Content-Type: multipart/mixed; boundary="-------------0001171755830" This is a multi-part message in MIME format. ---------------0001171755830 Content-Type: text/plain; name="00-28.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-28.keywords" center manifold, nonlinear Schroedinger equation ---------------0001171755830 Content-Type: application/x-tex; name="central.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="central.tex" %This is a Latex file. \documentstyle[12pt]{article} \setlength{\oddsidemargin}{-10mm} \setlength{\evensidemargin}{10mm} \setlength{\textwidth}{185mm} \setlength{\textheight}{244mm} \setlength{\topmargin}{-20mm} \newcommand{\sss}{\setcounter{equation}{0}} \newtheorem{theorem}{THEOREM}[section] \renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}.} \newtheorem{lemma}[theorem]{LEMMA} \renewcommand{\thelemma}{\arabic{section}.\arabic{lemma}.} \newtheorem{corollary}[theorem]{COROLLARY} \renewcommand{\thecorollary}{\arabic{section}.\arabic{lemma}.} \newtheorem{remark}[theorem]{REMARK} \renewcommand{\theremark}{\arabic{section}.\arabic{remark}.} \newtheorem{prop}[theorem]{PROPOSITION} \renewcommand{\theprop}{\arabic{section}.\arabic{prop}.} \renewcommand{\theequation}{\arabic{section}.\arabic{equation}} \newtheorem{definition}[theorem]{DEFINITION} \renewcommand{\thedefinition}{\arabic{section}.\arabic{definition}.} %%%%macros%%%%%% \def\ER{{\bf R}} \def\beq{\begin{equation}} \def\ene{\end{equation}} %\def\d{\dot{m}} \def\bull{\begin{flushright} \vrule height 6pt width 6pt depth -.pt \end{flushright}} \def \M{\hbox{BMO}} \def \a{W_{\pm}^{\ast}} \def\w{W_{\pm}} %%%%%%terminan macros%%%%%%%%%%%%%% \begin{document} \baselineskip=23.6pt \title{Center Manifold for Nonintegrable Nonlinear Schr\"{o}dinger Equations on the Line\thanks{{\sc ams} classification 35P, 35Q, 35R and 81U}} \author{Ricardo Weder\thanks{Fellow Sistema Nacional de Investigadores}\\ Instituto de Investigaciones en Matem\'aticas Aplicadas y en Sistemas,\\ Universidad Nacional Aut\'onoma de M\'exico, \\Apartado Postal 20-726, M\'exico D.F. 01000\\ E-Mail: weder@servidor.unam.mx\\} \date{} \maketitle \begin{center} \begin{minipage}{5.75in} \centerline{{\bf Abstract}}\bigskip In this paper we study the following nonlinear Schr\"{o}dinger equation on the line, $$ i\frac{\partial}{\partial t}u(t,x)= -\frac{ d^2}{ d x^2} u(t,x) + V(x) u(t,x) + f(x, |u|)\frac{u(t,x)}{|u(t,x)|}, u(0,x)=\phi (x), $$ where $f$ is real-valued, and it satisfies suitable conditions on regularity, on grow as a function of $u $ and on decay as $ x \rightarrow \pm \infty$. The {\it generic} potential, $ V$, is real-valued and it is chosen so that the spectrum of $H:= -\frac{d^2}{ d x^2} +V$ consists of one simple negative eigenvalue and absolutely-continuous spectrum filling $[0, \infty)$. The solutions to this equation have, in general, a localized and a dispersive component. The nonlinear bound states, that bifurcate from the zero solution at the energy of the eigenvalue of $H$, define an invariant center manifold that consists of the orbits of time-periodic localized solutions . We prove that all small solutions approach a particular periodic orbit in the center manifold as $t \rightarrow \pm \infty$. In general, the periodic orbits are different for $ t \rightarrow \pm \infty$. Our result implies also that the nonlinear bound states are asymptotically stable, in the sense that each solution with initial data near a nonlinear bound state is asymptotic as $ t \rightarrow \pm \infty $ to the periodic orbits of nearby nonlinear bound states that are, in general, different for $ t \rightarrow \pm \infty$. \end{minipage} \end{center} \newpage \section{Introduction}\sss We study below the small solutions to the nonintegrable, nonlinear Schr\"{o}dinger equation, \beq i\frac{\partial}{\partial t}u(t,x)= -\frac{ d^2}{ d x^2} u(t, x) + V(x) u(t,x) + f(x, |u|)\frac{u(t,x)}{|u(t,x)|}, u(0,x)= \phi(x), \label{1.1} \ene where $u$ is a complex-valued function defined for $t, x \in \ER$. For each fixed $x \in \ER$, $f(x, \cdot ) \in C^1( \ER, \ER), \linebreak \frac{\partial}{\partial x} f(x, \cdot) \in C( \ER, \ER), \, f(x,0)=0 $ and, $$ \left|\frac{\partial}{\partial u}f(x, u)\right| \leq C |u|^{p-1}, $$ \beq \left|\frac{\partial}{\partial x}f(x, u)\right| \leq C |u|^{p}, \hbox{for some} \, p >2. \label{1.2} \ene The potential, $V$, is a real-valued function. For any $\gamma \in \ER$ we denote by $L^1_{\gamma}$ the Banach space of all complex-valued measurable functions, $\phi $, defined on $\ER$ and such that \beq \left\|\phi \right\|_{L^1_{\gamma}}:= \int |\phi (x)| \, (1+|x|)^{\gamma} \, dx < \infty . \label{1.3} \ene If $V \in L^1_1$,\, $-\frac{ d^2}{ d x^2}+ V$ has a unique self-adjoint realization in $L^2$, that we denote by $H$ . Moreover, $H$ has no singular-continuous spectrum, and its absolutely-continuous spectrum is $[0, \infty )$. $H$ has no positive or zero eigenvalues and, in general, it has a finite number of negative eigenvalues that are simple. For these results see \cite{1} and \cite{11}. We assume that $H$ has only one negative eigenvalue, that we denote by $E_0$. We also suppose that, \beq N(V):= \sup_{x \in \ER} \int_x^{x+1} |V(y)|^2\, dy < \infty . \label{1.4} \ene It follows from Theorem 2.7.1 in page 35 of \cite{6} that $D(H)= H_2$, where, $H_n, n=1,2, \cdots $, denote the Sobolev spaces \cite{8}. To explain our results let us consider first the associated linear Schr\"{o}dinger equation with $f \equiv 0$, \beq i\frac{\partial}{\partial t}u(t,x)= -\frac{ d^2}{ d x^2} u(x,t) + V(x) u(t,x), u(0,x)= \phi (x). \label{1.4b} \ene Let $\Psi_0$ be the eigenvector associated to the eigenvalue $E_0$, with $L^2$ norm equal to one. Equation (\ref{1.4b}) has an invariant center manifold given by, \beq {\cal M}_0 := \left\{ r e^{i \theta}\Psi_0: r \geq 0, 0 \leq \theta < 2 \pi \right\}. \label{1.5} \ene The invariant manifold, ${ \cal M}_0$, consists of the orbits of periodic localized solutions to (\ref{1.4b}) of the form $e^{-it E_0} r e^{i \theta}\Psi_0: r \geq 0, 0 \leq \theta < 2 \pi$. Every solution to (\ref{1.4b}), $u= e^{-itH} \phi$, with initial data in $\phi \in L^2$, can be decomposed as follows, \beq e^{-itH} \phi = e^{-itE_0} P_0 \phi + e^{-itH} P_c \phi, \label{1.5b} \ene where $P_0$ denotes the orthogonal projector in $L^2$ onto the one-dimensional subspace generated by $\Psi_0$, $P_0 \phi := (\phi , \Psi_0 ) \Psi_0$. Moreover, $P_c:= I-P_0$ is the projector onto the subspace of continuity of $H$, ${\cal H}_c:= P_c L^2$. The component $ e^{-itH} P_c \phi$ is a scattering state that propagates out to infinity as $ t \rightarrow \pm\infty$. In fact, it is an easy consequence of the RAGE theorem \cite{10b} that, \beq \lim_{ t \rightarrow \pm \infty} \left\| e^{-itH} P_c \phi \right\|_{L^2_{-s}} =0, \,\, s > 0, \label{1.6} \ene where $L^2_s, s \in \ER,$ denotes the weighted $L^2$ space consisting of all functions, $f$, that are locally in $L^2$ and such that $(1+ |x|^2)^{s/2} f(x) \in L^2$, with norm, \beq \left\| f\right\|_{L^2_s} := \left\| (1+ |x|^2)^{s/2} f(x) \right\|_{L^2}. \label{1.7} \ene Equation (\ref{1.6}) is an expression of the fact that, as the scattering state propagates to infinity the local energies tend to zero. Note that (\ref{1.7}) does not holds with $s=0$ because $e^{-itH}$ is unitary in $L^2$. By (\ref{1.5b}) and (\ref{1.6}) every solution, $u$, to (\ref{1.4b}) approaches a periodic orbit in the invariant center manifold in the sense that for any $ s > 0$, \beq \lim_{ t \rightarrow \pm \infty} \left\| u(t)- e^{-it E_0} P_0 \phi \right\|_{L^2_{-s}} =0. \label{1.7b} \ene For functions $u(t,x)$ defined for $t, x \in \ER$, we denote $u(t)$ for $u(t, \cdot )$. Equation (\ref{1.7b}) gives us also a time-dependent characterization of the stability of the bound state $\Psi_0$. It tells us that given any initial state in $L^2$, $\Psi= r e^{i\theta} \Psi_0 + \phi $ with $\phi \in {\cal H}_c$, the solution to (\ref{1.4b}) is the sum of the periodic orbit of the bound state, $e^{-itE_0}r e^{i\theta} \Psi_0$, and a dispersive solution, $ e^{-itH} \phi$, that propagates out to infinity as $t \rightarrow \pm \infty$ and whose local energies tend to zero. As we show below, this situation persists in the nonlinear case. There is an invariant center manifold, consisting of the orbits of periodic localized solutions, such that all small solutions to (\ref{1.1}) approach particular orbits in the center manifold as $t \rightarrow \pm \infty$. Invariant manifold theorems have been extensively used in the analysis of the time evolution of dissipative equations. See for example, \cite{carr} and \cite{hen}. Equation (\ref{1.1}) is however, dispersive. A periodic localized solution to (\ref{1.1}) is a solution of the type $u(t,x):= e^{-it E} \Psi_E$, where the nonlinear bound state, $\Psi_E$, is a solution to the following nonlinear eigenvalue problem, \beq H \Psi_E+ f(x, |\Psi_E| ) \frac{\Psi_E}{|\Psi_E|} = E \Psi_E, \Psi_E \in H_2. \label{1.8} \ene It is a consequence of standard bifurcation theory (see Theorem 3.2.2 in page 77 of \cite{nir}) that the set of nontrivial solutions to (\ref{1.8}) near the trivial solution $\Psi_{E_0}=0 $ consists of exactly one continuous curve such that $ |E- E_0| < \epsilon$ for some $ \epsilon > 0$, and, \beq \lim_{ E \rightarrow 0} \left\| \Psi_E \right\|_{H_2}=0. \label{1.9} \ene In other words, the nonlinear bound states bifurcate from the zero solutions at $ E=E_0$. We prove in Appendix 1 ( see (\ref{1.21}) below )that for some constant $C$ \beq \left| \Psi_0 (x)\right| \leq C e^{-\sqrt{|E_0|} |x|}; \,\, | \Psi_E(x)| \leq C e^{-\sqrt{|E|} |x|}, |E-E_0| < \epsilon. \label{1.9b} \ene If follows from (\ref{1.9b}) that for any $s > 0$ there is a constant $C_s$ such that, \beq \left\| \Psi_E\right\|_{L^2_s} \leq C_s, |E-E_0| < \epsilon. \label{1.9c} \ene The invariant center manifold for the nonlinear Schr\"{o}dinger equation (\ref{1.1}) is given by \beq {\cal M}:=\left\{ e^{i \theta } \Psi_E: |E-E_0| < \epsilon, 0 \leq \theta < 2 \pi \right\}. \label{1.10} \ene ${\cal M}$ consists of the orbits of periodic solutions to (\ref{1.1}) of the form $ e^{-it E} e^{i \theta} \Psi_E$. On the spirit of the central manifold theorem, \cite{carr}, \cite{hen}, \cite{pw}, let us write ${\cal M}$ as the graph of a function from $P_0 L^2$ into its orthogonal complement ,${\cal H}_c $. The proof given in \cite{pw} in the case of three or more dimensions applies also in one dimension, and it follows that there is a $ \delta > 0$ and a function, $h$, from $\{ z \in {\bf C}: |z| < \delta\}$ into ${\cal H}_c \cap H_2 \cap L^2_s, s > 0 $ , such that, \beq {\cal M}= \left\{\Psi= z \Psi_0 + h( z): |z| < \delta \right\}. \label{1.11} \ene Moreover, $h$ is a $C^1$ in the real sense, i.e., as a function from ${\bf C}$, as a two-dimensional real space, into ${\cal H}_c \cap H_2 \cap L^2_s$. Furthermore, $h(0)=0$, and $h(e^{i \theta}z)= e^{i \theta} h(z)$. See Proposition 2.2 of \cite{pw} and its proof. Let us denote by $F(x, u):=\int_0^{u} f(x, v) \,dv$. It follows from (\ref{1.2}) and since $f(x,0)= 0$, that \beq F(x, |u|) \leq C |u|^{p+1}, \label{1.14} \ene for some constant $C$. It is a consequence of standard standard results ( see \cite{23} and \cite{28}) that there is a $\rho > 0$ such that the initial value problem (\ref{1.1}) has a unique solution in $ C(\ER, H_1)$ for every $\phi \in H_1$ such that $\| \phi \| _{H_1} < \rho$. If moreover, \beq F(x, |u|) \geq - C (|u|^2+ |u|^{q+1} ), \hbox{ for some}\, 1 < q < 5, \label{1.15} \ene then the initial value problem (\ref{1.1}) has a unique solution in $ C(\ER, H_1)$ for every $\phi \in H_1$. In both cases the $L^2$ norm and the energy are conserved quantities, \beq \| u(t) \|_{L^2} = \| \phi \|_{L^2}, \label{1.16} \ene $$ \int dx \left[ \frac{1}{2} \left(\left| \frac{\partial}{\partial x}{u}(t,x)\right|^2 + V(x) |u(t,x)|^2\right) + F(x,|u|) \right] = $$ \beq \int dx \left[ \frac{1}{2} \left(\left| \frac{d}{d x}{\phi}(x)\right|^2 + V(x) |\phi(x)|^2\right) + F(x,|\phi |) \right] . \label{1.17} \ene Moreover, for any $\epsilon > 0$ there is a $\nu > 0$ such that \beq \| u(t)\|_{H_1} < \epsilon, t \in \ER, \hbox{if} \|\phi \|_{H_1} < \nu. \label{1.18} \ene Before we state our main result we need to introduce some more standard notations. For any pair $u,v$ of solutions to the stationary Schr\"{o}dinger equation: \beq -\frac{d^2}{d x^2} u+V u = k^2 u,\, k \in \hbox{C}, \label{1.19} \ene let $[u,v]$ denotes the Wronskian of $u,v$: \beq [u,v]:= \left(\frac{d}{d x}u\right) v- u \frac{d}{d x}v. \label{1.20} \ene Let $f_j(x,k), j=1,2, \Im k \geq 0$, be the Jost solutions to (\ref{1.19}) ( see \cite{12}, \cite{13}, \cite{1} and \cite{4}). A potential $V$ is said to be {\it generic} if $[f_1(x,0),f_2(x,0)]\neq 0$ and $V$ is said to be {\it exceptional} if $[f_1(x,0),f_2(x,0)]=0$. If $V$ is {\it exceptional} there is a bounded solution to (\ref{1.19}) with $k^2=0$ (a half-bound state or a zero-energy resonance). Note that the trivial potential, $V=0$, is {\it exceptional}. We prove in Appendix 2 that the {\it generic} potentials are a dense open set in $L^1_1$. Our main result is the following theorem. \begin {theorem} Suppose that $V \in L^1_{2}$, that $N(V) < \infty$, that $V$ is a {\rm generic} potential and that $H:= -\frac{d^2}{d x^2} + V(x)$ has only one negative eigenvalue. We assume that for each fixed $x \in \ER$, $f(x, \cdot ) \in C^1( \ER, \ER), \frac{\partial}{\partial x} f(x, \cdot) \in C( \ER, \ER), \, f(x,0)=0 $ and, for some $ p > 2$, \beq \left|\frac{\partial}{\partial u}f(x, u)\right| \leq q(x) |u|^{p-1}, \label{1.21} \ene where $ (1+|x|)^{2s+4 \beta} q(x) \in L^{\infty}$, for some $ s > 1 $, \, and \,, $ 1/2 < \beta \leq 1$. Moreover, \beq \left|\frac{\partial}{\partial x}f(x, u)\right| \leq C |u|^{p}. \label{1.21b} \ene Then there is a $\eta > 0$, such that for all $ \phi \in H_1 \cap L^2_{s+2 \beta}$ with $\|\phi\|_{H_1} < \eta$, there exist functions, $E(t)$ and $\theta (t)$, in $C^1( \ER, \ER)$, such that for some constant $C$, \beq \left\| u(t) - e^{-i \int_0^t E(\rho ) d \rho} e^{i \theta (t)} \Psi_{E(t)} \right\|_{L^2_{-s-2 \beta}} \leq C (1+ |t|)^{-1/2-\beta} \| P_c \phi- h((\phi, \Psi_0))\|_{L^2_{1+2\beta}}, \label{1.22} \ene where $u(t)$ is the solution to (\ref{1.1}) with initial value $\phi$. Moreover, the following limits exists, \beq \lim_{t \rightarrow \pm \infty} E(t) := E_{\pm}; \,\, \lim_{t \rightarrow \pm \infty} \theta (t) := \theta_{\pm}. \label{1.23} \ene \end{theorem} Equation (\ref{1.22}) tells us that $u$ tends to the periodic orbit of $ e^{i\theta_{\pm}} \Psi_{E_{\pm}}$. In particular, solutions with initial data near a nonlinear bound state are asymptotic as $ t \rightarrow \pm \infty $ to the periodic orbits of nearby nonlinear bound states. Note that the dispersive part, $ u(t)- e^{-i \int_0^t E(\rho ) d \rho } e^{i \theta (t)} \Psi_{E(t)}$, tends to zero in $L^2_{-s-2 \beta}$, as $ t \rightarrow \pm \infty$, with the same rate that as the dispersive solutions to the associated linear Schr\"{o}dinger equation (\ref{1.4b}) ( see Theorem 1.2 below). A result as in Theorem 1.1 was proven in three or more dimensions in \cite{sw1}, \cite{sw2} and \cite{pw}. For results on the asymptotic stability of stationary (time \linebreak independent ) solutions to nonlinear evolution equations see \cite{kw1}, \cite{ks1} , \cite{kw2} and the references mentioned in these papers. For results on the asymptotic stability of solitons see \cite{bp1} and \cite{bp2}. For studies of the orbital stability of solitons see \cite{mw1} and \cite{mw2}. The proof of Theorem 1.1 is based in ideas from center manifold theory, \cite{carr}, \cite{hen} and \cite{pw}. The basic dynamical imput of the proof is the following $L^2_{s+2 \beta}-L^2_{-s-2 \beta}$ estimate. For any pair of Banach spaces, $X,Y$, ${\cal B}\left( X , Y \right)$ denotes the Banach space of all bounded operators from $X$ into $Y$. In the Theorem below we consider the general case where $H$ has a finite number of eigenvalues. \begin{theorem} Suppose that $V \in L^1_{2}$, and that $V$ is {\rm generic}. Then, for any $s > 1$ and $ 0 \leq \beta \leq 1$, there is a constant $C$ such that, \beq \left\| e^{-itH} P_c \right\|_{{\cal B}\left( L^2_{s+2 \beta} ,L^2_{-s-2 \beta}\right)} \, \leq C (1+|t| )^{-1/2- \beta}. \label{1.24} \ene \end{theorem} A decay estimate as (\ref{1.24}) was proven in Theorem 7.6 of \cite{mu} for potentials such that $ V$ is a compact operator from $H_l$ into $L^2_{\rho}$ for some $ l < 1$ and some $\rho > 4$. Murata's condition rougly means that $V$ decays as $|x|^{-\rho}$ as $|x| \rightarrow \infty , \, \rho > 4$, whereas our condition, $ V \in L^1_{2}$, requires $ \rho > 3$. We give below a rather simple proof of Theorem 1.2, quite different from the one in \cite{mu}, based on our results in \cite{4}. The key issue of estimate (\ref{1.24}) is that it is integrable for large times if $\beta > 1/2$ . For {\it exceptional} potentials the decay rate is $(1+|t)^{-1/2}$. We can prove this result with our methods, but we do not consider this issue here. See \cite{mu} on this point. The paper is organized as follows. We prove Theorem 1.1 in Section 2. Theorem 1.2 is proven in Section 3. In Appendix 1 we prove estimate (\ref{1.9b}) and in Appendix 2 we prove that the {\it generic}\, potentials are a dense open set in $L^1_1$. \section{ The Center Manifold}\sss In this Section we prove Theorem 1.1. Let us project the solution to (\ref{1.1}) along $P_0$ and $P_c$, i.e., $u(t)= u_p(t) \Psi_0 + u_c(t)$, where $u_c(t):= P_c u(t)$. Then, (\ref{1.1}) is equivalent to the following system, \beq i \frac{d}{ dt} u_p=E_0 u_p + g_p(u_p,u_c) ; \,\, i \frac{\partial}{\partial t} u_c= H u_c + g_c (u_p,u_c), \label{1.25} \ene where, denoting \, $ g(x,u):= f(x,|u|) \frac{u}{|u|}$, we have that, \beq g_p(u_p,u_c):= P_0 g(x, u_p \Psi_0+ u_c)= ( g(x, u_p \Psi_0+u_c) , \Psi_0 ) \Psi_0; \,\, g_c(u_p,u_c):= P_c g(x, u_p \Psi_0+u_c) . \label{1.26} \ene In a similar way, any point on the center manifold is written as, $ e^{i\theta} \Psi_E= u_p \Psi_0 + h(u_p)$, $ h(u_p) \in {\cal H}_c$, where $u_p, h(u_p)$ are the solution to the following system ( see \cite{pw}), \beq E_0-E =- \frac{g_p(u_p, h(u_p))}{u_p}; \,\, h(u_p) = - (H-E)^{-1} g_c( u_p, h(u_p)). \label{1.27} \ene We first prove that $u(t)$ approaches the center manifold. For this purpose, let us consider the vector in ${ \cal M}$ that has the same projection along $P_0$ that $u(t)$, i.e., $ \Psi(t):= u_p(t) \Psi_0 + h(u_p(t))$. We prove below that the difference, $v(t):= u(t)- \Psi(t) = u_c(t)- h(u_p(t))$ satisfies the estimate, \beq \left\| v(t) \right\|_{L^2_{-s_1}} \leq C (1+|t|)^{-1/2- \beta} \| v(0) \|_{L^2_{s_1}}, \label{1.28} \ene where, $s_1:= s +2 \beta$. By (\ref{1.25}), $v(t)$ is a solution of the following equation, \beq i \frac{ \partial}{\partial t} v(t)= H v(t) + Q(u_p(t), v(t)), \label{1.29} \ene where, \beq Q(u_p,u_c):= g_c(u_p,h(u_p)+v)- g_c(u_p,h(u_p))- (Dh)(u_p) [ g_p (u_p,h(u_p)+v)- g_p(u_p, h(u_p))], \label{1.30} \ene with $(Dh)$ the Frech\'et derivative of $h$. To verify (\ref{1.30}) we must prove that, \beq (Dh)(u_p) \left[ E_0 u_p + g_p(u_p, h(u_p))\right]= H h(u_p) + g_c(u_p, h(u_p)). \label{1.31} \ene We prove below this equation at $t=t_0$, for any $t_0 \in \ER$. We denote, $E:=E(u_p(t_0))$. Note that by (\ref{1.27}), $ [ e^{-it E} u_p(t_0), h(e^{-it E} u_p(t_0))]$ is a solution to (\ref{1.25}) ( recall that $h(e^{-i tE}u_p)= e^{-i tE} h( u_p)$). Then, using the equation for $u_p$ in (\ref{1.25}), \beq i \frac{\partial}{\partial t}h( e^{-it E} u_p(t_0))= (Dh)( e^{-it E} u_p(t_0)) \left[ E_0\, e^{-it E} u_p(t_0) + e^{-it E} g_p(u_p(t_0), h(u_p(t_0))) \right]. \label{1.32} \ene Moreover, by the equation for $u_c$ in (\ref{1.25}), \beq i \frac{\partial}{\partial t} h(e^{-it E}u_p(t_0))= H h(e^{-it E} u_p(t_0)) + e^{-it E} g_c( u_p(t_0), h(u_p(t_0))). \label{1.33} \ene Equation (\ref{1.31}) follows taking $t=0$ in (\ref{1.32}) and (\ref{1.33}). By (\ref{1.21}), $ |g(x,u+v)-g(x,u)| \leq \linebreak C q(x) (|u|^{(p-1)}+|v|^{(p-1)}) |v|$, and we have that, $$ \left\| g(x, u_p \Psi_0 + h(u_p) )- g(x, u_p \Psi_0 +h(u_p)+v) \right\|_{L^2_{s_1}} \leq $$ \beq C \|(1+|x|)^{2s_1} q(x)\|_{L^{\infty}} \left( \| ( u_p \Psi_0 + h(u_p) \|_{H_1} ^{(p-1)}+ \|v\|^{(p-1)}_{H_{1}}\right) \|v\|_{L^2_{-s_1}}, \label{1.34} \ene where we used Sobolev's \cite{8} theorem to bound the $L^{\infty}$ norms by the $H_1$ norms. By (\ref{1.9b}) $P_0$ and $P_c= I-P_0$ are bounded operators on $L^2_s, s \in \ER$, and it follows from (\ref{1.34}) that \beq \left\| g_p( u_p, h(u_p) )- g_p( u_p ,h(u_p)+v) \right\|_{L^2_{s_1}} \leq C \left(\| u_p \Psi_0 + h(u_p) \|_{H_1} ^{(p-1)}+ \|v\|^{(p-1)}_{H_{1}}\right) \|v\|_{L^2_{-s_1}}, \label{1.35} \ene \beq \left\| g_c( u_p, h(u_p) )- g_c( u_p ,h(u_p)+v) \right\|_{L^2_{s_1}} \leq C \left(\| u_p \Psi_0 + h(u_p) \|_{H_1} ^{(p-1)}+ \|v\|^{(p-1)}_{H_{1}}\right) \|v\|_{L^2_{-s_1}}. \label{1.36} \ene By (\ref{1.18}) given any $ \epsilon_1 > 0$ we can take $\eta $ so small that if $ \|\phi\|_{H_1} < \eta $, we have that, $|u_p(t)| = |( u(t), \Psi_0)| \leq \| u(t)\|_{H_1} < \epsilon_1$. Moreover, since $h$ is $C^1$ and $h(0)=0$, \beq \| h(u_p (t) ) \|_{H_1} \leq C |u_p| \leq C \epsilon_1, \label{1.37} \ene and we conclude that, \beq \| v(t)\|_{H_1} \leq C \epsilon_1. \label{1.38} \ene By (\ref{1.30}), (\ref{1.34}), (\ref{1.35}) and (\ref{1.36}), \beq \| Q(u_p (t), v(t)) \|_{L^2_{s_1}} \leq C \epsilon_1 \|v(t)\|_{L^2_{-s_1}}, \, \hbox{ if} \, \|\phi\|_{H_1} < \eta. \label{1.39} \ene We write (\ref{1.29}) as an integral equation, \beq v(t)= e^{-itH} v(0) +\frac{1}{i} \,\int_0^t ds \, e^{-i(t-s) H} Q(u_p(s), v(s)) . \label{1.40} \ene Let us denote $ v_T:= \max_{ |t| \leq T} (1+|t|)^{1/2+\beta} \|v(t)\|_{L^2_{-s_1}}$. By Theorem 1.2, and (\ref{1.39}), for $|t| \leq T$, $$ \|v(t)\|_{L^2_{-1/2-\beta}} \leq C (1+|t|)^{-1/2-\beta} \| v(0) \|_{L^2_{s_1}} + C \epsilon_1 \, \hbox{(sign t)}\, \int^{t}_0 ds \, (1+|t-s|)^{-1/2-\beta} (1+|s|)^{-1/2-\beta} v_T $$ \beq \leq C (1+|t|)^{-1/2-\beta} [ \| v(0) \|_{L^2_{s_1}} + C \epsilon_1 v_T] . \label{1.41} \ene Taking $\eta$ so small that $ C \epsilon_1 < 1/2$, we obtain that, \beq v_T \leq C \| v(0) \|_{L^2_{s_1}}, \label{1.42} \ene and since the constant $C$ is independent of $T$ equation (\ref{1.28}) follows. Equation (\ref{1.22}) follows from (\ref{1.28}) by the argument given in Section 4 of \cite{pw}. \section{ The $ L^2_{s+2\beta}-L^2_{-s-2\beta}$ Estimate} \sss The results on the spectral theorem for $H$ that we state below follow from the Weyl--Kodaira-Titchmarsch theory. See for example \cite{1}. For a version of the Weyl--Kodaira--Titchmarsch theory adapted to our situation see Appendix 1 of \cite{wi} and also the proof of Theorem 6.1 in page 78 of \cite{wi}. Let us denote for any $k \in \ER$ \beq \Psi_+ (x,k):= \cases{ \frac{1}{\sqrt{2 \pi }} T(k) f_1 (x,k),& $ k \geq 0$,\cr\cr \frac{1}{\sqrt{2\pi}} T(-k) f_2 (x,-k),& $k < 0$,} \label{2.1} \ene and $\Psi_- (x,k) := \overline{\Psi_+ (x,-k)}$. Recall that ${\cal H}_{c}$ is the subspace of continuity of $H$. Then the following limits, \beq \hat{\phi}_{\pm}(k):= s-\lim_{N \rightarrow \infty}\int_{-N}^{N} \overline{\Psi_{\pm}(x,k)} \,\phi (x) \, dx, \label{2.2} \ene exist in the strong topology in $L^2$ for every $\phi \in {\cal H}_{c} $ and the operators \beq \left(F_{\pm} \phi \right)(k):= \hat{\phi}_{\pm}(k), \label{2.3} \ene are unitary from ${\cal H}_{c}$ onto $L^2$. Moreover, the $F_{\pm}^{\ast}$ are given by \beq \left(F_{\pm}^{\ast} \phi \right)(x)= s-\lim_{N\rightarrow \infty} \int_{-N}^{N} \Psi_{\pm}(x,k) \,\phi (k) \, dk, \label{2.4} \ene where the limits exist in the strong topology in $L^2$. Furthermore, the operators $ F_{\pm }^{\ast}\,F_{\pm}$ are the orthogonal projector onto ${\cal H}_{c}$. For each eigenvalue of $H$, let $\Psi_j , 0=1,2,\cdots ,N$, be the corresponding eigenfunction normalized to one, i.e. $\|\Psi_j \|_{L^2}=1$. The operators: \beq F_j \phi := (\phi ,\Psi_j ) \Psi_j, j=0,2,\cdots , N, \label{2.5} \ene are unitary from the eigenspace generated by $\Psi_j$ onto $C$. The following operators \beq F^{\pm}= F_{\pm} \oplus^N_{j=0} F_j , \label{2.6} \ene are unitary from $L^2$ onto $L^2 \oplus^N_{j=0} C$ and for any $\phi \in D(H)$: \beq F^{\pm} H \phi= \left\{ k^2 (F_{\pm} \phi )(k) , E_0 F_0 \phi , \cdots , E_N F_N \phi \right\}. \label{2.7} \ene Moreover, for any bounded Borel function , $ \Phi$, defined on $\ER$ \beq F^{\pm} \Phi (H) \phi = \left\{ \Phi (k^2) (F_{\pm}\phi )(k), \Phi(E_0 )F_0 \phi ,\cdots , \Phi (E_N ) F_N \phi \right\}. \label{2.8} \ene The projector, $P_0$, onto the subspace of $L^2$ generated by the eigenvectors of $H$ is given by \beq P_0 \phi:= \sum_{j=0}^N (\phi ,\Psi_j ) \Psi_j . \label{2.9} \ene Since $H$ has no singular--continuous spectrum the projector onto the continuous subspace of $H$ is given by: $P_c := I-P_0$. It follows from (\ref{2.6}) that \beq e^{-it H}P_c = F_{\pm}^{ \ast} e^{-ik^2 t} F_{\pm}. \label{2.10} \ene Equation (\ref{2.10}) is the starting point of our proof of Theorem 1.2. \noindent{\it Proof of Theorem 1.2:}\,\,It follows from (\ref{2.10}) that for any $\phi \in L^2 \cap L^1$, \beq e^{-itH}P_c \phi (x)= \int \Phi_t(x,y) \phi(y) dy, \label{2.11} \ene where \beq \Phi_t(x,y):= \int e^{-ik^2t} \Psi_+ (x,k) \overline{\Psi_+ (y,k)} \, dk. \label{2.12} \ene The proofs of Lemmas 2.4 and 2.5 of \cite{4} imply that \beq \left| \Phi_t(x,y) \right| \leq C \,\frac{1}{ \sqrt{|t|}}. \label{2.13} \ene We prove below that, \beq (1+|x|)^{-2} \, \left| \Phi_t(x,y) \right( 1+|y|)^{-2} \, \leq C \,\frac{1}{ |t|^{3/2} }. \label{2.14} \ene By (\ref{2.13}) and (\ref{2.14}), for any $ s > 1$ and $ 0 \leq \beta \leq 1$, \beq (1+|x|)^{-s-2\beta} \, \left| \Phi_t(x,y) \right| (1+|y|)^{-s-2\beta} \, \leq C \,\frac{1}{ |t|^{1/2+\beta} }. \label{2.15} \ene Equation ({\ref{1.24}) follows from (\ref{2.15}), from Schur's criterion and from the unitarity in $L^2$ of $e^{-itH} P_c$. We now prove (\ref{2.14}). Let $ \chi_1 \in C^{\infty}_0(\ER)$ satisfy , $ \chi_1 (k^2)=1$ for $|k| \leq 1$ and let us denote $\chi_2:= 1-\chi_1$. Changing the variable of integration in (\ref{2.12}) to $\lambda:= k^2$ we obtain that, \beq \Phi_t = \Phi_{1,t} + \Phi_{2,t}, \label{2.16} \ene where \beq \Phi_{j,t}(x,y):= \int_0^{\infty}\frac{1}{\sqrt{\lambda}} \, \chi_j(\lambda) \, e^{-i\lambda t}\, \Re\left( \Psi_+ (x,\sqrt{\lambda}) \overline{ \Psi_+ (y, \sqrt{\lambda}})\right) d \lambda, \, j=1,2, \label{2.17} \ene where we used that $\Psi_+(x,-k)= \overline{ \Psi_+(x,k)}$. Let us denote, \beq h_j(\lambda, x,y):=\frac{1 }{\sqrt{\lambda}} \, \chi_j(\lambda) \, \, \Re \left( \Psi_+ (x,\sqrt{\lambda}) \overline{ \Psi_+ (y, \sqrt{\lambda}})\right), \, j=1,2. \label{2.18} \ene We first estimate $\Phi_{1,t}(x,y)$. The key issue is that in the {\it generic} case the transmission coefficient satisfies $T(k) = \alpha k + o(k)$, as $k \rightarrow 0$, where $ \alpha \neq 0$, and the reflection coefficients satisfy $R_j(0)=-1, j=1,2$ \linebreak ( see \cite{1}). Then, it follows from \cite{4}: equation (2.5), Lemma 2.1, equations (2.38) and (2.40-2.42), that \beq |h_1(\lambda , x,y)| \leq C \frac{\sqrt{\lambda}}{1+\sqrt{\lambda}} ;\,\, \left| \frac{\partial} {\partial \lambda} h_1(\lambda ,x,y)\right| \leq C \frac{1}{\sqrt{\lambda}} (1+|x|) (1+|y|). \label{2.19} \ene Let us extend $h_1(\lambda, x,y)$ to a function defined for $\lambda \in \ER$ by setting $h_1(\lambda, x,y):=0$ for $\lambda \leq 0$. Then, $$ \Phi_{1,t}(x,y) =\frac{1}{2}\left[ \int e^{-it \lambda} h_1(\lambda, x,y) d \lambda - \int e^{-it( \lambda- \pi /t)} h_1(\lambda, x,y) d \lambda \right] = $$ \beq \frac{1}{2} \int e^{-it \lambda}[ h_1(\lambda, x,y) -h_1(\lambda + \pi/t, x,y)] d \lambda. \label{2.20} \ene Hence, \beq |\Phi_{1,t}(x,y) | \leq C \int | h_1(\lambda, x,y) -h_1(\lambda + \pi/t, x,y)| \, d \lambda. \label{2.21} \ene If $t > 0$, $$ \int | h_1(\lambda, x,y) -h_1(\lambda + \pi/t, x,y)| \, d \lambda \leq 2 \int_0^{2\pi/t} |h_1(\lambda, x,y)| d \lambda + \int_{\pi/t}^{\infty} d \lambda \int_{\lambda}^{\lambda+ \pi/t} \left| \frac{\partial} {\partial \rho} h_1(\rho ,x,y)\right| \, d \rho $$ \beq \leq \frac{C}{ t^{3/2}} (1+|x|) (1+|y|) + \frac{\pi}{t} \int_{\pi/t}^{\infty} \left| \frac{\partial} {\partial \rho} h_1(\rho ,x,y) \right| \,d \rho \leq \frac{C}{t^{3/2}} (1+|x|)\,(1+|y|), \label{2.22} \ene where we used (\ref{2.19}). If $t < 0$ we change the variable of integration in (\ref{2.21}) to $\acute{\lambda}:= \lambda+\pi /t$, and we proceed as above. Let us denote, $m_1(x,k):= e^{-ikx} f_1(x,k)$ and $ m_2(x,k):= e^{ikx} f_2(x,k)$. We designate, $\ddot{m}_j (x,k):=\frac{\partial^2}{\partial k^2}{m}_j (x,k), j=1,2$, and $\acute{m}_j := \frac{\partial}{\partial x} m_j (x,k), j=1,2$ . We prove below that if $V \in L^1_2$, then, for any $x_0 \in \ER$, \beq |\ddot{m}_1 (x,k)| \leq C_{x_0}\, \frac{1+|k|}{|k|^2 } , x \geq x_0;\,\,\, | \ddot{m}_2 (x,k)| \leq C_{x_0} \, \frac{1+|k|}{|k|^2} , x \leq x_0 , \label{2.23} \ene \beq \left|\ddot{{\acute{m}}}_1 (x,k)\right|\leq C_{x_0} \left[ 1+ \frac{1+|k|} {|k|^2}\right], x \geq x_0 ; \,\,\, \left|\ddot{{\acute{m}}}_2 (x,k)\right| \leq C_{x_0} \, \left[1+\frac{1+|k|}{|k|^2}\right] ,\, x \leq x_0 . \label{2.24} \ene Integrating by parts twice with respect to $\lambda$ in (\ref{2.17}) with $j=2$ and using (\ref{2.23}), (\ref{2.24}) and \cite{4}: equation (2.5), Lemma 2.1, equations (2.38) and (2.40-2.42), we obtain that, \beq |\Phi_{2,t}(x,y)| \leq C \frac{1}{t^2} (1+|x|)^2 \, (1+|y|)^2. \label{2.25} \ene Equation (\ref{2.14}) follows from (\ref{2.16}), (\ref{2.21}), (\ref{2.22}) and (\ref{2.25}). We prove now (\ref{2.23}) and (\ref{2.24}) for $m_1$. The case of $m_2$ follows in a similar way. As is well known \linebreak ( see \cite{1}, \cite{4}), \beq m_1 (x,k)=\lim_{n \rightarrow \infty} m_{1,n}(x,k), \label{2.26} \ene where $m_{1,0}(x,k):=1$ and the $m_{1,n}$ satisfy the following equation for $n=0,1,\cdots$ \beq m_{1,n+1}(x,k)=1+\int_x^{\infty}D_k (y-x) V(y) m_{1,n} (y,k) dy, \label{2.27} \ene where, \beq D_k (x):=\int_0^x e^{2iky}dy = \cases{ \frac{1}{2ik} (e^{2ikx}-1),& $k\neq 0,$ \cr\cr x,& $k=0.$} \label{2.28} \ene Clearly, \beq \left| \ddot{D}_k(x) \right| \leq C \frac{(1+|x|)^2\, (1+|k|)}{|k|^2}. \label{2.29} \ene By (\ref{2.27}), \beq \ddot{m}_{1,n+1}(x,k)= \int_x^{\infty} D_k (y-x) V(y) \ddot{m}_{1,n} (y,k) \, dy + A_n( x,k), \label{2.30} \ene where, \beq A_n(x,k):= 2 \int_x^{\infty} \dot{D}_k (y-x) V(y) \dot{m}_{1,n} (y,k) \, dy + \int_x^{\infty} \ddot{D}_k (y-x) V(y) m_{1,n} (y,k) \, dy. \label{2.31} \ene Then by (\ref{2.29}) and \cite{4}: equations (2.10), (2.18) and (2.21) with $ \gamma =2$, for $|k| \leq 1$ and $ \gamma =1$ for $|k| \geq 1$, \beq \left| A_n(x,k)\right| \leq \,C_{x_0} \, \frac{1+|k|}{|k|^2} \, \| V\|_{L^1_2}, \, x \geq x _0, \label{2.32} \ene and it follows from (\ref{2.29}) and (\ref{2.30}) that, \beq |\ddot{m}_{1,n+1}(x,k)| \leq C_{x_0} \, \frac{1+|k|}{|k|^2}+ C_{x_0} \frac{1+|k|}{|k|^2} \int_x^{\infty} |y-x|^2 |V(y)| |\ddot{m}_{1,n}(y,k)| \, dy. \label{2.33} \ene Iterating (\ref{2.33}) $n+1$ times we prove that \beq |\ddot{m}_{1,n+1}(x,k)| \leq C_{x_0} \, \frac{1+|k|}{|k|^2}, \, x \geq x_0, \label{2.34} \ene and (\ref{2.23}) for $m_1$ follows taking the limit as $ n \rightarrow \infty$. Equation (\ref{2.24}) folows from (\ref{2.23}) and from \cite{4}: Lemma 2.1 and equation (2.23). \section{Appendix 1}\sss In this Appendix we prove equation (\ref{1.9b}). For this purpose it is enough to assume that $V \in L^1$, and that $N(V) < \infty$. Note that as $\Psi_E \in H_2$ if follows from Sobolev's theorem \cite{8} that $\| \Psi_E\|_{L^{\infty}} \leq C \| \Psi_E\|_{H_2}$. Then $ q_e:= V + f(x,|\Psi_E|) / |\Psi_E| \in L^1$ and $N(q_e) < \infty$. By Theorem 2.7.1 in page 35 of \cite{6} the differential expression $ -\frac{d^2}{d x^2} +q_e$ is essentially-self-adjoint on $C^{\infty}_0$. We denote by $H_e$ the unique self-adjoint realization in $L^2$. Moreover, $D(H_e)=H_2$. Let $g_j\left(x,i \sqrt{|E|}\right), j=1,2$, be the Jost solutions for the potential $q_e$ at energy $E$. They satisfy ( see Lemma 1 of \cite{1}), \beq \left| g_1\left(x, i \sqrt{|E|}\right) \right| \leq C e^{-\sqrt{|E|} |x|}, x \geq 0 ;\,\, \left| g_2\left(x, i \sqrt{|E|}\right) \right| \leq C e^{-\sqrt{|E|} |x|}, x \leq 0. \label{3.1} \ene But the differential expression $ -\frac{d^2}{d x^2} +q_e$ is in the limit-point case at $\pm \infty$ \cite{11}. Hence, $g_j(x,i \sqrt{|E|}), j=1,2$, are, respectively, the only independent solutions to the eigenvalue equation \beq -\frac{d^2}{d x^2} g+q_e g = E g, \label{3.2} \ene that are square integrable on a neighborhood of $ \pm \infty$. By (\ref{1.8}) $ \Psi_E $ is a solution in $H_2$ to the linear eigenvalue equation (\ref{3.2}). Then, $g_1$ and $g_2$ are linearly dependent, and \beq \Psi_E(x) = \alpha g_1\left(x, i \sqrt{|E|}\right) = \beta g_2\left(x,i \sqrt{|E|}\right), \label{3.3} \ene for some constants $ \alpha ,\beta$. Equation (\ref{1.9b}) follows from (\ref{3.1}) and (\ref{3.3}). Note that the argument above also implies that, \beq | \Psi_0(x) | \leq C e^{- \sqrt{|E_0|} |x|}. \label{3.4} \ene \section{Appendix 2} \sss In this Appendix we prove that the {\it generic} potentials are a dense open set in $L^1_1$. For each fixed $x \in \ER$ the functions $ V \in L^1_1 \hookrightarrow f_j(x,k), j=1,2$, are continuous \cite{1}. It follows that the set of {\it generic} potentials is an open set in $L^1_1$. Suppose that $V \in L^1_1 $ and denote, \beq W(\lambda ):= \left[ f_{1,\lambda} (x,0), f_{2, \lambda}(x,0)\right], \label{4.1} \ene where $f_{j, \lambda}, j=1,2$, are the Jost solutions for $\lambda V$ given by \cite{4} equations (2.11-2.13)) for $j=1$ and similar formulas for $j=2$. $W(\lambda)$ is an entire analytic function of $\lambda$ and there are two posibilities. (a) $W(\lambda)$ is not identically zero. Then the set of zeros of $W(\lambda)$ is discrete and there exists a sequence $\lambda_n$, with $W(\lambda_n) \neq 0$ and $\lim_{n \rightarrow \infty} \lambda_n =1$. It follows that $ \lambda_n V$ is {\it generic} and that $ \lim_{n \rightarrow \infty} \lambda_n V= V$ strongly in $L^1_1$. (b) $W(\lambda)$ is identically zero . In this case it follows from \cite{4}: equations (2.11-2.13) and (2.23) that \beq -\int V(y)\, dy= \frac{d}{d \lambda} W(\lambda)\bigg|_{\lambda =0}=0. \label{4.2} \ene Take any $q(x) \in L^1_1$ with $q(x) > 0$ and any sequence, $\epsilon_n > 0$, with $ \lim_{n \rightarrow \infty} \epsilon_n=0$. As $V+\epsilon_n q$ does not satisfies (\ref{4.2}) there are sequences $\lambda_{n,m}$ with $\lim_{ m \rightarrow \infty} \lambda_{n,m}=1$ such that $ V_{n,m}:= \lambda_{n,m} ( V +\epsilon_n q)$ are {\it generic}. We can always take a subsequence of the $V_{n,m}$ that converges strongly to $V$ in $L^1_1$. It follows that the set of {\it generic} potentials is dense in $L^1_1$. \begin{thebibliography}{99} \bibitem{8} R.A. Adams, ``Sobolev Spaces'', Academic Press, New York, 1975. \bibitem{bp1} V.S. Buslaev and G.S. Perelīman, Scattering states for the nonlinear Schr\"{o}dinger equation: states close to a soliton, Algebra i Analiz {\bf 4} (1992), 63-102 [ english translation in St. Petersburg Math. J. {\bf 4} (1993), 1111-1142]. \bibitem{bp2} V. S. Buslaev and G.S. Perelīman, On the stability of solitary waves for nonlinear Schro\"{o}dinger equation, Amer. Math. Soc. Transl. (2) {\bf 164} (1995), 75-98. \bibitem{carr} J. Carr, " Applications of Centre Manifold Theory", Applied Mathematical Sciences {\bf 35}, Springer-Verlag, New York, 1981. \bibitem {1}P. Deift and E. Trubowitz, Inverse scattering on the line, Commun. Pure Appl. Math. {\bf XXXII} (1979), 121-251. \bibitem{12} L.D. Faddeev, Properties of the S matrix of the one--dimensional Schr\"{o}dinger equation, Trudy Math. Inst. Steklov {\bf 73} (1964), 314-333 [ english translation American Mathematical Society Translation Series 2 {\bf 65} (1964), 139-166 ]. \bibitem{13} L.D. Faddeev, Inverse problems of quantum scattering theory, II, Itogi Nauki i Tekhniki Sovremennye Problemy Matematiki {\bf 3} (1974), 93-180 [ english translation J. Soviet Math. {\bf 5} (1976),\linebreak 334-396 ]. \bibitem{28} J. Ginibre, ``Introduction aux \'Equations de Schr\"{o}dinger non Lin\'eaires'', Onze \'Edition, Paris, 1998. \bibitem{hen} D. Henry, "Geometric Theory of Semilinear Parabolic Equations", Lecture Notes in Mathematics, {\bf 840}, Springer-Verlag, Berlin, 1981. \bibitem{23} T. Kato, Nonlinear Schr\"{o}dinger equations, {\it in} ``Schr\"{o}dinger Operators'', pp. 218--263, H. Holden and A. Jensen, eds., Lecture Notes in Physics {\bf 345}, Springer--Verlag, Berlin, 1989. \bibitem{kw1} A. Komech and B. Vainberg, On asymptotic stability of stationary solutions to nonlinear wave and Klein-Gordon equations, Arch. Rational Mechanics and Analysis, {\bf 134} (1996), 227-248. \bibitem{kw2} A. Komech, On transitions to stationary states in Hamiltonia nonlinear wave equations, Physics Letters A {\bf 241} ( 1998), 311-322. \bibitem{ks1} A. Komech, H. Spohn and M. Kunze, Long-time asymptotics for a classical particle interacting with a scalar field, Comm. Part. Diff. Equations, {\bf 22} (1997),307-335. \bibitem{mu} M. Murata, Asymptotic expansions in time for solutions of Schr\"{o}dinger-type equations, J. Funct. Analysis, {\bf 49} (1982), 10-56. \bibitem{nir} L. Nirenberg, "Topics in Nonlinear Functional Analysis", Courant Institute of Mathematical Sciences Lecture Notes, New York University, New York, 1974. \bibitem{pw} C.-A. Pillet and C.E. Wayne, Invariant manifolds for a class of dispersive, hamiltonian, partial differential equations, J. Differential Equations, {\bf 141} (1997), 310-326. \bibitem{10c}M. Reed and B. Simon, ``Methods of Modern Mathematical Physics I: Functional Analysis'', Academic Press, New York, 1972 \bibitem{10}M. Reed and B. Simon, ``Methods of Modern Mathematical Physics II: Fourier Analysis, Self--Adjointness'', Academic Press, New York, 1975. \bibitem{10b}M. Reed and B. Simon, ``Methods of Modern Mathematical Physics III: Scattering Theory'', Academic Press, New York, 1978. \bibitem{6} M. Schechter, ``Operator Methods in Quantum Mechanics'', North Holland, New York, 1981. \bibitem{sw1} A. Soffer and M.I. Weinstein, Multichannel nonlinear scattering for nonintegrable equations, Commun. Math. Phys. {\bf 133} (1990), 119-146. \bibitem {sw2} A. Soffer and M.I. Weinstein, Multichannel nonlinear scattering for nonintegrable equations II. The case of anisotropic potentials and data, J. Differential Equations, {\bf 98} (1992), 376-390. \bibitem{31}R. Weder, Inverse scattering for the nonlinear Schr\"{o}dinger equation, Commun. Part. Diff. Equations {\bf 22} (1997), 2089-2103. \bibitem{4} R. Weder, $L^p-L^{\acute{p}}$ estimates for the Schr\"{o}dinger equation on the line and inverse scattering for the nonlinear Schr\"{o}dinger equation with a potential, Preprint, 1998, 28 pages, to appear in J. Funct. Analysis. \bibitem{40}R. Weder, The $W_{k,p}$-continuity of the Schr\"{o}dinger wave operators on the line, Preprint, 1999, 16 pages, to appear in Comm. Math. Phys. \bibitem{11}J. Weidmann, ``Spectral Theory of Ordinary Differential Operators'', Lecture Notes in Mathematics {\bf 1258}, Springer--Verlag, Berlin, 1987. \bibitem{mw1} M.I. Weinstein, Modulation stability of ground states of nonlinear Schr\"{o}dinger equations , SIAM J. Math. Anal. {\bf 16} (1985), 472-491. \bibitem{mw2} M.I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Commun. Pure Appl. Math. {\bf 39} (1986), 51-67. \bibitem{wi} C. H. Wilcox, ``Sound Propagation in Stratified Fluids'', Applied Mathematical Sciences {\bf 50}, Springer--Verlag, New York, 1984. \end{thebibliography} \end{document} ---------------0001171755830--