\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2008(2008), No. 148, pp. 1--6.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2008 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2008/148\hfil Nonlinear Schr\"odinger systems] {A note on radial nonlinear Schr\"odinger systems with nonlinearity spatially modulated} \author[J. Belmonte-Beitia\hfil EJDE-2008/148\hfilneg] {Juan Belmonte-Beitia} \address{Departamento de Matem\'aticas, E. T. S. de Ingenieros Industriales and Instituto de Matem\'atica Aplicada a la Ciencia y la Ingenier\'{\i}a (IMACI), \\ Universidad de Castilla-La Mancha s/n, 13071 Ciudad Real, Spain} \email{juan.belmonte@uclm.es} \thanks{Submitted May 13, 2008. Published October 29, 2008.} \subjclass[2000]{35Q55, 34B15, 35Q51} \keywords{Nonlinear Schr\"odinger equation; nonlinearity spatially modulated; Ermakov-Pinney equation; fixed point theorem} \begin{abstract} First, we prove that for Schr\"odinger radial systems the polar angular coordinate must satisfy $\theta'= 0$. Then using radial symmetry, we transform the system into a generalized Ermakov-Pinney equation and prove the existence of positive periodic solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{Lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \section{Introduction} This note concerns the existence of solutions for the nonlinear Schr\"odinger systems with nonlinearity spatially modulated and radial symmetry in $1D$ \begin{subequations}\label{sist} \begin{eqnarray}\label{sistema} u_{1}''(x)+a(x)u_{1}(x)=b(x)f(u_{1}^2+u_{2}^2)u_{1}\\ \label{sistema1b} u_{2}''(x)+a(x)u_{2}(x)=b(x)f(u_{1}^2+u_{2}^2)u_{2} \end{eqnarray} \end{subequations} % where $f(u_{1}^2+u_{2}^2)$ is a positive continuous function with radial symmetry, and $a$ and $b$ are positive, continuous and $L$-periodic functions; i.e., % $$a(x)=a(x+L),\quad b(x)=b(x+L).$$ Such solutions satisfy the boundary conditions % \begin{subequations}\label{conditions} \begin{eqnarray}\label{condiciones} \lim_{|x|\to\infty}u_{1}(x)=\lim_{|x|\to\infty}u_{2}(x)=0,\\ \label{condicionesb} \lim_{|x|\to\infty}u_{1}'(x)=\lim_{|x|\to\infty}u_{2}'(x)=0 \end{eqnarray} \end{subequations} % The study of the existence of positive solutions for systems like \eqref{sistema}, \eqref{sistema1b} with one coupled lineal term has gained the interest of many mathematicians in recent years. We refer to the surveys \cite{Ambrosetti,Eduardo2,Eduardo}. In these papers, the authors show the existence of positive solutions for different systems, using critical point theory or a variational approach. Another different approximation to this kind of problems can be found in Ref. \cite{Nuestro}. From of physical point of view, this kind of systems has gained a lot of interest in the last years, in particular in the context of systems for the mean field dynamics of Bose-Einstein condensates \cite{Williams} and in applications to fields as nonlinear and fibers optics \cite{Malomed}. On the other hand, the existence of positive solutions for the nonlinear Schr\"odinger equation % $$u''+a(x)u=b(x)f(u(x))$$ % was proved in Ref. \cite{Pedro}. Thus, the existence of semitrivial solutions $(u_{1},0)$ and $(0,u_{2})$ of the system \eqref{sistema} is guaranteed by the Ref. \cite{Pedro}. We can transform the system \eqref{sistema} in a equation, doing $y=(u_{1},u_{2})$ % $$\label{auxiliar} y''+a(x)y=b(x)f(I)y$$ % with $I=u_{1}^2+u_{2}^2$. With the change of variable % $$\label{cambio} u_{1}=\rho\cos\theta,\quad u_{2}=\rho\sin\theta.$$ % equation (\ref{auxiliar}) becomes % $$\label{mean} \left[\rho''-\rho(\theta')^{2}+a(x)\rho \right] \cos\theta-\left[2\rho'\theta'+\rho\theta''\right] \sin\theta=b(x)f(\rho^{2})\rho\cos\theta$$ The aim of this paper is to show that for Schr\"odinger radial systems, as \eqref{sistema}, \eqref{sistema1b} with conditions \eqref{condiciones}, \eqref{condicionesb}, can only exist solutions with $\theta'=0$, specifically, we are thinking in the semitrivial solutions $(u_{1},0)$ and $(0,u_{2})$. On the other hand, for $\theta'\neq 0$, there not exist solutions of the system \eqref{sistema}, \eqref{sistema1b} with conditions \eqref{condiciones}, \eqref{condicionesb}. Moreover, we can transform the system, by using the radial symmetry, to a generalized Ermakov-Pinney equation and study positive periodic solutions for this equation. The rest of the papers is organized as follows. In section 2 we prove that the only solutions of system \eqref{sistema}, \eqref{sistema1b} with conditions \eqref{condiciones}, \eqref{condicionesb}, if they exist, are given by solutions which verify $\theta'=0$. In section 3, we prove the existence of positive periodic solutions of the system \eqref{sistema}, \eqref{sistema1b}, with periodic conditions. In this note, $\|\cdot\|$ denotes the supremum norm. \section{Nonexistence of solutions for $\theta'\neq 0$ and existence for $\theta'=0$} Physically, when a physical system possesses a symmetry, it means that a physical quantity is conserved. As the system \eqref{sistema}, \eqref{sistema1b} has radial symmetry, the conserved quantity is the angular momentum. In polar coordinates, the conservation of the angular momentum is given by % $$\label{conservacion} \rho^{2}\theta'=\mu,$$ % where $\mu$ is a constant. Using this fact, \eqref{mean} becomes % $$\label{EP} \rho''+a(x)\rho=b(x)f(\rho^{2})\rho+\frac{\mu^{2}}{\rho^{3}},$$ % which can be taken as a generalized Ermakov-Pinney \cite{Ermakov,Pinney}. Now, it is easy to prove that, if there exist solutions of the system \eqref{sistema}, \eqref{sistema1b}, with the boundary conditions \eqref{condiciones}, \eqref{condicionesb}, they must satisfy the condition $\theta'=0$: for these solutions, $\theta$ is constant and these solutions can be solutions of (\ref{auxiliar}). In fact, we can find two examples of solutions for this case: the semitrivial solutions $(u_{1},0)$ and $(0,u_{2})$ are solutions of the system \eqref{sistema}, \eqref{sistema1b}, with conditions \eqref{condiciones}, \eqref{condicionesb} (see Ref. \cite{Pedro}). On the other hand, for one solution $(u_{1},u_{2})$ with $\theta'\neq 0$, one has $\mu\neq 0$. Thus, if would exist a solution $(u_{1},u_{2})$ of the system \eqref{sistema}, \eqref{sistema1b} with the boundary conditions \eqref{condiciones}, \eqref{condicionesb} it would exist a solution $\rho$ that would verify $\rho\to 0$ as $|x|\to\infty$. But it is impossible, by the singularity of (\ref{EP}). Thus, we are in disposition to formulate the following theorem. % \begin{theorem} \label{thm1} Let system \eqref{sist} be with conditions \eqref{conditions} where $a(x)$ and $b(x)$ are positive, continuous and $L$-periodic functions. Then, if there exist solutions of the system \eqref{sist}, with the conditions \eqref{conditions}, different of the trivial solution, they must satisfy the condition $\theta'=0$, where $\theta$ is the polar angular coordinate in \eqref{cambio}. \end{theorem} \begin{remark} \label{rmk1} \rm Specifically, for $\theta=k\pi$ or $\theta=\frac{k}{2}\pi$, for any $k\in\mathbb{Z}$, we obtain the semitrivial solutions. These solutions are called bright solitons in the physical literature. The dark solitons are also solutions of the system \eqref{sistema}, \eqref{sistema1b} but with different boundary conditions \cite{Kivshar}. It is straightforward to prove that, for this case, the only solutions are the former with $\theta'=0$, provided that $a(x)$ is different to $b(x)$. \end{remark} \begin{remark} \label{rmk2} \rm We can use another approximation, where one can see the universality of the method exposed here. Thus, let the nonlinear Schr\"odinger equation be % $$\label{NLSE} iu_{t}+u_{xx}+b(x)f(|u|^2)u+V(x)u=0$$ % with $V(x)$ a $L$-periodic function. If we have the change of variable $u(t,x)=\left(v(x)+iw(x)\right)e^{i\lambda t}$ and if we separate in real and imaginary part, we obtain % \begin{gather*} v''+\left(V(x)-\lambda\right)v+b(x)f(v^2+w^2)v=0\\ w''+\left(V(x)-\lambda\right)w+b(x)f(v^2+w^2)w=0 \end{gather*} % which is similar to the system \eqref{sistema}, \eqref{sistema1b} for $a(x)=V(x)-\lambda$. \end{remark} \section{Periodic Solutions} As we showed in the previous section, system \eqref{sistema} \eqref{sistema1b}, or equation (\ref{NLSE}), can be reduced to (\ref{EP}). Thus, we can describe the behaviour of solutions of \eqref{sistema}--\eqref{sistema1b} (or (\ref{NLSE})) using (\ref{EP}) Then, the aim of this section is to provide some existence result for the periodic boundary-value problem % $$\label{EP2} \rho''+a(x)\rho=b(x)f(\rho^{2})\rho+\frac{\mu^{2}}{\rho^{3}},$$ % with $\rho(0)=\rho(L)$, $\rho'(0)=\rho'(L)$, where $a(x)$ and $b(x)$ are positive, continuous and $L$-periodic functions. To do it, we will use the following fixed-point theorem for a completely continuous operator in a Banach space, due to Krasnoselskii \cite{Krasnoselskii}. \begin{theorem}\label{KN} Let $X$ be a Banach space, and let $P\subset X$ be a cone in $X$. Assume $\Omega_{1}, \Omega_{2}$ are open subsets of $X$ with $0\in\Omega_{1},\overline{\Omega}_{1}\subset\Omega_{2}$ and let $T: P\cap(\overline{\Omega}_{2}\backslash\Omega_{1})\to P$ be a completely continuous operator such that one of the following conditions is satisfied % \begin{enumerate} \item $\|Tu\|\leq\|u\|$, if $u\in P\cap\partial\Omega_{1}$, and $\|Tu\|\geq\|u\|$, if $u\in P\cap\partial\Omega_{2}$. \item $\|Tu\|\geq\|u\|$, if $u\in P\cap\partial\Omega_{1}$, and $\|Tu\|\leq\|u\|$, if $u\in P\cap\partial\Omega_{2}$. \end{enumerate} % Then, $T$ has at least one fixed point in $P\cap(\overline{\Omega}_{2}\backslash\Omega_{1})$. \end{theorem} From the physical explanation, \eqref{EP2} has a repulsive singularity at $x=0$. In order to apply Theorem \ref{KN}, we need some information about the properties of the Green's function. Thus, let us consider the linear equation % $$\label{eqhom} \rho''+a(x)\rho=0,$$ % with periodic conditions % $$\label{condition} \rho(0)=\rho(L),\quad \rho'(0)=\rho'(L)$$ % In this section, we assume conditions under which the only solution of problem (\ref{eqhom})-(\ref{condition}) is the trivial one. As a consequence of Fredholm's alternative, the nonhomogeneous equation % $$\label{eqnonhom} \rho''+a(x)\rho=h(x),$$ % admits a unique $T$-periodic solution which can be written as % $$\rho(x)=\int_{0}^{L}G(x,s)h(s)ds,$$ % where $G(x,s)$ is the Green's function of problem (\ref{eqhom})-(\ref{condition}). Following \cite{Multiplicity}, we assume that problem (\ref{eqhom}) satisfies that the Green function, $G(x,s)$, associated with problem (\ref{eqnonhom}), is positive for all $(x,s)\in[0,L]\times[0,L]$. Moreover, following \cite{Pedro2}, we denote % $$M=\max_{x,s\in[0,L]}G(x,s),\quad m=\min_{x,s\in[0,L]}G(x,s)$$ % where $M>m>0$. \begin{theorem} Let us assume the following hypotheses \begin{itemize} \item[(i)] $a(x)$ and $b(x)$ are continuous and $L$-periodic functions with $a>0, b>0$. \item[(ii)] $f(s)\geq 0$ for every $s\geq 0$. \item[(iii)] There exists $r>0$ such that \begin{equation*} A_{r}\max_{x\in[0,L]}\int_{0}^{L}G(x,s)b(s)ds+B_{r} \max_{x\in[0,L]}\int_{0}^{L}G(x,s)ds\leq r \end{equation*} % for $A_{r}=\max_{s\in[0,r]}f(s^2)s$ and $B_{r}=\max_{s\in[0,r]}\mu^{2}/s^3$. \item[(iv)] There exist $R>r>0$ such that \begin{equation*} A_{R}\min_{x\in[0,L]}\int_{0}^{L}G(x,s)b(s)ds +B_{R}\min_{x\in[0,L]}\int_{0}^{L}G(x,s)ds\geq \frac{M}{m}R \end{equation*} for $A_{R}=\min_{s\in[R,(M/m)R]}f(s^2)s$ and $B_{R}=\min_{s\in[R,(M/m)R]}\mu^{2}/s^3$. \end{itemize} Then, \eqref{EP2} has a positive periodic solution $\rho$ with $\frac{m}{M}r\leq\rho(x)\leq\frac{M}{m}R$. \end{theorem} \begin{proof} Let $X=C[0,L]$ with the supremum norm $\|\cdot\|$. We define the open sets % \begin{gather*} \Omega_{1}=\{\rho\in X: \|\rho\|