\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 217, pp. 1--16.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2014 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2014/217\hfil Stability of traveling-wave solutions] {Stability of traveling-wave solutions for a Schr\"odinger system with power-type nonlinearities} \author[N. V. Nguyen, R. Tian, Z.-Q. Wang \hfil EJDE-2014/217\hfilneg] {Nghiem V. Nguyen, Rushun Tian, Zhi-Qiang Wang} % in alphabetical order \address{Nghiem V. Nguyen \newline Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA} \email{nghiem.nguyen@usu.edu} \address{Rushun Tian \newline Academy of Mathematics and System Science, Chinese Academy of Sciences, Beijing 100190, China} \email{rushun.tian@amss.ac.cn} \address{Zhi-Qiang Wang \newline Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA} \email{zhiqiang.wang@usu.edu} \thanks{Submitted November 22, 2013. Published October 16, 2014.} \subjclass[2000]{35A15, 35B35, 35Q35, 35Q55} \keywords{Solitary wave solutions; stability; nonlinear Schr\"odinger system; \hfill\break\indent traveling-wave solutions} \begin{abstract} In this article, we consider the Schr\"odinger system with power-type nonlinearities, \[ i\frac{\partial}{\partial t}u_j+ \Delta u_j + a|u_j|^{2p-2} u_j + \sum_{k=1, k\neq j}^m b |u_k|^{p}|u_j|^{p-2} u_j=0; \quad x\in \mathbb{R}^N, \] where $j=1,\dots ,m$, $u_j$ are complex-valued functions of $(x,t)\in \mathbb{R}^{N+1}$, $a,b$ are real numbers. It is shown that when $b>0$, and $a+(m-1)b>0$, for a certain range of $p$, traveling-wave solutions of this system exist, and are orbitally stable. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction} It is well-understood that the nonlinear Schr\"odinger (NLS) equation \begin{equation} iu_t+ \Delta u \pm |u|^2 u=0 \label{NLS} \end{equation} where $u$ is a complex-valued function of $(x,t) \in \mathbb{R}^{N+1}$, arises in a generic situation. The equation describes evolution of small amplitude, slowly varying wave packets in a nonlinear media \cite{BN}. Indeed, it has been derived in such diverse fields as deep water waves \cite{Z1}, plasma physics \cite{Z2}, nonlinear optical fibers \cite{HT1,HT2}, magneto-static spin waves \cite{ZP}, to name a few. The $m$-coupled nonlinear Schr\"odinger (CNLS) system \begin{equation} i\frac{\partial}{\partial t}u_j+ \Delta u_j + a_{j}|u_j|^{2p-2} u_j + \sum_{k=1, k\neq j}^m b_{kj} |u_k|^{p}|u_j|^{p-2} u_j=0; \quad x\in \mathbb{R}^N, \label{CNLS} \end{equation} for $j=1,\dots ,m$, where $u_j$ are complex-valued functions of $(x,t)\in \mathbb{R}^{N+1}$, $a_j$ and $b_{jk}=b_{kj}$ are real numbers, arise physically under conditions similar to those described by \eqref{NLS}. The CNLS system also models physical systems in which the field has more than one components; for example, in optical fibers and waveguides, the propagating electric field has two components that are transverse to the direction of propagation. When $m=2$, the CNLS system also arises in the Hartree-Fock theory for a double condensate; i.e., a binary mixture of Bose-Einstein condensates in two different hyperfine states. Readers are referred to the works \cite{BN,HT1,HT2,Z1,Z2} for the derivation as well as applications of this system. The system admits the conserved quantities \begin{gather*} \begin{aligned} &E(u_1,u_2, \dots, u_m)\\ &= \int_{\mathbb{R}^N} \Big[\sum_{j=1}^{m}\big(|\nabla u_j(x)|^2 - \frac{a_{j}}{p}|u_j(x)|^{2p}\big) -\sum_{j,k=1;j\not=k}^m \frac{b_{jk}}{p}|u_k(x)|^p|u_j(x)|^p\Big]dx, \end{aligned}\\ Q(u_j)= \int_{\mathbb{R}^N} |u_j(x,t)|^2dx, \end{gather*} for $j=1,2,\dots,m$. We are interested in traveling-wave solutions for \eqref{CNLS} of the form $u(x,t) = (u_1,u_2, \dots, u_m)$, where for $j=1,2,\dots, m$, \begin{equation} u_j(x,t) = e^{i[(\omega_j -\frac{1}{4}|\theta|^2) t +\frac{1}{2}\theta x +m_j]} \varphi_{j,\omega_j}(x-\theta t) \label{0} \end{equation} for $m_j, \omega_j$ real constants, $\theta \in \mathbb{R}^N$ with $\omega_j-\frac{1}{4}|\theta|^2>0$ and $\varphi_{j,\omega_j}: \mathbb{R}^N \to \mathbb{R}$ are functions of one variable whose values are small when $|\xi|=|x-\theta t|$ is large. An important special case arises when $m_j=0$, $\theta =\vec{0}$ and $\omega_j=\Omega_j>0$. These special solutions (where, to emphasize the dependence on the parameters, we write $\varphi_{j,\omega_j}$ as $\phi_{j, \Omega_j}$) \begin{equation} u_j(x,t) = e^{i\Omega_j t} \phi_{j,\Omega_j}(x) \label{00} \end{equation} are often referred to as \textit{standing waves}. It is easy to see that, for example, standing waves are solutions of \eqref{CNLS} if and only if $(u_1, u_2,\dots, u_m)$ is a critical point for the functional $E(u_1,u_2,\dots, u_m)$, when the functions $u_j(x)$ are varied subject to the $m$ constraints that $Q(u_j)$ be held constant. If $(u_1,u_2,\dots,u_m)$ is not only a critical point but in fact a global minimizer then the standing wave is called a \textit{ground-state solution}. In some cases, namely when $p=2$, $N=1$ and certain conditions on $a_j, b_{jk}$ (see, for example, \cite{NW,NW2,NVN}), it is possible to show further that the ground-state solutions are solitary waves with the usual $\operatorname{sech}$-profile. One question unique to such type of nonlinear systems as \eqref{CNLS} is to study the existence and stability of {\it nontrivial solutions} $(u_1, \dots , u_m)$; that is, all components of the solutions are non-zero, and they may be refereed to as co-existing solutions or vector solutions. For the system \eqref{CNLS}, there are many semi-trivial solutions (or collapsing solutions) which are solutions with at least one component being zero. In those cases, the system collapses into system of lower orders. For example, our result in \cite{NW} shows that for the 2-coupled system, (that is, when $m=p=2$ and $N=1$) there are obstructions to the existence and stability of nontrivial solutions with all components being positive. Roughly speaking, our result says that in order to have positive non-trivial solutions, the nonlinear couplings have to be either small or large. Thus this is a situation where multiple solutions exist and classifying and distinguishing the solutions becomes an important and difficult issue. Intensive work has been done in the last few years, see \cite{amco1, BW, BWW, dww, lw1, lw2, liuw, MMP, pom, si, ww1, ww2}. All these works have been mainly on $2$-systems or with small couplings. Despite the partial progress made so far, many difficult questions remain open and little is known for $m-$systems for $m\geq 3$. \section{Statement of results} In this work, we concentrate on the case when $a_{j}=a$ and $b_{kj}=b_{jk}=b$. However, we also discuss how the method can be extended to include the general case. In particular, we will employ the techniques used in \cite{NW,NW2,O} to show the existence and stability of ground state solutions to the system \begin{equation} \begin{gathered} i\frac{\partial}{\partial t}u_j+ \Delta u_j + a|u_j|^{2p-2} u_j + \sum_{k=1, k\neq j}^m b |u_k|^{p}|u_j|^{p-2} u_j=0; \quad x\in \mathbb{R}^N, \\ \big(u_1(\cdot,0),\dots,u_m(\cdot,0)\big) = (u_{10},\dots,u_{m0}) \end{gathered} \label{NNLS} \end{equation} for $j=1,\dots ,m$ and for a certain range of $p$. Logically, prior to a discussion of stability in terms of perturbations of the initial data should be a theory for the initial-valued problem itself. This issue has been studied in a previous work of ours (\cite{NTDS}). In that work, the contraction mapping technique based on Strichartz estimates was used to first establish local well-posedness in $H^1_{\mathbb C}(\mathbb{R}^N)\times \dots \times H^1_{\mathbb C}(\mathbb{R}^N): \equiv X^{(m)}$ for $2\leq p0$, let \[ \phi_{\Omega,a+(m-1)b}(x) =\Big(\frac{\Omega}{a+(m-1)b}\Big)^{\frac{1}{2(p-1)}}\phi(\sqrt \Omega x). \] In Section 3, we establish the stability result for ground-state solutions of \eqref{NNLS}. \begin{proposition} \label{thm1} %\label{prop2.1} For $2\leq p<3$ and $N=1$, let $a$,$b \in \mathbb{R}$ such that $b>0$ and $a+(m-1)b>0$. Then for any $\Omega>0$, the ground-state solutions \[ \big(e^{i\Omega t} \phi_{\Omega,a+(m-1)b}(x), \dots, e^{i\Omega t}\phi_{\Omega,a+(m-1)b}(x)\big) \] of \eqref{NNLS} are orbitally stable in the following sense: for any $\epsilon>0$, there exists $\delta>0$ such that if $(u_{10},\dots, u_{m0})\in X^{(m)}$ with $$ \inf_{\gamma_i,y \in \mathbb{R}}\big\{ \sum_{j=1}^m \big\|u_{j0} - e^{i\gamma_j}\phi_{\Omega,a+(m-1)b}(\cdot+y) \big\|_{H^1} \big\} <\delta. $$ The solution $\big(u_1(x,t),\dots,u_m(x,t)\big)$ with $\big(u_1(\cdot,0),\dots,u_m(\cdot,0)\big) =(u_{10},\dots,u_{m0})$ satisfies $$ \inf_{\theta_i,y \in \mathbb{R}} \big\{\sum_{j=1}^m \big\|u_j - e^{i\theta_j}\phi_{\Omega,a+(m-1)b}(\cdot+y)\big\|_{H^1} \big\} <\epsilon $$ uniformly for all $t\geq 0$. \end{proposition} \begin{remark} \label{rmk1} \rm (1) As mentioned above, when $10$ and $a+(m-1)b>0$. Then for any $\Omega>0$, the ground-state solutions \[ \big(e^{i\Omega t} \phi_{\Omega,a+(m-1)b}(x), \dots, e^{i\Omega t}\phi_{\Omega,a+(m-1)b}(x)\big) \] of \eqref{NNLS} are orbitally stable in the sense that for any $\epsilon>0$, there exists $\delta>0$ such that if $(u_{10}, \dots, u_{m0})\in X^{(m)}$ satisfies \begin{equation}\label{initialdata} \inf_{\theta_j, \eta\in\mathbb{R}}\Big\{\sum_{j=1}^m\|u_{j0}-e^{i\theta_j} \phi_{\Omega,a+(m-1)b}(\cdot+\eta)\|_{H^1}. \Big\}<\delta \end{equation} There exist $C^1$ mappings $\theta_j, \eta: \mathbb{R}\to\mathbb{R}$ for which the solution $(u_1(x, t), \dots, u_m(x, t))$ with initial data $(u_1(\cdot, 0), \dots, u_m(\cdot, 0))= (u_{10}, \dots, u_{m0})$ satisfies \begin{equation}\label{continueextention} \sum_{j=1}^m\|u_j(\cdot, t) - e^{i\theta_j(t)}\phi_{\Omega,a+(m-1)b}(\cdot+\eta(t))\|_{H^1}<\epsilon \end{equation} for all $t\geq0$. Moreover, \begin{equation}\label{thm:estimates} \eta'(t)=O(\epsilon),\quad \theta_j'(t)=\Omega+O(\epsilon), \end{equation} for $j=1, \dots, m$ as $\epsilon\to0$, uniformly in $t$. \end{theorem} This result is then extended in Section 4 to include traveling-wave solutions. For $\theta\in\mathbb{R}$, define the operator $T_\theta: H_\mathbb{C}^1(\mathbb{R})\to H_\mathbb{C}^1(\mathbb{R})$ by \[ (T_\theta u)(x)=\exp\Big(\frac{i\theta x}{2}\Big)u(x). \] For any pair $(\omega,\theta)\in\mathbb{R}\times\mathbb{R}$ such that $\Omega=\omega-\frac{1}{4}\theta^2>0$, let $\varphi_\omega=T_\theta\phi_\Omega$. It is straightforward to see that if $(e^{i\Omega t}\phi_{\Omega},\dots,e^{i \Omega t}\phi_{\Omega})$ is a ground-state solution of \eqref{NNLS}, then $(e^{i\omega t}\varphi_{\omega},\dots,e^{i\omega t}\varphi_{\omega})$ is a traveling-wave solution of \eqref{NNLS}. \begin{corollary} \label{thm1B} For $2\leq p<3$ and $N=1$, let $a$ and $b \in \mathbb{R}$ such that $b>0$ and $a+(m-1)b>0$. The traveling-wave solutions $(e^{i\omega t}\varphi_\omega, \dots, e^{i\omega t}\varphi_\omega)$ are orbitally stable in the sense that for any $\epsilon>0$ given, there exists $\delta=\delta(\varphi)>0$ such that if \[ \inf_{\vec{\gamma},y}\Big\{\sum_{j=1}^m\|u_{0j}-e^{i\gamma_j}\phi(\cdot+y) \|_{H^1}\Big\}<\delta \] then there are $C^1$ mappings $p_j, q: \mathbb{R}\to\mathbb{R}$ for which the solution $\vec{u}=(u_1,\dots,u_m)$ with initial data $\vec{u}_0=(u_{01},\dots,u_{0m})$ satisfies, for all $j=1,2,\dots,m$ \[ \sum_{j=1}^m\|u_j(\cdot,t)-e^{ip_j(t)}\varphi_\omega(\cdot+q(t)) \|_{H^1}\leq\epsilon \] for all $t\geq0$. Moreover, $p_j$ and $q$ are close to $\omega$ and $\theta$ in the sense that \[ p_j'(t)=\omega+O(\epsilon)\hspace{1cm} q'(t)=\theta+O(\epsilon) \]as $\epsilon\to0$, uniformly in $t$. \end{corollary} \begin{remark}\label{rmk2} \rm It follows immediately from Remark 1 that Theorem \ref{thm1A} and Corollary \ref{thm1B} hold in $Y^{(m)}$ for the case $10$ and $a+(m-1)b>0$. \begin{remark} \label{rmk3} \rm As mentioned previously, to establish stability results as well as to extend the local existence to a global one, all that is needed is $p<1+2/N$. This condition when coupled with $2\leq p< N/(N-2)$ for local existence implies that $N=1$. However, there are instances when $p<2$ is permissible (see, for example Remark 1). Thus, to allow for the adaptability of the proofs obtained when $2\leq p<3$ to those instances, we refrain from taking $N=1$ directly, with the understanding that when $2\leq p <3$ then $N=1$. \end{remark} For $u\in H^1_{\mathbb{C}}(\mathbb{R}^N)$, define \begin{equation} E^{(m)}_1(u) = \int_{\mathbb{R}^N}\big( |\nabla u(x)|^2 - \frac{a+(m-1)b}{p} |u(x)|^{2p}\big)dx. \label{E1m} \end{equation} It is clear that for any $\Omega>0$, \[ \phi_{\Omega,a+(m-1)b}(x)=\Big(\frac{\Omega}{a+(m-1)b}\Big) ^{\frac{1}{2(p-1)}}\phi(\sqrt \Omega x) \] is the unique positive, spherically symmetric and decreasing solution in $H^1_{\mathbb C}(\mathbb{R}^N)$ of \[ -\Delta f + \Omega f = \big(a+(m-1)b\big) |f|^{2p-2} f, \] and $$ \|\phi_{\Omega, a+(m-1)b}\|_{L^2}= \big(a+(m-1)b\big)^{-\frac{1}{2(p-1)}} \Omega^{\frac{1}{2(p-1)}-\frac{N}{4}}\|\phi\|_{L^2}. $$ Fix an $\Omega>0$ and let \begin{equation} \lambda= \big(a+(m-1)b\big)^{-\frac{1}{(p-1)}} \Omega^{\frac{1}{(p-1)}-\frac{N}{2}} \|\phi\|^2_{L^2}. \label{lambda(m3)} \end{equation} For fixed $\Omega>0$ (hence $\lambda>0$ is also fixed) and any $\mu_1,\dots, \mu_{m-1} >0$, define the real numbers $I^{(m)}, I^{(m)}_1$ as follows: \begin{align*} &I^{(m)}(\lambda, \mu_1,\dots, \mu_{m-1})\\ & = \inf \Big\{E^{(m)}: u_1,\dots, u_m \in H^1_{\mathbb C}(\mathbb{R}^N), \|u_1\|_{L^2}^2 =\lambda, \|u_j\|^2_{L^2}=\mu_{j-1}, \ j=2,\dots,m \Big\}, \end{align*} and \[ I^{(m)}_1(\lambda) = \inf\{E^{(m)}_1(u): u\in H^1_{\mathbb C}(\mathbb{R}^N), \|u\|_{L^2}^2 =\lambda\}. \] The sets of minimizers for $I^{(m)}(\lambda,\mu_1,\dots,\mu_{j-1})$ and $I^{(m)}_1(\lambda)$ for $j=2,\dots,m$ are, respectively, \begin{gather*} \begin{aligned} &G^{(m)}(\lambda,\mu_1,\dots,\mu_{j-1}) \\ &= \Big\{(u_1,\dots, u_m) \in X^{(m)}: I^{(m)}(\lambda,\mu_1,\dots,\mu_{j-1})= E^{(m)}(u_1,\dots, u_m),\\ & \quad \|u\|^2_{L^2}=\lambda, \|u_j\|^2_{L^2}=\mu_{j-1}\Big\}, \end{aligned}\\ G^{(m)}_1(\lambda) = \big\{u \in H^1_{\mathbb C}(\mathbb{R}^N): I^{(m)}_1(\lambda)= E^{(m)}_1(u), \|u\|^2_{L^2}=\lambda \big\}. \end{gather*} The following 2 Lemmas are clear. (For example, the proofs can be easily modified from those presented in \cite{NW}.) \begin{lemma} \label{lem5.1} For all $\lambda>0$, one has $-\infty 0$ and $\delta>0$ such that \begin{itemize} \item[(i)] $\sum_{j=1}^m \|u_{jn}\|_{H^1} \leq B$ for all $n$, and \item[(ii)] $\sum_{j=1}^m \|u_{jn}\|_{L^{2p}}^{2p}\geq \delta$ for all sufficiently large $n$. \end{itemize} \end{lemma} Let $\{(u_{1n},\dots,u_{mn})\}\in X^{(m)}$ be a minimizing sequence for $E^{(m)}$ and consider a sequence of nondecreasing functions $M_n: [0,\infty) \to [0,m\lambda]$ as follows $$ M_n(s)=\sup_{y\in \mathbb{R}^N} \int_{|x-y|0$, there exists $s<\infty$ such that \[ \int_{|x-y_n|0$. If $-\Delta f +\Omega f= a |f|^{2p-2} f$, then \begin{gather*} %(a) \int_{\mathbb{R}^N} (|\nabla f|^2 +\Omega |f|^2 )dx = a \int_{\mathbb{R}^N} |f|^{2p}dx, \\ %(b) (N-2) \int_{\mathbb{R}^N} |\nabla f|^2dx + N\Omega \int_{\mathbb{R}^N} |f|^2dx = \frac{Na}{p} \int_{\mathbb{R}^N} |f|^{2p}dx. \end{gather*} \end{lemma} Using the above identities, the next Lemma can be derived easily. \begin{lemma} \label{lem5.4} The following statements hold : \begin{itemize} \item[(1)] for any $\lambda,\mu_{j-1} \geq 0$ and $j=2,\dots,m$, \[ I^{(m)}(\lambda, \mu_1,\dots, \mu_{m-1}) \geq I^{(m)}_1(\lambda) +\sum_{j=2}^m I^{(m)}_1(\mu_{j-1}); \] \item[(2)] \[ I^{(m)}_1\big(\lambda\big)= E^{(m)}_1(\phi_{\Omega,a+(m-1)b}) =-\frac{N+2-Np}{N+2p-Np}\lambda \Big(\frac{\lambda \big(a+(m-1)b \big)^{\frac{1}{p-1}}}{\|\phi\|^2_{L^2}}\Big)^{\frac{2(p-1)}{2-N(p-1)}}; \] \item[(3)] $I^{(m)}(\lambda,\dots,\lambda) = mI^{(m)}_1(\lambda)$ for $\lambda >0$, and $(\phi_{\Omega,a+(m-1)b}, \dots, \phi_{\Omega,a+(m-1)b}) \in G^{(m)}(\lambda,\dots,\lambda)$; \item[(4)] $ I^{(m-k)}_1(\lambda)>I^{(m)}_1(\lambda)$, for all $k\in (0,m)$ and $\lambda>0$. \end{itemize} \end{lemma} \begin{corollary} \label{cor5.A} For any $\Omega>0$ fixed, $$ \Big \{ \Big(e^{i\alpha_1} \phi_{\Omega,a+(m-1)b}(\cdot+y), \dots, e^{i\alpha_m} \phi_{\Omega,a+(m-1)b}(\cdot+y)\Big)\Big \} \subset G^{(m)}\big(\lambda(\Omega),\dots,\lambda(\Omega)\big) $$ where $\alpha_j \in \mathbb{R}$, $j=1,2,\dots, m$; $y\in \mathbb{R}^N$. \end{corollary} The following Lemma provides \textit{strict sub-additivity} of the function $I^{(m)}$ needed to rule out the dichotomy of minimizing sequences. \begin{lemma} \label{lem5.6} For any $\beta_j \in [0,\lambda]$, $j=1, \dots, m$ satisfying $0<\sum_{j=1}^m \beta_i< m\lambda$, we have $$ I^{(m)}(\lambda, \dots,\lambda) < I^{(m)}(\beta_1, \dots,\beta_m) +I^{(m)} (\lambda-\beta_1, \lambda-\beta_2,\dots,\lambda -\beta_m). $$ \end{lemma} \begin{proof} We consider separately the following cases. \textit{Case 1:} $\beta_j\in (0,\lambda)$ for $j=1,\dots, m$. From (3) in Lemma \ref{lem5.4}, we have $$ I^{(m)}(\lambda, \dots, \lambda) =mI^{(m)}_1(\lambda); $$ and from \cite[Theorem II.1]{CL} we have, for any $\beta_j \in (0, \lambda)$ $$ I^{(m)}_1(\lambda)I^{(m)}_1(\beta_j), $$ we obtain that $I^{(m)}(\beta_1,\dots,\beta_{m-k},0,\dots,0) >\sum_{j=1}^{m-k}I^{(m)}_1(\beta_j)$. Thus, \begin{align*} &I^{(m)}(\lambda,\dots,\lambda)\\ &= mI^{(m)}_1(\lambda) \leq k I^{(m)}_1(\lambda) + \sum_{j=1}^{m-k}\Big(I^{(m)}_1(\beta_j)+I^{(m)}_1(\lambda-\beta_j) \Big)\\ &\leq I^{(m)}(\lambda-\beta_1,\dots, \lambda-\beta_{m-k},\lambda,\dots, \lambda) + \sum_{j=1}^{m-k} I^{(m)}_1(\beta_j)\\ &0$ given, there exists $\delta>0$ such that if $$ \inf_{(\Phi_1,\dots,\Phi_m)\in G^{(m)}} \|(u_{10},\dots,u_{m0}) -(\Phi_1,\dots,\Phi_m)\|_{X^{(m)}} <\delta, $$ then the solution $\big(u_1(x,t),\dots,u_m(x,t)\big)$ of \eqref{NNLS} with $\big(u_1(x,0),\dots,u_m(x,0)\big) =(u_{10},\dots,u_{m0})$ satisfies $$ \inf_{(\Phi_1,\dots,\Phi_m) \in G^{(m)}} \|\big(u_1(\cdot,t), \dots, u_m(\cdot,t)\big) -(\Phi_1,\dots,\Phi_m)\|_{X^{(m)}} <\epsilon $$ for all $t\in \mathbb{R}$. \end{lemma} \subsection{Stability of ground-state solutions} In this subsection, we will show that the set of minimizers $G^{(m)}(\lambda,\dots,\lambda)$ contains just a single $m-$tuple of functions (modulo translations and phase shifts), and that this $m-$tuple of functions is indeed a ground-state solution of \eqref{NNLS} given by \[ \big(\Phi_1(x,t),\dots, \Phi_m(x,t)\big) =\Big(e^{i\Omega t} \phi_{\Omega,a+(m-1)b}(x), \dots ,e^{i\Omega t} \phi_{\Omega,a+(m-1)b}(x) \Big). \] Proposition \ref{thm1} then follows directly from this fact and Lemma \ref{thm5.2}. We start first with the following Lemma that relates the functions $\Phi_1,\dots,\Phi_m$ whenever $(\Phi_1,\dots,\Phi_m) \in G^{(m)}(\lambda,\dots,\lambda)$. \begin{lemma} \label{lem5.6b} Let $(\Phi_1,\dots,\Phi_m) \in G^{(m)}(\lambda,\dots,\lambda)$. Then for any $x\in \mathbb{R}^N$, $$ |\Phi_1(x)| = |\Phi_2(x)|= \dots = |\Phi_m(x)|. $$ \end{lemma} \begin{proof} It follows from Lemma \ref{lem5.4} that for any $(\Phi_1,\dots,\Phi_m) \in G^{(m)}(\lambda, \dots, \lambda)$ $$ I^{(m)}(\lambda, \dots,\lambda)= E^{(m)}(\Phi_1,\dots,\Phi_m) \geq \sum_{j=1}^m E^{(m)}_1(\Phi_j) \geq m I^{(m)}_1(\lambda) =I^{(m)}(\lambda,\dots,\lambda). $$ Thus, \[ \frac{a}{p}\sum_{j=1}^m \|\Phi_j\|_{L^{2p}}^{2p} + \frac{b}{p}\int_{\mathbb{R}^N}\sum_{i,j=1; i\not=j}^m |\Phi_i|^p|\Phi_j|^p dx = \frac{a+(m-1)b}{p}\sum_{j=1}^m \|\Phi_j\|^{2p}_{L^{2p}} \] which implies that \begin{equation} \int_{\mathbb{R}^N}\sum_{i,j=1; i\not=j}^m |\Phi_i|^p|\Phi_j|^p dx = (m-1)\sum_{j=1}^m \|\Phi_j\|^{2p}_{L^{2p}}. \label{5.8} \end{equation} We can rewrite \eqref{5.8} as $$ \int_{\mathbb{R}^N} \sum_{i,j=1;i\not=j}^m \Big||\Phi_i(x)|^p- |\Phi_j(x)|^p\Big|^2 dx =0, $$ from which the statement of the Lemma immediately follows. \end{proof} Next, we show the following. \begin{lemma} \label{thm5.3} For any $\Omega>0$ fixed, $$ \Big \{ \Big(e^{i\alpha_1} \phi_{\Omega,a+(m-1)b}(\cdot+y), \dots, e^{i\alpha_m} \phi_{\Omega,a+(m-1)b}(\cdot+y)\Big)\Big \}= G^{(m)} \big(\lambda(\Omega),\dots,\lambda(\Omega)\big) $$ where $\alpha_j \in \mathbb{R}$, $j=1,2,\dots,m$; $y\in \mathbb{R}^N$. \end{lemma} \begin{proof} It has been established (Corollary \ref{cor5.A}) that for any $\Omega>0$ fixed and for $\alpha_j \in \mathbb{R}$, $j=1,2,\dots,m$ and $y\in \mathbb{R}^N$, $$ \Big(e^{i\alpha_1} \phi_{\Omega,a+(m-1)b}(\cdot+y), \dots, e^{i\alpha_m} \phi_{\Omega, a+(m-1)b}(\cdot+y)\Big)\in G^{(m)}\big(\lambda(\Omega), \dots,\lambda(\Omega)\big). $$ Hence, the Lemma is proved if we show that any minimizer in $G^{(m)}(\lambda(\Omega),\dots,\lambda(\Omega))$ must be of the form given above. Now, since the constrained minimizer for the variational problem exists, there are Lagrange multipliers $\Omega_1, \dots ,\Omega_m\in \mathbb{R}$ such that for $j=1,2,\dots,m$ \begin{equation} -\Delta\Phi_j +\Omega_j \Phi_j = a|\Phi_j|^{2p-2}\Phi_j +b\sum_{i=1;i\not=j}^m |\Phi_i|^{p}|\Phi_j|^{p-2}\Phi_j. \label{Lagrange5} \end{equation} Using Lemma \ref{lem5.6}, we can rewrite this system as $m$-uncoupled equations \begin{equation} -\Delta \Phi_j +\Omega_j \Phi_j = \big(a+(m-1)b\big) |\Phi_j|^{2p-2} \Phi_j. \label{5.9} \end{equation} A bootstrap argument shows that any $m$-tuple $L^2$-distribution solution of \eqref{5.9} must indeed be smooth and given by (see, for example, \cite{Ca1}) \[ \Phi_j(x)= e^{i\alpha_j}\phi_{\Omega_j,a+(m-1)b}(x+y_j), \] where $\alpha_j \in \mathbb{R}$, $y_j \in \mathbb{R}^N$ and $\Omega_j>0$ for $j=1,2,\dots,m$. Now recall that for any $x\in \mathbb{R}^N$, we must have $$ |\Phi_1(x)| = |\Phi_2(x)|=\dots = |\Phi_m(x)|, $$ and $$ \|\Phi_1\|_{L^2}^2= \|\Phi_2\|_{L^2}^2=\dots = \|\Phi_m\|_{L^2}^2=\lambda = \big(a+(m-1)b\big)^{-\frac{1}{(p-1)}} \Omega^{\frac{1}{(p-1)} -\frac{N}{2}} \|\phi\|^2_{L^2}. $$ It is easy to see then that $y_1=y_2=\dots=y_m$, and $$ \Omega=\Omega_1=\Omega_2=\dots=\Omega_m>0. $$ The Lemma is thus established. \end{proof} The above proposition follows from Lemmas \ref{thm5.2} and \ref{thm5.3}. Next, we will show that instead of allowing the ground-state solutions to wander around at random, one can pick unique trajectory and phase shifts that the ground-state solutions must follow. Denote $\vec{\theta}=(\theta_1, \dots, \theta_m)$. Following the idea used in \cite{BoSo,W}, the functions $\theta_j$ ($j=1, 2, \dots, m$) and $\eta$ are found through minimizing the function $R=R(\vec{\theta},\eta):\mathbb{R}^{m+1} \to \mathbb{R}$, \begin{equation} \begin{aligned} R(\vec{\theta},\eta) &=\sum_{j=1}^m\Big[\Omega\|u_j(x)-e^{i\theta_j}\phi_{\Omega,a+(m-1)b} (x+\eta)\|_{L^2}^2 \\ &\quad + \|u_j'(x)-e^{i\theta_j}\phi_{\Omega,a+(m-1)b}'(x+\eta)\|_{L^2}^2\Big]. \end{aligned} \end{equation} From now on, denote $\phi(x)=\phi_{\Omega, a+(m+1)b}(x)$ for simplicity. Due to the symmetry, we only need to consider one component \begin{equation} \begin{aligned} R_j(\theta_j, \eta) &=\Omega\|u_j(x)-e^{i\theta_j}\phi(x+\eta)\|_{L^2}^2 + \|u_j'(x)-e^{i\theta_j}\phi'(x+\eta)\|_{L^2}^2\\ &=\int_{\mathbb{R}^N}\Big(\Omega|u_j(x)-e^{i\theta_j}\phi(x+\eta)|^2+|(u_j)_x - e^{i\theta_j}\phi'(x+\eta)|^2\Big)dx. \end{aligned} \end{equation} Then \begin{equation} \begin{aligned} \frac{\partial R_j}{\partial\eta} &=\int_{\mathbb{R}^N}\big[-2\Omega\operatorname{Re}(u_j(x)e^{-i\theta_j})\phi'(x+\eta) -2\operatorname{Re}((u_j)_xe^{-\theta_j}\phi''(x+\eta)\big]dx\\ &=2\operatorname{Re}\int_{\mathbb{R}^N} u_j(x)e^{-i\theta_j}(\phi''(x+\eta)-\Omega\phi(x+\eta))'dx\\ &=-2[a+(m-1)b]\operatorname{Re}\int_{\mathbb{R}^N} u_j(x)e^{-i\theta_j}(\phi^{2p-1}(x+\eta))'dx\\ &=-2(2p-1)[a+(m-1)b]\int_{\mathbb{R}^N} \operatorname{Re}\big(u_j(x)e^{-i\theta_j}\big) \phi^{2p-2}(x+\eta)\phi'(x+\eta)dx, \end{aligned} \end{equation} and \begin{equation} \frac{\partial R_j}{\partial\theta_j}=i[a+(m-1)b] \int_{\mathbb{R}^N}\operatorname{Im}\big(u_j(x)e^{-i\theta_j}\big)\phi^{2p-1}(x+\eta)dx. \end{equation} Define vector-valued function $Q: X\times\mathbb{R}^{m+1}\to\mathbb{R}^{m+1}$, \[ Q(\vec{\psi},\vec{\theta},\eta)=(F(\vec{\psi},\vec{\theta},\eta), \vec{G}(\vec{\psi},\vec{\theta},\eta)) \] where \begin{equation} \begin{gathered} F(\vec{\psi},\vec{\theta},\eta)=\sum_{j=1}^m\int_{\mathbb{R}^N} \operatorname{Re}\big(\psi_je^{-i\theta_j}\big)\phi^{2p-2}(x+\eta) \phi'(x+\eta)dx;\\ G_j(\vec{\psi},\vec{\theta},\eta)=\int_{\mathbb{R}^N} \operatorname{Im}\big(\psi_je^{-i\theta_j}\big)\phi^{2p-1}(x+\eta)dx. \end{gathered} \end{equation} Next, we verify the conditions needed for using the Implicit Function Theorem. \begin{lemma}\label{lemma:groundproperty} Denote $\vec\phi=(\phi,\dots,\phi)$. Then: \begin{itemize} \item[(i)] $Q(\vec\phi,\vec{0},0)=(0,\vec{0})$. \item[(ii)] $|\nabla Q|<0$. \end{itemize} \end{lemma} \begin{proof} Statement (i) follows from the facts that $-\Delta\phi + \Omega\phi= (a+(m-1)b)\phi^{2p-1}$ and \begin{gather*} F(\vec\phi,\vec{0},0)=\sum_{j=1}^m\int_{\mathbb{R}^N}\phi^{2p-2}\phi'dx=0;\\ G_j(\vec\phi,\vec{0},0)=\int_{\mathbb{R}^N}\operatorname{Im}(\phi\cdot1)\phi^{2p-1}dx=0. \end{gather*} To prove (ii), notice that \begin{align*} \frac{\partial F}{\partial\theta_j}\Big|_{(\vec\phi,\vec{0},0)} &=\operatorname{Re}\Big(-i\int_{\mathbb{R}^N}\operatorname{Im}\big(\psi_j(x) e^{-i\theta_j}\big)\phi^{2p-2}(x+\eta)\phi'(x+\eta)dx\Big)\Big|_{(\vec\phi,\vec{0},0)} =0,\\ \frac{\partial F}{\partial\eta}\Big|_{(\vec\phi,\vec{0},0)} &=\frac{\partial}{\partial\eta}\Big[ \sum_{j=1}^m \int_{\mathbb{R}^N}\operatorname{Re}\big(\psi_j(x)e^{-\theta_j} \big)\phi^{2p-2}(x+\eta)\phi'(x+\eta)dx\Big]\Big|_{(\vec\phi,\vec{0},0)}\\ &=\frac{\partial}{\partial\eta}\Big[ \sum_{j=1}^m \int_{\mathbb{R}^N}\operatorname{Re}\big(\psi_j(x-\eta)e^{-\theta_j}\big) \phi^{2p-2}(x)\phi'(x)dx\Big]\Big|_{(\vec\phi,\vec{0},0)}\\ &=\sum_{j=1}^m\int_{\mathbb{R}^N}-\phi'(x)\phi^{2p-2}(x)\phi'(x)dx\\ &=-m\int_{\mathbb{R}^N}\phi^{2p-2}(x)\big[\phi'(x)\big]^2dx,\\ \frac{\partial G_j}{\partial\eta}\Big|_{(\vec\phi,\vec{0},0)} &=(2p-1)\int_{\mathbb{R}^N}\operatorname{Im}\big(\psi_j(x)e^{-i\theta_j}\big) \phi^{2p-2}(x+\eta)\phi'(x+\eta)dx\Big|_{(\vec\phi,\vec{0},0)}=0,\\ \frac{\partial G_j}{\partial\theta_j}\Big|_{(\vec\phi,\vec{0},0)} &=\int_{\mathbb{R}^N} i\big(\psi_j(x)e^{-\theta_j}- \overline{\psi}_j(x)e^{i\theta_j}\big)\phi^{2p-1}(x+\eta)dx \Big|_{(\vec\phi,\vec{0},0)}=\int_{\mathbb{R}^N} \phi^{2p}(x)dx. \end{align*} Thus, \[ \det(\nabla G)=-m\Big(\int_{\mathbb{R}^N}\phi^{2p-2}(x)\big[\phi'(x)\big]^2dx\Big) \Big(\int_{\mathbb{R}^N} \phi^{2p}(x)dx\Big)^{m-1}<0. \] \end{proof} Define an equivalent norm in $X^{(m)}$ by \begin{equation} \|\vec{u}\|_{X^{(m)}}^2=\sum_{j=1}^m\big[\Omega\|u_j(x)\|_{L^2}^2 + \|u_j'(x)\|_{L^2}^2\big];\quad \vec{u}=(u_1, \dots, u_m)\in X^{(m)}. \end{equation} Let the $X^{(m)}$-neighborhood of the trajectory of $(e^{i\theta_1}\phi, \dots, e^{i\theta_m}\phi)$ be defined by \[ \mathcal{U}_\beta=\Big\{\vec{\phi}\in X^{(m)} : \inf_{\vec{\theta},\eta}\big\{ \|\vec{\psi}(x)-e^{i\vec{\theta}_j}\phi(x+\eta)\|_{X^{(m)}}<\beta\big\}\Big\}. \] \begin{lemma}\label{lemma:c1maps} There exist a $\beta>0$ and $C^1$ maps $\vec{\theta}, \eta: \mathcal{U}_\beta\to\mathbb{R}$ such that \[ F(\vec{\psi}, \vec{\theta}(\vec{\psi}), \eta(\vec{\psi}))\equiv0,\quad G_j(\vec{\psi}, \vec{\theta}(\vec{\psi}), \eta(\vec{\psi}))\equiv0, \] for all $\vec{\psi}\in\mathcal{U}_\beta$. \end{lemma} \begin{proof} It is easy to see that the function $Q$ is $C^1$ on its' domain. Lemma \ref{lemma:groundproperty} verifies all the conditions needed to apply the Implicit Function Theorem. Thus Lemma \ref{lemma:c1maps} follows provided $\beta>0$ is small enough. \end{proof} Because of the stability result stated in Proposition \ref{thm1}, $\vec{u}(\cdot,t) \in \mathcal U_{\beta}$ and hence the corresponding functions $\vec{\theta}$ and $\eta$ are defined on $\vec{u}(\cdot,t).$ One can therefore consider the functions $\vec{\theta}$ and $\eta$ from $ \mathbb{R} \to \mathbb{R}$ as \[ \eta(t)= \eta\big(\vec{u}(\cdot,t)\big), \] and for $i=1,2,\dots,m$ \[ \theta_i(t)= \theta_i\big(\vec{u}(\cdot,t)\big). \] The next Lemma is clear. (See, for example, \cite{NVN2}.) \begin{lemma}\label{lemma:QC1} The function $Q$ is continuously differentiable with respect to $t$. \end{lemma} We are now ready for the following proof. \begin{proof}[Proof of Theorem \ref{thm1A}] The first part of the Theorem is an immediate consequence of Lemma \ref{lemma:c1maps} and Proposition \ref{thm1}. Indeed, for fixed $\epsilon>0$, one can first apply Lemma \ref{lemma:c1maps} to find a proper $\beta>0$ such that the continuous maps exist. Then, the result of \cite{NTDS} implies the existence of some $\delta>0$ such that when the initial data satisfies assumption \eqref{initialdata}, the resulting perturbations from $\eta$ and $\theta_j$'s for all $t\geq0$ will remain in the ball $\mathcal{U}_\beta$. Thus the estimate \eqref{continueextention} holds. It is left to show \eqref{thm:estimates}. Define the $m$ functions \[ h_j(x, t)=e^{-i\theta_j(t)}u(x, t)-\phi(x+\eta(t))=h_{j1}+ih_{j2} \] for $j=1,\dots,m$. According to \eqref{continueextention}, $\sum_{j=1}^m\|h_{j1}\|_{H^1}+ \sum_{j=1}^m\|h_{j2}\|_{H^1}=O(\epsilon)$ for all $t\geq0$. Differentiating $F$ with respect to $t$, we obtain \begin{align*} F_t&=\sum_{j=1}^m\int_{\mathbb{R}^N}\operatorname{Re}\Big([(u_j)_t-(u_j)_x\eta'(t)]e^{-i\theta_j(t)} -i\theta_j'u_j e^{-i\theta_j(t)} \Big) \phi^{2p-2}\phi'dx\\ &=\operatorname{Re}\sum_{j=1}^m\int_{\mathbb{R}^N}\Big[(u_j)_te^{-i\theta_j(t)} \phi^{2p-2}\phi'- (u_j)_x\eta'(t)e^{-i\theta_j(t)} \phi^{2p-2}\phi'\\ &\quad - i\theta_j'(t)u_je^{-i\theta_j(t)}\phi^{2p-2}\phi'\Big]dx\\ &=\sum_{j=1}^m(I_{j1}-I_{j2}-I_{j3})=0. \end{align*} Notice that \begin{gather*} \begin{aligned} I_{j1}&=\operatorname{Re}\int_{\mathbb{R}^N} i\Big[(u_j)_{xx}+\big(a|u_j|^p+b\sum_{k\neq j}|u_k|^p\big)|u_j|^{p-2}u_j\Big]e^{-i\theta_j(t)} \phi^{2p-2}\phi'dx\\ &=-\int_{\mathbb{R}^N}\Big[(h_2)_{xx}+\big(a|u_j|^p+b\sum_{k\neq j}|u_k|^p\big)|u_j|^{p-2}h_2\Big]\phi^{2p-2}\phi'dx=O(\epsilon), \end{aligned} \\ I_{j2}=\int_{\mathbb{R}^N}\Big(\phi^{2p-2}(\phi')^2\eta'(t)+(h_1)_x\eta'(t)\phi^{2p-2} \phi'\Big)dx=c\eta'(t)+O(\epsilon), \\ I_{j3}=\int_{\mathbb{R}^N}\theta_j'(t)h_2\phi^{2p-2}\phi'dx=O(\epsilon)\theta_j'(t). \end{gather*} Thus we have \[ \eta'(t)=O(\epsilon)+O(\epsilon)\sum_{j=1}^m\theta_j'(t). \] Similarly, the other $m$ equations for $\vec{\theta}$ give \[ \theta_j'(t)=\Omega+O(\epsilon)+O(\epsilon)\eta'(t),\quad j=1,\dots, m. \] The statement \eqref{thm:estimates} follows immediately. Thus, the Theorem is proved. \end{proof} \section{Stability of traveling-wave solutions} The result obtained in Section 3 is now broadened to include traveling-wave solutions and improved by providing a more detailed view of the connection between the functions $\eta$ and $\theta_i$. For $\theta\in\mathbb{R}$, define the operator $T_\theta: H_\mathbb{C}^1(\mathbb{R})\to H_\mathbb{C}^1(\mathbb{R})$ by \[ (T_\theta u)(x)=\exp\Big(\frac{i\theta x}{2}\Big)u(x). \] For any pair $(\omega,\theta)\in\mathbb{R}\times\mathbb{R}$ such that $\Omega=\omega-\frac{1}{4}\theta^2>0$, let $\varphi_\omega=T_\theta\phi_\Omega$. The following Lemma is straightforward. \begin{lemma} \label{lem4.1} If $(e^{i\Omega t}\phi_{\Omega},\dots,e^{i \Omega t}\phi_{\Omega})$ is a ground-state solution of \eqref{NNLS}, then \\ $(e^{i\omega t}\varphi_{\omega},\dots,e^{i\omega t}\varphi_{\omega})$ is a traveling-wave solution of \eqref{NNLS}. \end{lemma} \begin{proof}[Proof of Corollary \ref{thm1B}] Similar arguments as used in \cite{BoSo,NVN,W} allow us to extend the stability result obtained above to include traveling-wave solutions as well. Readers are referred to, for example, \cite{NVN} for the proof of this. \end{proof} \section{Conclusion} The traveling-wave solutions of \eqref{NNLS} have been shown to be orbitally stable in $X^{(m)}$ when $2\leq p<3$ and $N=1$ and orbitally stable in $Y^{(m)}$ when $1