phi_{j,\gamma_1,\ldots,\gamma_j,\pm}$in (D.67) (unless$\lambda_j0. \tag {E.16} Abelian differentials of the second kind (DSK) $\omega^{(2)}$ are characterized by their vanishing residues and the normalization $$\int\limits_{a_j}\omega^{(2)}=0,\ 1\leq j\leq g. \tag {E.17}$$ \begin{lem} Let $\omega^{(2)}_{P_1}$ be a DSK on $K_g,\ g\in \N$ whose only pole is $P_1\in \hat K_g$ with principal part $\zeta^{-n}_\alpha d\zeta_\alpha,\ n\geq 2$ and write $$\omega_j=(\sum\limits^\infty_{m=0}c_{j,m}\zeta^m_\alpha)d\zeta_\alpha =f_{\alpha, j}(\zeta_\alpha)d\zeta_\alpha. \tag {E.18}$$ Then $$\int\limits_{b_j}\omega^{(2)}_{P_1}=2\pi i(n-1)^{-1}c_{j,n-2} =\dfrac {2\pi i}{(n-1)!}\dfrac {d^{n-2}f_{\alpha,j}}{d\zeta^{n-2}_\alpha}(0), \ n\geq 2,\ 1\leq j\leq g. \tag {E.19}$$ \end{lem} A basis for DSKs at $P_\infty=(\infty,\infty)$ on $K_g,\ g\in \N_0$ holomorphic on $K_g\backslash\{P_\infty\}$ is provided by $$\omega^{(2)}_{2k}=R_0(z)^{-1/2}z^{g+k}dz=[\prod\limits^{2g}_{n=0} (1-E_n\zeta^2_\alpha)]^{-1/2}2i(-1)^g \zeta^{-2-2k}_\alpha d\zeta_\alpha,\ k\in \N_0. \tag {E.20}$$ In the following we denote by $\M(K_g)$ and $\M^1(K_g)$ the set of meromorphic functions (0-forms) and meromorphic 1-forms on $K_g,\ g\in \N_0$. Let $P_0\in K_g$ and $(U_\alpha, z_\alpha)$ be a chart near $P_0$ with $z_\alpha(P_0)=0$ and $f\in \M(K_g),\ \omega\in \M^1(K_g)$ of the type $$(f\circ z^{-1}_\alpha)(\zeta_\alpha)=\sum\limits^\infty_{n=m_0} a_n\zeta^n_\alpha,\qquad \omega=(\sum\limits^\infty_{n=m_1}b_n\zeta^n_\alpha)d\zeta_\alpha \tag {E.21}$$ near $P_0$ for some $m_0, m_1\in \Z$ (which turn out to be chart independent). Then the order $\nu_f(P_0)$ of $f$ at $P_0$ and the order of $\nu_\omega(P_0)$ of $\omega$ at $P_0$ are defined by $$\nu_f(P_0)=m_0,\ \nu_\omega(P_0)=m_1. \tag {E.22}$$ One defines $\nu_f(P)=\infty, P\in K_g$ iff $f\equiv 0$ on $K_g$. \begin{defn} %e.2 Let $g\in \N_0$.\hfill\break (i) A divisor $\D$ on $K_g$ is a map $\D:\ K_g\rightarrow \Z$, where $\D (P)\neq 0$ for only finitely many $P\in K_g$. On the set of all divisors $\Div (K_g)$ on $K_g$ one introduces the partial ordering $$\D\geq \E \text { iff } \D(P)\geq \E(P),\ P\in K_g. \tag {E.23}$$ (ii) The degree $\deg (\D)$ of $\D\in \Div (K_g)$ is defined by $$\deg (\D)=\sum\limits_{P\in K_g}\D(P). \tag {E.24}$$ (iii) $\D\in \Div (K_g)$ is called positive (or effective) iff $$\D\geq \O, \tag {E.25}$$ where $\O$ is the zero divisor $$\O:\ \cases K_g\rightarrow \Z\\ P\phantom{_g}\,\rightarrow 0. \endcases \tag {E.26}$$ (iv) $\D,\E\in \Div (K_g)$, then $\D$ is called a multiple of $\E$ iff $$\D\geq \E \tag {E.27}$$ (i.e., iff $\D-\E\geq \O$ is positive).\hfill\break (v) If $f\in \M(K_g)\backslash \{0\},\ \omega\in \M^1(K_g)\backslash \{0\}$ then the divisor $(f)$ of $f$ and $(\omega)$ of $\omega$ are defined by $$(f):\cases K_g\rightarrow \Z\\ P\rightarrow \nu_f(P)\endcases,\qquad (\omega):\cases K_g\rightarrow \Z\\ P\rightarrow \nu_\omega(P). \endcases \tag {E.28}$$ (f) is called a principal divisor, $(\omega)$ a canonical divisor.\hfill\break (vi) $\D, \E\in \Div (K_g)$ are called equivalent, $\D\sim \E$ iff $$\D-\E=(f)\text { for some }f\in \M(K_g)\backslash \{0\}. \tag {E.29}$$ \end{defn} The divisor class [$\D$] of $\D$ is defined by $$[\D]=\{\E\in \Div (K_g)|\E\sim \D\}. \tag {E.30}$$ $\Div (K_g)$ forms an Abelian group with respect to addition of divisors and $\sim$ in (E.29) defines an equivalence relation. \begin{lem} %lem-e.3 Let $f\in \M(K_g)\backslash \{0\},\ \omega\in \M^1(K_g)\backslash \{0\},\ g\in \N_0$. Then \begin{align*} \deg ((f))&=0, \tag {E.31}\\ \deg ((\omega))&=2(g-1). \tag {E.32} \end{align*} \end{lem} \begin{defn} Let $g\in \N$ and define the period lattice $L_g$ in $\C^g$ by $$L_g=\{\underline z\in \C^g|\underline z=\underline N+(\tau \underline M),\ \underline N, \underline M\in \Z^g\}. \tag {E.33}$$ Then the Jacobi variety $J(K_g)$ of $K_g$ is defined by $$J(K_g)=\C^g/L_g \tag {E.34}$$ and the Abel map is defined by $$\underline A_{P_0}:\cases K_g\rightarrow J(K_g)\\ P\rightarrow \underline A_{P_0}(P)=\{A_{P_0,j}(P) =\int\limits^P_{P_0}\omega_j\}^g_{j=1}(\text{mod } L_g), \endcases \tag {E.35}$$ respectively by $$\underline \alpha_{P_0}:\ \cases \Div (K_g)\rightarrow J(K_g)\\ \D@>{\hskip .50truein}>> \underline \alpha_{P_0}(\D)=\sum\limits_{P\in K_g}\D(P)\underline A_{P_0}(P), \endcases \tag {E.36}$$ where $P_0\in K_g$ is a fixed base point and (for convenience only) the same path from $P_0$ to $P$ is chosen for all $1\leq j\leq g$ in (E.35). \end{defn} Then Abel's theorem and Jacobi's inversion theorem read \begin{thm} Let $g\in \N$. Then $\D\in \Div (K_g)$ is principal iff \seq{36} \begin{align} \begin{split} &(i).\ \deg (\D)=0. \\ &(ii).\ \underline \alpha_{P_0}(\D)=\underline 0. \lb{E.37} \end{split} \end{align} \end{thm} \begin{thm} \lb{te.6} Let $g\in \N$. Then $\underline \alpha_{P_0}$ is surjective. \end{thm} The Riemann theta function associated with $K_g$ is defined as $$\theta(\underline z)=\sum\limits_{\underline m\in \Z^g}\exp [2\pi i(\underline m,\underline z)+\pi i(\underline m, \tau\underline m)],\ \underline z\in \Z^g \tag {E.38}$$ (with $(.,.)$ the scalar product in $\C^g$). Some of its elementary properties are, e.g., $$\theta(z_1,\ldots,z_{j-1},\ -z_j,\ z_{j+1},\ldots, z_g)=\theta (\underline z),\ \underline z=(z_1,\ldots,z_g)\in \C^g, \tag {E.39}$$ \seq{39} \begin{align} \begin{split} \theta (\underline z+\underline e+\tau \underline e') & =\exp [-2\pi i(\underline e', \underline z)-\pi i (\underline e', \tau \underline e')]\theta (\underline z),\\ & \hspace*{.7in} \underline e, \underline e'\in \Z^g,\ \underline z\in \C^g. \lb{E.40} \end{split} \end{align} We recall \begin{lem} Let $\underline \xi \in \C^g,\ g\in \N$, and define $$F:\cases \hat K_g\rightarrow \C\\ P\rightarrow \theta (\underline {\hat A}_{P_0}(P)-\underline \xi), \endcases \tag {E.41}$$ where $$\underline {\hat A}_{P_0}:\cases \hat K_g\rightarrow \C^g\\ P\rightarrow \underline {\hat A}_{P_0}(P)=\{\hat A_{P_0,j}(P)=\int\limits^P_{P_0}\omega_j\}^g_{j=1}.\endcases \tag {E.42}$$ Suppose $F\not\equiv 0$ on $\hat K_g$. Then $F$ has precisely $g$ zeros on $\hat K_g$ counting multiplicity. \end{lem} \begin{thm} Let $\underline \xi\in \C^g,\ g\in \N$ and define $F$ as in (E.41). Suppose $F\not\equiv 0$ on $K_g$ and let $P_1,\ldots,\, P_g\in \hat K_g$ be the zeros of $F$ (counting multiplicity) given by Lemma E.7. Defining $$\D_{P_1+\ldots +P_g}:\cases K_g\rightarrow \N_0\\ P\rightarrow \D_{P_1+\ldots+P_g}(P)=\cases m & \text{ if } P \text{ occurs } m \text{--times in } \{P_1,\ldots, P_g\}\\ 0 & \text{ if } P\in K_g\backslash \{P_1,\ldots, P_g\}\endcases \endcases \tag {E.43}$$ there exists a vector $\underline \zeta_{P_0}\in \C^g$, Riemann's vector, such that $$\underline \alpha_{P_0}(\D_{P_1+\ldots+P_g})=[\underline \xi-\underline \zeta_{P_0}](\text {mod } L_g). \tag {E.44}$$ $\underline \zeta_{P_0}=(\zeta_{P_{0,1}},\ldots,\zeta_{P_0,g})$ is given by $$\zeta_{P_{0,j}}=2^{-1}(1+\tau_{j,j})-\sum\limits^g_{\Sb \ell=1\\ \ell\neq j\endSb}\int\limits_{a_\ell}\omega_\ell(P) \int\limits^P_{P_0}\omega_j,\ 1\leq j\leq g. \tag {E.45}$$ \end{thm} \begin{rem} While $\theta(\underline {.})$ is well-defined (in fact entire) on $\C^g$, it is not well-defined on $J(K_g)=\C^g/L_g$ because of (E.40). Nevertheless $\theta$ is a multiplicative function'' on $J(K_g)$ since the multipliers in (E.40) are exponentials (that cannot vanish). In particular, if $\underline z_1=\underline z_2(\text {mod } L_g)$ then $\theta(\underline z_1)=0$ iff $\theta (\underline z_2)=0$. Hence it is meaningful to state that $\theta$ vanishes at points of $J(K_g)$. Since the Abel map $\underline A_{P_0}$ maps $K_g$ into $J(K_g),\ \theta(\underline A_{P_0}(.)-\underline \xi),\ \underline \xi\in \C^g$ becomes a multiplicative function on $K_g$. Again it makes sense to speak of the vanishing of $\theta(\underline A_{P_0}(.)-\underline \xi)$ at points of $K_g$. \end{rem} In the following we shall use the standard notation $$X\pm Y=\{(\underline x\pm \underline y)\in J(K_g)|\,\underline x\in X,\ \underline y\in Y\} \tag {E.46}$$ for $X,Y\subseteq J(K_g)$ and identify $\sigma^nK_g$, the n-th symmetric power of $K_g$, with the set of positive divisors of degree $n,\, n\in \N$ on $K_g$. \begin{defn} Let $g\in \N$.\hfill\break (i) We define $$\underline W_0=\{\underline 0\}\subset J(K_g),\ \underline W_n=\underline \alpha_{P_0}(\sigma^nK_g),\ n\in \N. \tag {E.47}$$ (ii) A positive divisor $\D\in \Div (K_g)$ is called special iff $$i(\D)=\dim_{\C}\{\omega\in \M^1(K_g)|\ (\omega)\geq \D\}\geq 1, \tag {E.48}$$ otherwise $\D$ is called nonspecial.\hfill\break (iii) $\tilde P\in K_g$ is called a Weierstrass point of $K_g$ iff $i(\D_{g\tilde P})\geq 1$, where $$\D_{g\tilde P}:\cases K_g\rightarrow \N_0\\ P\rightarrow \D_{g\tilde P}(P)=\cases g,&P=\tilde P\\ 0,&P\in K_g\backslash \{\tilde P\}.\endcases \endcases \tag {E.49}$$ \end{defn} Since $i(\D_P)=0$ for all $P\in K_1,\ K_1$ has no Weierstrass points. For $g\geq 2$ the Weierstrass points of $K_g$ are precisely the branch points (E.6). Although $\sigma^mK_g\not\subset \sigma^nK_g$ for m0. \lb{E.54} \end{split} \end{align} (iii) If\theta (\underline \xi)=0$and$g\geq 2$, let$1\leq s\leq (g-1),\ s\in \N$be the smallest integer such that$\theta(\underline W_s-\underline W_s-\underline \xi)\neq 0$(i.e., there exist$\E, \F\in \sigma^sK_g,\ \E\neq F$with$\theta (\underline \alpha_{P_0}(\E)-\underline \alpha_{P_0}(\F)-\underline \xi)\neq 0$). Then there exists a$\D\in \sigma^{g-1}K_g$such that $$\underline \xi=[\underline \alpha_{P_0}(\D)+\underline \zeta_{P_0}](\text {mod } L_g) \tag {E.55}$$ and $$i(\D)=s. \tag {E.56}$$ \end{thm} All partial derivatives of$\theta$with respect to$\underline A_{P_0,j},\ 1\leq j\leq g$of order strictly less than$s$vanish at$\underline \xi$, whereas at least one partial derivative of$\theta$of order$s$is nonzero at$\underline \xi$. Moreover,$1\leq s\leq (g+1)/2$and the integer$s$is the same for$\underline \xi$and$-\underline \xi$. \begin{rem} Theorem E.12 yields a constructive approach to Jacobi's inversion problem which may be stated as follows:\ \ Given$\underline \xi \in \C^g$find a positive divisor$\D_{P_1+\ldots+P_g}\in \Div (K_g)$of the type (E.43) such that $$\underline \alpha_{P_0}(\D_{P_1+\ldots+P_g})=\underline \xi(\text {mod } L_g). \tag {E.57}$$ Indeed, given$\underline {\tilde \xi}=(\underline \xi+\underline \zeta_{P_0})\in \C^g$, the divisors$\D$in (E.52), (E.55) (resp.$\D=\D_{P_0}$if$g=1$) solve Jacobi's inversion problem for$\underline \xi\in \C^g$. In particular, this yields a proof of Theorem E.6. \end{rem} Finally we mention \begin{thm} Let$g\geq 2$and$P_1,\ldots, P_g\in K_g$. Then$\D_{P_1+\ldots+P_g}$is special iff there is at least one pair$(P,P^*)\subseteq \{P_1,\ldots,P_g\}$or (as a limiting case) a Weierstrass point occurs at least twice in$\{P_1,\ldots,P_g\}$. 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