Content-Type: multipart/mixed; boundary="-------------1202170308174" This is a multi-part message in MIME format. ---------------1202170308174 Content-Type: text/plain; name="12-25.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="12-25.keywords" NLS, graphs, ground state ---------------1202170308174 Content-Type: application/x-tex; name="saddle13-2-12-bis.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="saddle13-2-12-bis.tex" \documentclass[reqno]{amsart} %\usepackage{amsfonts} \usepackage{amsmath, amssymb, amsfonts, amsthm} %\usepackage{epsfig} %\input{psfig} %\usepackage[notref,notcite]{showkeys} %\usepackage[notref,notcite]{showkeys} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \setlength{\oddsidemargin}{+5pt} \setlength{\evensidemargin}{+5pt} \setlength{\textwidth}{150mm} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{xca}[theorem]{Exercise} \newtheorem{cor}[theorem]{Corollary} \newtheorem{prop}[theorem]{Proposition} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\bes}{{\begin{split}}} \newcommand{\ees}{{\end{split}}} \newcommand{\bees}{{\begin{equation}\begin{split}}} \newcommand{\es}{{\end{split}\end{equation}}} \newcommand{\erre}{{\mathbb R}} \newcommand{\enne}{{\mathbb N}} \newcommand{\natu}{{\mathbb N}} \newcommand{\comple}{{\mathbb C}} \newcommand{\inte}{{\mathbb Z}} \newcommand{\bea}{\begin{eqnarray}} \newcommand{\eea}{\end{eqnarray}} \newcommand{\be}{\begin{equation}} \newcommand{\ee}{\end{equation}} %\newcommand{\me1}{\left<} %\newcommand{\me2}{\right<} \newcommand{\n}{\noindent} \newcommand{\sob}{{\mathcal L}^{1,2} (L^2 (\erre^n))} \newcommand{\sobb}{{\mathcal L}^{1,2} (L^2 (\erre^{n+1}))} \newcommand{\norm}[1]{\left\vert\!\left\vert #1 \right\vert\!\right\vert} \newcommand{\fine}{\hfill \mbox{$\Box$} \vskip .3cm} \newcommand{\x}{{\bf{x}}} \newcommand{\y}{{\bf{y}}} \newcommand{\ku}{{\bf{k}}} \newcommand{\F}{{\cal{F}}} \newcommand{\f}{\frac} \newcommand{\al}{\alpha} \newcommand{\la}{\lambda} \newcommand{\ve}{\varepsilon} \newcommand{\om}{\omega} \newcommand{\ov}{\overline} \newcommand{\U} {\mathcal{U}} \newcommand{\nnn} {\nonumber} \newcommand{\tal}{\widetilde \alpha} \newcommand{\mc}{\mathcal} \numberwithin{equation}{section} % Absolute value notation \newcommand{\abs}[1]{\lvert#1\rvert} % Blank box placeholder for figures (to avoid requiring any % particular graphics capabilities for printing this document). \newcommand{\blankbox}[2]{% \parbox{\columnwidth}{\centering % Set fboxsep to 0 so that the actual size of the box will match the % given measurements more closely. \setlength{\fboxsep}{0pt}% \fbox{\raisebox{0pt}[#2]{\hspace{#1}}}% }% } \usepackage{graphicx} \begin{document} \large \title{On the structure of critical energy levels \\ for the cubic focusing NLS on star graphs} % Information for first author \author{Riccardo Adami} \address{Adami and Noja: Dipartimento di Matematica e Applicazioni, Universit\`a di Milano Bicocca, via R. Cozzi 53, 20125 Milano, Italy and Istituto di Matematica Applicata e Tecnologie Informatiche ``Enrico Magenes'', CNR, via Ferrata, 1 27100 Pavia, Italy \\ e-mail: riccardo.adami@unimib.it, diego.noja@unimib.it} \address{ Cacciapuoti: Hausdorff Center for Mathematics, Institut f\"ur Angewandte Mathematik, 60, Endenicher Allee, 53115 Bonn, Germany \\ e-mail: cacciapuoti@him.uni-bonn.de} \address{ Finco: Facolt\`a di Ingegneria, Universit\`a Telematica Internazionale Uninettuno, Corso Vittorio Emanuele II 39, 00186 Roma, Italy \\ e.mail: d.finco@uninettunouniversity.net} %``La Sapienza''} % \thanks will become a 1st page footnote. % Information for second author \author{Claudio Cacciapuoti} \author{Domenico Finco} \author{Diego Noja} \date{February 11, 2012} %\dedicatory{This paper is dedicated to our advisors.} \keywords{} \begin{abstract} We provide information on a non trivial structure of phase space of the cubic NLS on a three-edge star graph. We prove that, contrarily to the case of the standard NLS on the line, the energy associated to the cubic focusing Schr\"odinger equation on the three-edge star graph with a free (Kirchhoff) vertex does not attain a minimum value on any sphere of constant $L^2$-norm. We moreover show that the only stationary state with prescribed $L^2$-norm is indeed a saddle point. \end{abstract} \maketitle \section{Introduction} %The definition of nonlinear evolutions on graphs has %recently drawn an increasing interest %(see e.g. \cite{accn,bellazzini,matrasulov}). %and for more information %on linear dynamics on graphs see \cite{kostrykin,kuchment}). A major issue in nonlinear dynamics consists in the search for stationary solutions and in the study of their stability properties. In the case of hamiltonian systems, a first picture of the phase space can be drawn by identifying critical points of the energy and their nature. In particular it is important to know if a {\em ground state} exists, where a ground state is defined as the minimizer of the energy functional, possibly restricted to suitable submanifolds. In the context of the nonlinear Schr\"odinger (NLS) equation, one typically restricts the energy functional to a manifold on which a second conserved quantity (sometimes called mass, or charge) is constant. From a physical point of view, such a restriction is meaningful, as the extra conserved quantity often represents a physical characteristic of the system (e.g., the mass, or the number of particles). In the one dimensional case, the ground states of the NLS with power nonlinearity on the line are well known and completely described in the classical paper \cite{BL1}, where more general nonlinearities are also treated. It turns out that on the line, the NLS energy constrained to the manifold of the states of constant mass, attains its minimum value in correspondence of a unique (up to translation) positive, symmetric and decreasing (for $x>0$) function. No other critical points of the energy exist. \par\noindent In this paper we are interested in the case of the focusing nonlinear cubic Schr\"odinger equation on a three-edge star graph, sometimes called in the physical literature a Y junction. To put the issue in a physical context, we begin by recalling that the {\it linear} Schr\"odinger equation on a graph is a well developed subject, as an effective description of dynamics of many mesoscopic systems such as, for example, quantum nanowires (see \cite{Ex, kostrykin, Kuchment} and references therein). It is of interest to extend the analysis to the nonlinear wave propagation on networks. In particular, it is well known that the NLS appears as an effective equation in several different areas: the description of Bose condensates, the propagation of electromagnetic pulses in nonlinear (Kerr) media and Langmuir waves in plasma physics. In many situations it seems of interest to treat the propagation of NLS solutions associated to such phenomena in one dimensional ramified structures, the prototype of which is a three-edge star graph, or Y-junction. The subject is at its beginnings and for some preliminary experimental, numerical and analytical works see \cite{Smi, Kevr, Miro, Sob, TOD, Wan}. A first rigorous analysis of nonlinear stationary (or bound) states for NLS on a star graphs is outlined in \cite{ACFN2}, where solitary waves for a star graph with delta vertex are constructed, including the case of a free or Kirchhoff vertex. The Kirchhoff vertex is the closest analogue to the free particle on the line, to which it reduces when the line is considered as a graph with two edges. In the linear case the Kirchhoff laplacian on a graph, in analogy with the laplacian on the line, has only absolutely continuous spectrum, and so only scattering states are possible for the Schr\"odinger dynamics. However, in the presence of a nonlinearity a three-edge star graph with Kirchhoff conditions at the vertex, admits a unique stationary state, which is quite simply described: it is the state on the graph which coincides on every edge with half a soliton. One could suspect that this stationary state is the ground state. On the contrary, we show that, perhaps unexpectedly, this is not the case. This fact highlights that the NLS on graphs, even on the simplest one, exhibits remarkable differences w.r.t. the same evolution equation on the line. In this letter we consider the case of the cubic focusing NLS and analyze the energy constrained to a fixed sphere in $L^2$, i.e. the set of states of prescribed mass. We show that the constrained energy is bounded from below (a fact that can be shown similarly to the case of ${\erre}^n$ and subcritical nonlinearity, see \cite{C} and \cite{ACFN2}), but it approaches the infimum without attaining a minimum value. The more, the only nonlinear stationary state is a saddle point of the constrained energy functional. The existence of a saddle point of the energy is a remarkable feature, inducing on the phase space of the system stable and unstable manifolds which, in the absence of other critical points, are the main structural properties of the dynamics. The consequences of this fact will be investigated in a subsequent paper. \par\noindent \vskip5pt \section{Preliminaries} Before giving the main results, we start by fixing the framework, the notation, and recalling some basic results. \medskip \n 1. A three-edge star graph ${\mathcal G}$ can be thought of as composed by three halflines with a common origin, called {\em vertex}. A {\em state} or {\em wavefunction} on the graph is an element of the Hilbert space $L^2 ({\mathcal G}) \ = \ \bigoplus_{i=1}^3 L^2 (\erre^+; dx_i)$, and can be represented as a column vector, namely $$ \Psi \ = \ \left( \begin{array}{c} \Psi_1 \\ \Psi_2 \\ \Psi_3 \end{array} \right), \qquad \Psi_i \in L^2 (\erre^+). $$ The space $L^2 ({\mathcal G})$ is naturally endowed with the hermitian product $$ ( \Phi, \Psi)_{L^2 ({\mathcal G})} \ = \ \sum_{i=1}^3 \int_0^{+ \infty} \overline{\Phi_i (x_i)}\Psi_i (x_i) \, dx_i. $$ Sobolev and $L^p$-spaces on ${\mathcal G}$ are defined analogously, namely $$ H^s ({\mathcal G}) \ = \ \ \bigoplus_{i=1}^3 H^s (\erre^+), \qquad L^p ({\mathcal G}) \ = \ \ \bigoplus_{i=1}^3 L^p (\erre^+). $$ In the following, we denote by $\| \Psi \|_p$ the norm of the function $\Psi$ in the space $L^p ({\mathcal G})$. When $p=2$ we shall simply write $\| \Psi \|$. \medskip \noindent 2. The dynamics of the system is described by the Schr\"odinger equation \be \label{schrod} i \partial_t \Psi (t) \ = \ - \Delta \Psi (t) - |\Psi (t)|^2\Psi (t), \ee where: \begin{itemize} \item The operator $-\Delta$ acts on the domain \be \nonumber %\label{domkirch} D (- \Delta ) \ : = \ \{ \Psi \in H^2 ({\mathcal G}), \ \Psi_1 (0) = \Psi_2 (0) = \Psi_3 (0), \ \Psi^{\prime}_1 (0) +\Psi^{\prime}_2 (0) + \Psi^{\prime}_3 (0) = 0 \} \ee and its action reads $$ - (\Delta \Psi)_i \ = \ - \Psi_i^{\prime \prime}, \qquad i = 1,2,3. $$ The condition at the vertex is usually referred to as the {\em Kirchhoff's boundary condition}. The operator $- \Delta$ is selfadjoint on $L^2 ({\mathcal G})$. Notice that, on the graph, the laplacian with Kirchhoff boundary conditions is the natural generalization of the free laplacian on the line, as it is easily seen by considering the line as a two-edge star graph, and noticing that the boundary conditions reduces to continuity of the wavefunction and continuity of the derivative at the vertex, i.e. $\Psi\in H^2(\erre)$. \item The nonlinear term in \eqref{schrod} is defined componentwise, namely $$ (|\Psi|^2 \Psi)_i : = |\Psi_i|^2 \Psi_i.$$ \end{itemize} \medskip \n 3. The problem \eqref{schrod} is globally well-posed in $H^1 ({\mathcal G})$ (see \cite{ACFN1}). The $L^2$-norm and the energy \be \begin{split} \label{energy} E (\Psi) & \ = \ \f 1 2 \| \Psi^\prime \|^2 - \f 1 4 \| \Psi \|_4^4 \ = \ \sum_{i=1}^3 \left( \f 1 2 \| \Psi_i^\prime \|_{L^2(\erre^+)}^2 - \f 1 4 \| \Psi_i \|_{L^4(\erre^+)}^4 \right) \end{split} \ee are conserved by time evolution. In the sequel we use the following notation: \be \label{energy1} E_1 (\psi) \ = \ \f 1 2 \| \psi^\prime \|_{L^2(\erre^+)}^2 - \f 1 4 \| \psi \|_{L^4(\erre^+)}^4, \qquad \psi \in H^1 (\erre^+)\ee and \be \label{energy2} E_2 (\psi) \ = \ \f 1 2 \| \psi^\prime \|_{L^2(\erre)}^2 - \f 1 4 \| \psi \|_{L^4(\erre)}^4, \qquad \psi \in H^1 (\erre). \ee \medskip \n 4. Let us recall two well-known results on minimization for the cubic NLS on one- and two-edge star graphs (i.e., on the halfline and on the line): \begin{enumerate} \item The minimum of the functional $E_1$ on the functions in $H^1 (\erre^+)$ with squared $L^2$-norm equal to $m > 0$ is achieved on the function (up to a phase factor) $$ \phi_m (x) \ = \ \f m {\sqrt 2} \cosh^{-1} \left( \f m 2 x \right), \qquad x \geq 0,$$ and gives \be \label{min1} E_1 (\phi_m ) \ = \ - \f {m^3} {24}. \ee \item The minimum of the functional $E_2$ on the functions in $H^1 (\erre)$ with squared $L^2$-norm equal to $m > 0$ is achieved on the functions (up to a phase factor) \be \label{solit} \varphi_m^y (x) \ = \ \f m {2 \sqrt 2} \cosh^{-1} \left( \f m 4 (x-y) \right), \qquad x \in \erre \ee and gives \be \label{min2} E_2 (\varphi_m^y ) \ = \ - \f {m^3} {96}. \ee \end{enumerate} \section{Results} As recalled in the introduction the energy functional \eqref{energy} at fixed $L^2$ norm is bounded from below for subcritical nonlinearity, which is our case.\par\noindent Let us introduce the following family of states: \begin{definition} We call {\em sesquisoliton} (i.e. ``one and half" soliton) any function of the form \be \label{sesqui} \Phi_{m_1, m_2}^x (x_1,x_2,x_3) : = \left( \begin{array} {c} \f{m_1}{\sqrt 2} \cosh^{-1} \left(\f{m_1} 2 x_1\right) \\ \f{m_2}{2 \sqrt 2} \cosh^{-1} \left(\f{m_2}{4 } (x_2-x)\right) \\ \f{m_2}{2 \sqrt 2} \cosh^{-1} \left(\f{m_2}{4 } (x_3+x)\right) \end{array} \right) \ee where $0 < m_1 \leq m_2$, $x \geq 0$, and the following condition of ``continuity at the vertex'' holds: \be \label{kirch} m_1 \ = \ \f {m_2} 2 \cosh^{-1} \left( \f{m_2} 4 x\right). \ee \end{definition} \begin{figure}[h!] \begin{center} \includegraphics[width=0.75\textwidth]{sesqui-11feb12.pdf} \caption{\label{fig1} A sesquisoliton on the three-edge star graph.} \end{center} \end{figure} \n Notice that the sesquisolitons are obviously elements of $H^1 ({\mathcal G})$. As a matter of fact they also belong to $D (- \Delta )$. \par\noindent Moreover, in the case $x=0$ one has $m_1=\frac{m_2}{2}$, and one obtains a symmetric configuration with three half solitons concurring at the vertex. In \cite{ACFN2} it is shown that this is a standing wave for NLS equation \eqref{schrod}. \par\noindent Now, we define the manifold of the sequisolitons with fixed $L^2$-norm as follows: \be \nonumber %\label{mani-sesqui} {\mathcal S}_M : = \{ \Phi_{m_1, m_2}^x, \ \| \Phi_{m_1, m_2}^x \|^2 = M \}. \ee \begin{theorem} \label{result} For any $\Psi$ such that $\| \Psi \|^2 = M$, the following chain holds: \be \label{inf} E (\Psi) > \inf_{\| \Psi \|^2 = M} E (\Psi) = \inf_{\Psi \in {\mathcal S}_M} E (\Psi) = - \f {M^3} {96}. \ee \end{theorem} \begin{proof} Given $\Psi \in L^2 ({\mathcal G})$, it is possible to construct a sesquisoliton with the same $L^2$-norm but with lower energy. We proceed as follows. Let us suppose that \be \label{masses} \| \Psi_1 \| \leq \min (\| \Psi_2 \|, \| \Psi_3 \|). \ee Then, consider the sesquisoliton \eqref{sesqui} with $m_1 = \| \Psi_1 \|^2, m_2 = \| \Psi_2 \|^2 + \| \Psi_3 \|^2$, and $x \geq 0$ chosen in order to satisfy the condition \eqref{kirch}. Notice that such a choice is always possible since $2 m_1 \leq m_2$, and is unique. If the condition \eqref{masses} is not fulfilled, then one first relabels the edges in order to have the minimal mass on the first one, and thus proceeds as before. It is immediately seen that $ \| \Psi_1 \|^2 = \| (\Phi_{m_1, m_2}^x )_1 \|^2 = m_1$ and $ \| \Psi_2 \|^2 + \| \Psi_3 \|^2 = \| (\Phi_{m_1, m_2}^x )_2 \|^2 + \| (\Phi_{m_1, m_2}^x )_3 \|^2 = m_2$, thus $\| \Phi_{m_1, m_2}^x \|_2^2 = M$. \n Let us define the following function on the real line: \be \label{raddrizza} \begin{split} \psi (\xi) \ : = & \ \left\{ \begin{array}{c} \Psi_2 (- \xi), \qquad \xi < 0 \\ \Psi_3 (\xi), \qquad \xi > 0 \end{array} \right. \end{split} \ee and notice that, by \eqref{solit} and \eqref{sesqui}, \be \label{raddrizza2} \begin{split} \varphi_{m_2}^{-x} (\xi) \ : = & \ \left\{ \begin{array}{c} (\Phi_{m_1,m_2}^x)_2 (- \xi), \qquad \xi < 0 \\ (\Phi_{m_1,m_2}^x)_3 (\xi), \qquad \xi > 0 \end{array} \right. \end{split} \ee Furthermore, by \eqref{min1}, \eqref{min2}, \eqref{energy}, \eqref{energy1}, \eqref{energy2}, \eqref{raddrizza}, and \eqref{raddrizza2} one immediately has the following chain of inequalities \be \nonumber \begin{split} E (\Psi) & \ = \ E_1 (\Psi_1) + E_2 (\psi) \ \geq \ E_1 ( \phi_{m_1} ) + E_2 (\varphi_{m_2}^{-x}) \ = \ E (\Phi_{m_1, m_2}^x ) %+ E_2 (\phi_{m_2}^x) \\ %& \ = \ E_1 ( \phi %E_1 ((\Phi_{m_1, m_2}^x )_1) + E_2 (\chi_- (\Phi_{m_1, m_2}^x )_2 + %\chi_+ (\Phi_{m_1, m_2}^x )_3) \ = \ E_1 (\phi_M) + E_2 (\varphi_M) \end{split} \ee so the first of the two identities in \eqref{inf} is proven. To prove the second, we use \eqref{min1} and \eqref{min2} and obtain \be \nonumber %\label{e-sesqui} E (\Phi_{m_1, m_2}^x) \ = \ - \f {m_1^3}{24} - \f {m_2^3}{96}, \ee so, noting that $M = m_1 + m_2$, we obtain \be \label{e-sesqui-2} E (\Phi_{m_1, m_2}^x) \ = \ - \f {m_1^3}{24} + \f {(m_1-M)^3}{96}, \ee where $m_1$ plays the role of a parameter. We stress that, due to the constraint of the mass of the soliton, $m_1$ can vary in the interval $(0, M/3]$. Differentiating \eqref{e-sesqui-2}, one immediately has that $E (\Phi_{m_1, m_2}^x)$ is monotonically increasing in such interval, so $$ \inf_{\Psi \in {\mathcal S}_M} E (\Psi) \ = \ \lim_{m_1 \to 0+} E (\Phi_{m_1, m_2}^x) \ = \ - \f {M^3}{96}. $$ To complete the proof, we must show that, for any $\Psi$ in $H^1({\mathcal G})$, $E(\Psi)$ is strictly larger than $- M^3/96$. To this aim, we notice that such an infimum cannot be achieved, as for $m_1 =0$ the condition \eqref{kirch} does not correspond to an admissible sesquisoliton, so it cannot be fulfilled. \end{proof} \begin{cor} The sesquisoliton $\Phi_{M/3, 2M/3}^0$ is a saddle point for the energy functional. \end{cor} \begin{proof} First, we notice that $\Phi_{M/3, 2M/3}^0$ is a critical point. Indeed, it satisfies the Euler-Lagrange equation for the energy functional constrained on the manifold $\| \Psi \|^2 = M$, namely $$ - \Delta \Psi - | \Psi |^2 \Psi + \omega \Psi \ = \ 0, $$ where $\omega$ is a Lagrange multiplier, coinciding with $\f {M^2}{36}$. In order to prove that $\Phi_{M/3, 2M/3}^0$ is a saddle point, it is sufficient to show that it maximizes the energy restricted to a curve to which it belongs in the constraint manifold, and minimizes the energy when restricted to a different curve. By the proof of theorem \eqref{result} we have a curve on which $\Phi_{M/3, 2M/3}^0$ is a maximum of the energy, that is, the curve made of sesquisolitons parametrized by $m_1$. On the other hand, by \eqref{min1} we know that $\Phi_{M/3, 2M/3}^0$ minimizes the energy at fixed mass on any edge. Then, it is a minimum of the energy restricted to the submanifold of function with $\Psi_1 = \Psi_2 = \Psi_3$. \end{proof} \vskip 10pt \begin{figure}[h!] \begin{center} \includegraphics[width=0.75\textwidth]{3halfsol-11feb12.pdf} \caption{\label{fig2}The unique stationary state of the cubic NLS on the three-edge star graph.} \end{center} \end{figure} \n The previous result can be extended to all star graphs with a similar construction, and a more systematic analysis of the character of stationary states on star graphs will be given in a future work. \bigskip \n {\bf Acknowledgements.} R. A. is partially supported by the PRIN2009 grant ``Critical Point Theory and Perturbative Methods for Nonlinear Differential Equations''. \vskip20pt \begin{thebibliography}{99} \bibitem{ACFN1}Adami R., Cacciapuoti C., Finco D., Noja D.: Fast solitons on star graphs, Rev Math. Phys. {\bf 23}, 4, 409-451 (2011) \bibitem{ACFN2}Adami, R. Cacciapuoti C., Finco D., Noja D.: Stationary states of NLS on star graphs arXiv:1104.3839v2 (2011) \bibitem{BL1} Berestycki H., Lions P.L.: Nonlinear scalar field equations I and II, Arch. Rat. Mech. Anal. {\bf 82}, 313-375 (1983) \bibitem{C} Cazenave T.: Semilinear Schr\"odinger equations, AMS (2003) \bibitem{Ex} Exner P., Keating J.P., Kuchment P., Sunada T., and Teplyaev A.: Analysis on graphs and its applications, AMS (2008). \bibitem{Smi}Gnutzmann, S. Smilansky, U., Derevyanko S.: Stationary scattering from a nonlinear network, Phys. Rev. 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E., Molina M.I., Kivshar Y.S.: Localized modes and bistable scattering in nonlinear network junctions, Phys. Rev. E {\bf 75}, 046602 (2007) \bibitem{Wan} Wan, W. Muenzel S. Fleischer, W.: Wave Tunneling and Hysteresis in Nonlinear Junctions, Phys. Rev. Lett. {\bf 104}, 073903 (2010) \end{thebibliography} \end{document} \end{document} In recent years, the spectacular development of experimental techniques in condensed matter and ultracold gases, has magnified the interest in the mathematical modeling of Bose-Einstein condensates, i.e., in the Gross-Pitaevskii equation, namely \begin{equation} \label{gp} i \partial_t \psi_t (x) \ = \ - \Delta \psi_t (x) \pm \alpha | \psi_t (x) |^2 \psi_t (x) + V (x) \psi_t (x), \qquad t \in \erre, \, x \in \erre^d \end{equation} where the unknown $\psi_t (x)$ is the wavefunction of the condensate, $V(x)$ models the action of an external potential, and the cubic term summarizes the effects of the two-body interactions among the particles in the condensate. It is well-known that the coupling constant $\alpha$ is proportional to the scattering length of such a two-body interaction. The function $V$ can be carachterized by the same lengthscale of the condensate, for instance when it models the trap confining the system, or by a much shorter lengthscale, for instance when it describes the effect of an inhomogeneity, or of an impurity. In this paper we focus on the latter case, restrict our study to the so-called cigar-shaped condensates, i.e., effectively one-dimensional systems, and restrict to the choice $$V (x) = \delta^\prime_0,$$ where, according to the theory of Schr\"odinger operators with point interactions in dimension one (\cite{albeverio, dombroski}), the symbol $\delta^\prime_0$ does not actually refers to a distributional derivative of a Dirac's delta potential, but rather to the choice of a particular, self-adjoint boundary condition at the origin of the real axis. Moreover, we restrict our study to the focusing case (i.e. with the minus sign in the nonlinearity in \eqref{gp}) and put equal to one the qualitatively irrelevant constant $\alpha$. Summarizing all the hypotheses, we aim at explicitly finding all stationary solutions to the equation \begin{equation} \label{problem} i \partial_t \psi_t (x) \ = \ H \psi_t (x) \pm \alpha | \psi_t (x) |^2 \psi_t (x), \qquad t \in \erre, \, x \in \erre, \end{equation} where $H$ is the so-called ``Schr\"odinger operator with a $\delta^\prime$ potential'', i.e., the self-adjoint operator defined on the domain \be D (H) \ : = \ \{ \psi \in H^2 (\erre \backslash \{0 \}, \ \psi (0+) - \psi (0-) = - \psi^\prime (0+) = - \psi^\prime (0-) \}, \ee whose action reads \be (H \psi) (x) \ = \ \psi^{\prime \prime} (x). \ee It is well-known that the standard nonlinear Schr\"odinger equation in dimension one is an integrable system (\cite{magri}) and that, adding a $\delta$-interaction, the stationary states of the system can still be exactly computed (\cite{fukuzumi, fukujean, lacozza}). In analogy with these two cases, the stationary states for the problem \eqref{problem} prove exactly solvable. In addition, as shown in \cite{an2}, the structure of the family of ground states is here much richer, as it exhibits a sypontaneous symmetry-breaking bifurcation. Preliminarily, let us fix some notation. NOTATION Furthermore, we recall some basic results on well-posedness and conservation laws WP and CL In the next section we explicitly find the ground states, recall results on stability contained in \cite{an2}, and show the appearence of the bifurcation. In the last section we explicitly compute the excited states. \section{Stationary states} We call {\em stationary state} any solution to equation \eqref{problem} of the type \be \label{stat} \psi_\om (t,x) \ = \ e^{i \om t} \phi_\om (x). \ee As a consequence, the square-integrable function $\phi_\om$ must solve the {\em stationary Schr\"odinger equation} \be \label{stat-eq} H \phi_\om - | \phi_\om |^2 \phi_\om + \om \phi_\om \ = \ 0. \ee Equation \eqref{stat-eq} can be rephrased as \be \label{stat-eq} \begin{split} & - \phi_\om^{\prime \prime} - | \phi_\om |^2 \phi_\om + \om \phi_\om \ = \ 0, \qquad x \neq 0 \\ & \phi_\om (0+) - \phi_\om (0-) \ = \ - \phi_\om^\prime (0+) \ = \ - \phi_\om^\prime (0-). \end{split} \ee In the following we reduce equation \eqref{stat-eq} to a couple of systems of two equations in two real unknowns. For the convenience of the reader, we preliminary solve such systems. \begin{prop} The solutions $(t_1, t_2)$ to the system \be \label{tipiu} T_+ : = \left\{ \begin{array}{ccc} t_1^2 - t_1^4 & = & t_2^2 - t_2^ 4 \\ & & \\ t_1^{-1} - t_2^{-1} & = & \sqrt \om \end{array} \right. \ee with the condition $0 \leq t_1, t_2 \leq 1$, are given by \be \begin{split} \label{solt+} t_1 \ = & \ \f{1 - \sqrt{1 + \om} + \sqrt{ \om - 2 + 2 \sqrt{1 + \om}}} {2 \sqrt \om} \\ t_2 \ = & \ \f{- 1 + \sqrt{1 + \om} + \sqrt{ \om - 2 + 2 \sqrt{1 + \om}}} {2 \sqrt \om} \end{split} \ee Such solutions exist for any $\om > 0$. \end{prop} \begin{proof} the first equation in \eqref{tipiu} rewrites as $$ (t_1^2 - t_2^2) (t_1^2 + t_2^2 - 1) \ = \ 0, $$ so the system splits into two subsystems $T_{1,+}$ and $T_{2,+}$, defined by \be T_{1,+} : = \left\{ \begin{array}{ccc} t_1 & = & t_2 \\ & & \\ t_1^{-1} - t_2^{-1} & = & \sqrt \om \end{array} \right. \nonumber \ee and \be T_{2,+} : = \left\{ \begin{array}{ccc} t_1^2 + t_2^2 & = & 1 \\ & & \\ t_1^{-1} - t_2^{-1} & = & \sqrt \om \end{array} \right. \nonumber \ee The system $T_{1,+}$ can be solved only for $\om = 0$, so we are not interested in such a solution Let us consider the system $T_{2,+}$. Multiplying the second equation by $t_1t_2$ one obtains \be \label{summa} t_1 - t_2 \ = \ \sqrt \om t_1 t_2, \ee thus squaring both sides, using the first equation of $T_2$ and imposing positivity, we obtain $$ t_1 t_2 \ = \ \f{\sqrt{1 + \om} - 1} {\om} $$ and so, by \eqref{summa} \be \label{strauss} t_2 - t_1 \ = \ \f{\sqrt{1 + \om} - 1}{\sqrt \om}. \ee By \eqref{strauss} and the first equation of $T_{2,+}$, we finally get \eqref{solt+}. % %\be \label{kahn} %t_{1,2} \ = \ \f{ \beta \pm \sqrt{2 - \beta^2}} 2 %\ee %with $\beta : = \f{1 + \sqrt{1 + \gamma^2 \om}}{\gamma \sqrt % \om}$. The positivity of the quantity under square yields %the condition $\om > \om^*$. By \eqref{kahn} the proof of 2. is complete. \end{proof} \begin{prop} The solutions $(t_1, t_2)$ to the system \be \label{timeno} T_- : = \left\{ \begin{array}{ccc} t_1^2 - t_1^4 & = & t_2^2 - t_2^ 4 \\ & & \\ t_1^{-1} + t_2^{-1} & = & \sqrt \om \end{array} \right. \ee with the condition $0 \leq t_1, t_2 \leq 1$, can be classified as follows: \begin{enumerate} \item For $0 < \om \leq 4$ there are no solutions. \item For $\om > 4$ there esists the solution \be \label{t-symm} t_1 \ = \ t_2 \ = \ \f 2 {\sqrt \om}. \ee \item For $\om > 8$ there exist the two solutions \be \begin{split} \label{solt-asym} t_1 \ = & \ \f{1 + \sqrt{1 + \om} \pm \sqrt{ \om - 2 - 2 \sqrt{1 + \om}}} {2 \sqrt \om} \\ t_2 \ = & \ \f{ 1 + \sqrt{1 + \om} \mp \sqrt{ \om - 2 - 2 \sqrt{1 + \om}}} {2 \sqrt \om} \end{split} \ee \end{enumerate} \end{prop} \begin{proof} As in the previous proof, we rewrite the first equation in \eqref{solt-asym} as $$ (t_1^2 - t_2^2) (t_1^2 + t_2^2 - 1) \ = \ 0, $$ so the system \eqref{solt-asym} is equivalent to the union of the systems \be T_{1,-} : = \left\{ \begin{array}{ccc} t_1 & = & t_2 \\ & & \\ t_1^{-1} + t_2^{-1} & = & \sqrt \om \end{array} \right. \nonumber \ee and \be T_{2,-} : = \left\{ \begin{array}{ccc} t_1^2 + t_2^2 & = & 1 \\ & & \\ t_1^{-1} + t_2^{-1} & = & \sqrt \om \end{array} \right. \nonumber \ee The system ${T_{1,-}}$ gives the solution \eqref{t-symm}. The condition $\om > 4$ emerges by the prescription $t_1, t_2 < 1$, so point {\em 1.} is proven. Let us consider the system $T_{2,-}$. Multiplying the second equation by $t_1t_2$ one obtains \be \label{summa2} t_1 + t_2 \ = \ \sqrt \om t_1 t_2, \ee thus squaring both sides, using the first equation of $T_{2,-}$ and imposing positivity, we have $$ t_1 t_2 \ = \ \f{1 + \sqrt{1 + \om}} { \om} $$ and so, by \eqref{summa2} \be \label{strauss2} t_1 + t_2 \ = \ \f{1 + \sqrt{1 + \om}}{ \sqrt \om}. \ee By \eqref{strauss2} and the first equation of ${T_{2,-}}$, we finally get \eqref{solt-asym}. The positivity of the quantity under the square root of \eqref{solt-asym} gives the condition $\om > 8$, so the proof is complete. \end{proof} Now we can explicitly give all the solutions to \eqref{eq-stat}. \begin{theorem}\label{statstat} The stationary states of equation \eqref{problem} can be classified as follows: \begin{enumerate} \item The unperturbed symmetric solitary wave \be \label{unperturbed} \phi_{\om}^{0,0} \ : = \ \sqrt{2 \om} \cosh^{-1} (\sqrt\om x). \ee Such solutions are present for any $\om > 0$. \item The asymmetric non-changing sign solutions %, as either $\phi_{\om,+}^{x_1, x_2}$ or % $\phi_{\om,-}^{x_1, x_2}$, where \be \label{stazzionario2} \phi_{\om, \pm}^{x_1, x_2} \ : = \ \begin{array}{ccc} \sqrt{2 \om} \cosh^{-1} (\sqrt\om (x-x_1)), & & x< 0 \\ \sqrt{2 \om} \cosh^{-1} (\sqrt\om (x-x_2)), & & x< 0, \end{array} \ee \be \begin{split} x_1 \ = & \ \f 1 {2 \sqrt \om} \log \f{1+ t_1} {1 - t_1}, \quad x_2 \ = \ \f 1 {2 \sqrt \om} \log \f{1+ t_2} {1 - t_2} \\ \\ x_1 \ = & \ - \f 1 {2 \sqrt \om} \log \f{1+ t_2} {1 - t_2}, \quad x_2 \ = \ - \f 1 {2 \sqrt \om} \log \f{1+ t_1} {1 - t_1} \end{split} \ee, where the couple $(t_1, t_2)$ solves the system \eqref{t1}. \noindent Such solutions are present for any $\om > 0$. \item The changing sign solutions, which, in turn, can be classified as \begin{enumerate} \item the antisymmetric solutions \be \label{stazzionario} \phi_{\om, -}^{\bar x, - \bar x} \ : = \ \begin{array}{ccc} \sqrt{2 \om} \cosh^{-1} (\sqrt\om (x- \bar x)), & & x< 0 \\ - \sqrt{2 \om} \cosh^{-1} (\sqrt\om (x + \bar x)), & & x > 0, \end{array} \ee where \be \label{xbar} \bar x = \f 1 {2 \sqrt \om} \log \f{\sqrt \om + 2} {\sqrt \om - 2}. \ee Such solutions are present for $\om > 4$. \item \be \label{stazzionario2} \phi_{\om, \pm}^{x_1, x_2} \ : = \ \begin{array}{ccc} \sqrt{2 \om} \cosh^{-1} (\sqrt\om (x-x_1)), & & x< 0 \\ - \sqrt{2 \om} \cosh^{-1} (\sqrt\om (x-x_2)), & & x< 0, \end{array} \ee \be \begin{split} x_1 \ = & \ \f 1 {2 \sqrt \om} \log \f{1+ t_1} {1 - t_1}, \quad x_2 \ = \ - \f 1 {2 \sqrt \om} \log \f{1+ t_2} {1 - t_2} \\ x_1 \ = & \ \f 1 {2 \sqrt \om} \log \f{1+ t_2} {1 - t_2}, \quad x_2 \ = \ - \f 1 {2 \sqrt \om} \log \f{1+ t_1} {1 - t_1} \end{split} \ee, where the couple $(t_1, t_2)$ solves the system \eqref{timeno} \noindent Such solutions are present for any $\om > 8$. \end{enumerate} \end{enumerate} \end{theorem} \begin{proof} Equation \eqref{eqstat} can be separately solved in the positive and in the negative halfline. It is well-known that the whole family of $L^2$-solutions is given by \be \phi_\om^{x_1, x_2, \theta_1, \theta_2} \ : = \ \begin{array}{ccc} e^{i \theta_1} \sqrt{2 \om} \cosh^{-1} (\sqrt\om (x-x_1)), & & x< 0 \\ e^{i \theta_2} \sqrt{2 \om} \cosh^{-1} (\sqrt\om (x-x_2)), & & x< 0. \end{array} \ee Since both equations \eqref{problem}, \eqref{stat-eq} are invariant under multiplication by a phase factor, one can always fix $\theta_1 = 0$. Furthermore, the matching conditions at the origin can hold only if either $\theta_2 = 0$ or $\theta_2 = \pi$, so the family of solution to \eqref{eq-stat} simplifies to \be \label{stazzionario} \phi_{\om, \pm}^{x_1, x_2} \ : = \ \begin{array}{ccc} \sqrt{2 \om} \cosh^{-1} (\sqrt\om (x-x_1)), & & x< 0 \\ \pm \sqrt{2 \om} \cosh^{-1} (\sqrt\om (x-x_2)), & & x< 0. \end{array} \ee To fix parameters $x_1$ and $x_2$ one imposes the matching conditions, and obtains that the couples $(x_1, x_2)$ are to be determined as follows: \noindent -- For the family of changing-sign solutions $\phi_{\om,+}^{x_1, x_2}$, one solves the system \be T_+ : = \left\{ \begin{array}{ccc} t_1^2 - t_1^4 & = & t_2^2 - t_2^ 4 \\ & & \\ t_1^{-1} - t_2^{-1} & = & \sqrt \om \end{array} \right. , \qquad 0 \leq t_1, t_2 \leq 1. \nonumber \ee Then, the possible couples $(x_1, x_2)$ are two, and they are determined by \be \begin{split} x_1 \ = & \ \f 1 {2 \sqrt \om} \log \f{1+ t_1} {1 - t_1}, \quad x_2 \ = \ \f 1 {2 \sqrt \om} \log \f{1+ t_2} {1 - t_2} \\ x_1 \ = & \ - \f 1 {2 \sqrt \om} \log \f{1+ t_2} {1 - t_2}, \quad x_2 \ = \ - \f 1 {2 \sqrt \om} \log \f{1+ t_1} {1 - t_1} \end{split} \ee \noindent -- For the family of changing sign solutions $\phi_{\om,-}^{x_1, x_2}$, one solves the system \be T_- : = \left\{ \begin{array}{ccc} t_1^2 - t_1^4 & = & t_2^2 - t_2^ 4 \\ & & \\ t_1^{-1} + t_2^{-1} & = & \sqrt \om \end{array} \right. , \qquad 0 \leq t_1, t_2 \leq 1. \nonumber \ee Then, the possible couples $(x_1, x_2)$ are two, and they are determined by \be \begin{split} x_1 \ = & \ \f 1 {2 \sqrt \om} \log \f{1+ t_1} {1 - t_1}, \quad x_2 \ = \ - \f 1 {2 \sqrt \om} \log \f{1+ t_2} {1 - t_2} \\ x_1 \ = & \ \f 1 {2 \sqrt \om} \log \f{1+ t_2} {1 - t_2}, \quad x_2 \ = \ - \f 1 {2 \sqrt \om} \log \f{1+ t_1} {1 - t_1} \end{split} \ee \end{proof} By a direct comptutation, one can prove the following \begin{proposition} The $L^2$-norm, the $L^4$-norm, and the energy of the stationary states found in Theorem \ref{statstat}, are explicitly given by \begin{eqnarray} \| \psi_{\om,+}^{0,0} \|_2^2 \ = \ 4 \sqrt \om & \| \psi_{\om,+}^{0,0} \|_4^4 \ = \ \f {16} 3 \om^{\f 3 2} & E ( \psi_{\om,+}^{0,0} ) \ = \ - \f 2 3 \om^{\f 3 2} \\ S_\om ( \psi_{\om,+}^{0,0} ) \ = \ \f 4 3 \om^{\f 3 2} & & \\ \| \psi_{\om,+}^{x_1,x_2} \|_2^2 \ = \ 2 \sqrt \om \left( 2 + t_2 - t_1 \right) & & \| \psi_{\om,+}^{x_1,x_2} \|_4^4 \ = \ 4 \om^{\f 3 2} \left[ \f 4 3 + (t_2 - t_1) - \f 1 3 (t_2^3 - t_1^3) \\ E (\psi_{\om,+}^{x_1,x_2}) \ = \ - \f 2 3 \om^{\f 3 2} + \f {2 - \om} 3 (\sqrt{\om + 1} - 2 ) & & S_\om (\psi_{\om,+}^{x_1,x_2}) \ = \ \f 4 3 \om^{\f 3 2} + \f 2 3 (\om + 1)^{\f 3 2} - \om - \f 2 3 \\ \| \psi_{\om,-}^{\bar x, - \bar x} \|_2^2 \ = \ 4 \sqrt \om - 8 & \| \psi_{\om,+}^{\bar x, - \bar x} \|_4^4 \ = \ \f {16} 3 \om^{\f 3 2} - 16 \om + \f {64} 3 \\ E (\psi_{\om,+}^{\bar x, - \bar x}) \ = \ \f {16} 3 - \f 2 3 \om^{\f 3 2} & S_\om (\psi_{\om,+}^{\bar x, - \bar x}) \ = \ \f {16} 3 - 4 \om + \f 4 3 \om^{\f 3 2} + \f 2 3 (\om + 1)^{\f 3 2} - \om - \f 2 3 \\ \| \psi_{\om,-}^{x_1,x_2} \|_2^2 \ = \ 2 \sqrt \om \left( 2 + t_2 - t_1 \right) & & \| \psi_{\om,+}^{x_1,x_2} \|_4^4 \ = \ 4 \om^{\f 3 2} \left[ \f 4 3 + (t_2 - t_1) - \f 1 3 (t_2^3 - t_1^3) \\ E (\psi_{\om,+}^{x_1,x_2}) \ = \ - \f 2 3 \om^{\f 3 2} + \f {2 - \om} 3 (\sqrt{\om + 1} - 2 ) & & S_\om (\psi_{\om,+}^{x_1,x_2}) \ = \ \f 4 3 \om^{\f 3 2} + \f 2 3 (\om + 1)^{\f 3 2} - \om - \f 2 3 \\ \section{Energy of the stationary states} \section{Bifurcations} ere we specialize to the case with $\mu = 1$, i.e., the cubic nonlinearity. As widely known, the unperturbed problem gives an integrable system. The presence of a defect prevents the system from being fully integrable, nevertheless it remains possible to perform exact computations. The results can be summarized as follows \begin{theorem} \label{cubicat} Fix $\mu = 1$. Then, the solution to the system \eqref{tsystem} are \begin{enumerate} \item The symmetric one $t_1 = t_2 = \f 2 {\gamma \sqrt \om}$, for any $\om > \om_0$. \item The asymmetric one \be \label{cubicasimm} \begin{split} t_1 \ : = \ & \f {1 + \sqrt{1 + \gamma^2 \om} - \sqrt{\gamma^2 \om - 2 - 2 \sqrt{1 + \gamma^2 \om}}} {2 \gamma^2 \om} \\ t_2 \ : = \ & \f {1 + \sqrt{1 + \gamma^2 \om} + \sqrt{\gamma^2 \om - 2 - 2 \sqrt{1 + \gamma^2 \om}}} {2 \gamma^2 \om}, \end{split} \ee and the solution obtained exchanging $t_1$ and $t_2$, for any $\om > \om^*$. \end{enumerate} \end{theorem} \begin{proof} Fixed $\mu = 1$, the first equation in \eqref{tsystem} rewrites as $$ (t_1^2 - t_2^2) (t_1^2 + t_2^2 - 1) \ = \ 0, $$ so the system splits into two subsystems $T_1$ and $T_2$, defined by \be T_1 : = \left\{ \begin{array}{ccc} t_1 & = & t_2 \\ & & \\ t_1^{-1} + t_2^{-1} & = & \gamma \sqrt \om \end{array} \right. \nonumber \ee and \be T_2 : = \left\{ \begin{array}{ccc} t_1^2 + t_2^2 & = & 1 \\ & & \\ t_1^{-1} + t_2^{-1} & = & \gamma \sqrt \om \end{array} \right. \nonumber \ee The system $S_1$ gives the symmetric solution. The condition $\om > \om_0$ emerges by the prescription $t_1, t_2 < 1$, so point {\em 1.} is proven. Let us consider the system $T_2$. Multiplying the second equation by $t_1t_2$ one obtains \be \label{summa} t_1 + t_2 \ = \ \gamma \sqrt \om t_1 t_2, \ee thus squaring both sides, using the first equation of $T_2$ and imposing positivity, we have $$ t_1 t_2 \ = \ \f{1 + \sqrt{1 + \gamma^2 \om}} {\gamma^2 \om} $$ and so, by \eqref{summa} \be \label{strauss} t_1 + t_2 \ = \ \f{1 + \sqrt{1 + \gamma^2 \om}}{\gamma \sqrt \om}. \ee By \eqref{strauss} and the first equation of $S_2$, we finally get \be \label{kahn} t_{1,2} \ = \ \f{ \beta \pm \sqrt{2 - \beta^2}} 2 \ee with $\beta : = \f{1 + \sqrt{1 + \gamma^2 \om}}{\gamma \sqrt \om}$. The positivity of the quantity under square yields the condition $\om > \om^*$. By \eqref{kahn} the proof of 2. is complete. \end{proof} The evaluation of the action $S_\om$ on the ground states too can be explicitly computed. One obtains \begin{prop} Fix $\mu = 1$. Then, \be \label{dcubic} d (\om) \ = \ \left\{ \begin{array} {ccc} \f {2 \lambda} 3 \om^{\f 3 2} \left( 2 - \f 6 {\gamma \sqrt \om} + \f 8 {\gamma^3 \om^{\f 3 2}} \right), & & \om_0 < \om \leq \om^* \\ & & \\ \f \lambda 3 \om^{\f 3 2} \left( 4 - \f{1 + \sqrt{1 + \gamma \sqrt \om}} {\gamma \sqrt \om} \left( 2 + \f {1 + \sqrt{1 + \gamma \sqrt \om}} {\gamma^2 \om} \right)\right), & & \om > \om^*. \end{array} \right. \ee \end{prop} \begin{proof} We preliminarily recall that \be \label{intbase} \int_0^\infty \f{dx}{\cosh^4( \sqrt \om (x-z))} \ = \ \f 1 {3 \om^{\f 32 }} ( 2 + 3t - t^3), \qquad t : = \tanh (\sqrt \om z). \ee Since $d (\om) \ = \ \f \lambda 4 \| \psi_\om \|_4^4$, where $\psi_\om$ is any ground state with frequency $\om$, the computation for the case $\om \leq \om^*$ can be immediately performed and provides the first formula in \eqref{dcubic}. For $\om > \om^*$, using \eqref{intbase} gives \be d (\om) \ = \ \lambda \om^{\f 3 2} \left[ \f 4 3 - (t_1 + t_2) + \f{(t_1 + t_2)^3} 3 \right]. \ee By \eqref{strauss} the computation can be completed up to obtaining the second formula in \eqref{dcubic}, and this completes the proof. \end{proof} \end{document} exact solvability of the model holds for the standard nonlinear Sc One of the most celebrated features of the point interactions (PI) lies in their capability of supplying exactly solvable models (\cite{[AGHH],[AK]}). For this reason, PI have been widely employed to construct toy models and for pedagogical purposes. Nonetheless, their exactness proves useful also when PI are involved in models of treal physical systems: the more a physically relevant quantity (e.g. energy spectrum or time evolution) is explicitly known, the more information can be extracted. In particular, PI fit well the needs of modeling the so-called defects, namely, small inhomogeneities in a medium where a wave propagates, under the hypothesis that the details of the internal structure of the inhomogeneity are not relevant, so that its action can be modeled as concentrated at a point. More precisely, the smallness of the inhomogeneity is to be evaluated with respect to the typical wavelength of the incoming waves, or equivalently, in the case of a quantum system, to the width of the wave function. In this paper we would like to address the analysis of effects of the interaction between nonlinearity and point defects in the behaviour of solutions of nonlinear Schr\"odinger (NLS) equation. We prefer not to enter in a description of the vast field of application of the NLS equation, from the theory of integrable systems and inverse scattering to the propagation of amplitude envelope of waves. We quote just two relevant applications of the NLS as an effective model for real physical systems: dynamics of Bose-Einstein condensates and laser beam propagation in nonlinear (Kerr) media. In both cases it is physically meaningful to consider the propagation of NLS waves in the presence of defects. In particular, the recent spectacular development of both theoretical research and experimental technology involving Bose-Einstein condensates (BEC) (see \cite{stringari} and references therein, and \cite{[BKP], [BKP2],bcstv}) provides point interactions with a wide range of applications. As widely known, in current experiments the formation of a BEC is induced in bounded region of spaces, usually delimited by magnetic and/or optical traps. In such situations, the condensate lies in a one-particle quantum state, whose corresponding wavefunction is characterized as the minimizer of the Gross-Pitaevskii functional. When the trap is removed, the wavefunction of the BEC spreads out according to the evolution prescribed by the cubic Schr\"odinger equation \be \label{gp} i \partial_t \psi (t,x) \ = \ - \partial_x^2 \psi (t,x) + \alpha | \psi (t,x) |^2 \psi (t,x), \ee where we denoted by $\psi$ the wave function of the condensate, and the space variable $x$ belongs to $\erre$, $\erre^2$ or $\erre^3$ according to the fact that we are modeling a {\em cigar-shaped} or a {\em disc-shaped} or a genuinely three-dimensional BEC. We recall that the nonlinearity carries the information that, even though its state is an actual one-particle state, the condensate consists of a large number of interacting particles (in the experimentally realized condensates, at least thousands); the fact that the nonlinearity is cubic means that the dynamical effects of the two-particle interactions overwhelm the effects of many-body collisions. The strength of the nonlinear term, given by the constant $\alpha$, is proportional to the scattering length of the two-body interaction between the particles. Here we do not summarize the progress in the derivation of \eqref{gp} as an effective equation for a many-body quantum system. The interested reader is referred to \cite{esy1,esy2,esy3} for the three-dimensional problem, \cite{kss} for the two-dimensional case, and to \cite{agt,abgt,ab} for the case of cigar-shaped condensates. In the following we focus on this last case, in which, on one hand, the nonlinearity is milder, while, on the other hand, the family of point interactions is richer. A natural question in this context is the following: what happens when a wave (i.e. a condensate) is sent against a defect? One would guess (and it has been shown for some models, see e.g. \cite{gow,hmz,p}) that the incoming wave (think of a solitary wave) splits into a reflected wave, a transmitted wave and a captured component. Similar results have been proven for propagation on graphs also (\cite{[ACFN1]}), in the case of a repulsive vertex, where no capture occurs. The regime of fast soliton has been also investigated in \cite{dh}. Indeed, it seems reasonable to conjecture that a capture can occur only if a nonlinear stationary state exists. Since equation \eqref{gp} is dispersive, the presence of a nonlinear stationary state (or more than one) must be related to the defect. This is the reason why such possible stationary states are called {\em defect modes}. Even though at this stage it is an unproven fact, it is plausible to link the persistence of a captured wave with some sort of stability (in a sense to be made precise) of the defect mode. For this reason the interest in determining the stability of the defect modes lies not only in the problem itself, but extends to models of reality too. As a short review on results on stability and instability of defect modes in the presence of a power nonlinearity $| \psi |^{2 \mu} \psi$, we recall results proved in \cite{reika,giannigianni,lacozza}, where the effects of a $\delta$-like defect are analysed. The first cited work deals with an attractive defect, and shows that, for any frequency $\om$ above the proper frequency of the unique bound state of the delta potential, there is a unique defect mode that oscillates at frequency $\om$. It turns out that the wavefunction of such a defect mode is nothing but the nonlinear deformation of the linear bound state. The stability (more exactly, the {\em orbital} stability, see Definition \ref{os}) of such a mode depends on $\mu$ and $\om$: if $\mu \leq 2$, then the defect mode is stable for any $\om$; if $\mu > 2$, then it becomes unstable at high frequencies. References \cite{giannigianni,lacozza} extend the analysis to a repulsive delta-like defect. The situation becomes more involved in the case of a more singular defect, for instance, the so-called $\delta^\prime$ defect. The following sections are devoted to this case. For more details see also the comprehensive review \cite{[ANS]} and the forthcoming paper \cite{[AN2]}. The established theoretical framework for the study of stability is provided by Grillakis-Shatah-Strauss' theory (\cite{gss1,gss2}) or, alternatively, by Lions' concentration-compactness method (\cite{l1,l2} and \cite{c} for a review). The occurrence of bifurcation in the ground state has been investigated in \cite{jw} and more recently in \cite{supersacchetti,fukusacchetti,kirr,kevrekidis,tuoc}. \section{Results} \subsection{The $\delta^\prime$ "potential".} The so-called $\delta^\prime$-defect, with strength $- \gamma$, located (just to be definite) at zero, is defined imposing the boundary condition \be \label{bc} \psi (0+) - \psi (0-) = - \gamma \psi^\prime (0+) = -\gamma \psi^\prime (0-) \ee to the solutions to \eqref{gp} (see \cite{abd,[EG]}). The parameter $\gamma$ is real; when positive, the defect is called {\em attractive}, otherwise {\em repulsive}. More formally, one defines a {\em linear Hamiltonian operator} $H_\gamma$ as the operator that acts as $- \partial_x^2$ on functions in $H^2(\erre^+) \oplus H^2(\erre^+)$ satisfying \eqref{bc}. Note that the only continuous elements of the domain of $H_{\gamma}$ have a vanishing derivative at the origin. The operator $H_\gamma$ is a self-adjoint operator with the following spectral features: singular continuous spectrum is empty, absolutely continuous spectrum is given by the positive halfline and point spectrum is empty in the repulsive case, and coincides with $\{ - 4 / \gamma^2 \}$ in the attractive case. In the last case the corresponding (non-normalized) eigenfunction is $$ \varphi_\gamma (x) \ = \ \epsilon (x) e^{-\f 2 \gamma |x|},$$ where we denoted the sign function by $\epsilon$. Notice that $\varphi_\gamma$ is odd. The quadratic form $F_\gamma$ associated to $H_\gamma$ is defined on the domain $Q : = H^1 (\erre^+) \oplus H^1 (\erre^-)$ (we stress that $Q$ is independent of $\gamma$) and reads $$ F_\gamma (\psi) \ = \ \| \psi^\prime \|^2 - \gamma^{-1} | \psi (0+) - \psi (0-) |^2, $$ where we made the following abuse of notation $$ \| \psi^\prime \|^2 : = \lim_{\ve \to 0+} \int_\ve^{+\infty} | \psi^\prime (x) |^2 dx + \lim_{\ve \to 0+} \int^{-\ve}_{-\infty} | \psi^\prime (x) |^2 dx, $$ that will be extensively repeated. At variance with the delta potential, the Schr\"odinger operator with a $\delta^\prime$ interaction cannot be derived from a form sum, because the $\delta^\prime$ is not small with respect to the laplacian. Nevertheless it can be obtained as the norm-resolvent limit of the sum of three $\delta$ potentials (\cite{cs}, \cite{enz}) with a fine tuned rescaling, defined as follows $$[H_\gamma + \nu]^{-1} \ = \ \lim_{\ve \to 0} \left[- \partial_x^2 - \left(\f 1 \gamma + \f 1 {2 \ve} \right) \delta (x - \ve) - \left( \f 1 \ve + \f \gamma {2 \ve^2} \right) \delta (x) - \left(\f 1 \gamma + \f 1 {2 \ve} \right) \delta (x - \ve) + \nu \right]^{-1} $$ for any $- \nu$ in the resolvent set of $H_\gamma$.\par\noindent Moreover, since any delta potential, in its turn, can be approximated by a strong limit of rescaled regular potentials, then it is possible to interpret a $\delta$-prime potential as the suitable limit of rescaled, well-behaved potentials. \begin{figure} \begin{center} %{}{}\scalebox{0.40}{\includegraphics{approx.pdf}} {}{}\scalebox{0.40}{\includegraphics{approx.eps}} \caption{A regular approximation for an attractive $\delta$-prime potential centred at zero. To obtain an approximation for a repulsive $\delta$-prime, one must reverse the central well.} \end{center} \end{figure} %\end{minipage} %\begin{minipage}{0.48\textwidth} Let us remark that if $\psi$ belongs to the {\it operator} domain of $H_\gamma$, then the form associated to $H_{\gamma}$ has the expression $$ (\psi, H_\gamma \psi) \ = \ \| \psi^\prime \|_2^2 - \gamma | \psi^\prime (0+)|^2 , $$ which explains the questionable name of $\delta^\prime$. %\end{minipage} \subsection{Combining nonlinearity and defect} Once constructed the operator $H_\gamma$, the evolution in the presence of both a generic power nonlinearity and a defect is defined by \be \label{gpdefect} i \partial_t \psi (t,x) \ = \ H_\gamma \psi (t,x) + \alpha | \psi (t,x) |^{2 \mu} \psi (t,x). \ee For such equation it is possible to prove global well-posedness if $\mu < 2$ (\cite{an1}), local well-posedness if $\mu \geq 2$ (\cite{[AN2]}), and to provide examples of blow-up for this last case (\cite{as}). However, until the solution exists, $L^2$-norm and energy $$ {\mathcal E}(\psi) = \f 1 2 \| \psi^\prime \|^2 - \f 1 {2 \gamma} | \psi (0+) - \psi (0-) |^2 - \f \lambda {2 \mu + 2} \| \psi \|_{2 \mu + 2}^{2 \mu + 2} $$ are conserved by time evolution. Thanks to the existence of a conserved energy it is possible to introduce a notion of {\em nonlinear ground state}: intuitively, one would define it as a minimizer of the energy among the wavefunctions endowed with the same $L^2$-norm, as this is the definition that naturally extends the more familiar notion of linear ground state. As in the linear case, it is meaningful to search for {\it stationary states} of \eqref{gpdefect}, i.e. solutions of the form \be \psi(x,t) = e^{i\omega t} \psi_{\omega}(x)\ . \ee The amplitudes $\psi_{\omega}$ are solutions of the stationary equation \be \label{stateq} H_\gamma \psi_{\omega} + \om \psi_{\omega} - \lambda | \psi_{\omega} |^{2 \mu} \psi_{\omega} \ = \ 0. \ee This leads to the introduction of the so-called {\em action functional} \be \label{som} S_\om ( \psi ) \ = \ {\mc E} (\psi) + \f \om 2 \| \psi \|^2, \ee defined on the energy domain $Q$. It is immediate indeed that Euler-Lagrange equations for $S_{\om}$ are given just by \eqref{stateq}. Note that the action (and the energy as well) is not bounded from below on $Q$. To overcome this problem, a {\em ground state} $\psi_{\om}$ is usually defined as a minimizer of $S_\om$ constrained on the {\em Nehari manifold} $$ I_\om (\psi) = S_\om^\prime ( \psi ) \psi = ( \psi, H_\gamma \psi - \lambda | \psi |^{2 \mu} \psi + \om \psi) \ = \ 0. $$ The above set is a codimension one manifold that obviously contains all stationary points of $S_\om$, and it tuns out that on it the action is bounded from below.\par\noindent It is a byproduct of the Grillakis-Shatah-Strauss theory on stability of stationary states (\cite{gss1}, \cite{gss2}) applied to minimizers of $S_\om$ the relation between the constrained variational problem for ${\mathcal E}$ and $S_\om$: a minimizer $\psi_\omega $ of the action on the Nehari manifold is a minimizer of the energy among the function with the same $L^2$-norm $\|\psi_\omega\|^2$ {\em if and only if it is stable} (in the sense of Definition \eqref{os}). The following preliminary result is obtained through variational techniques (we remove the subscript $\omega$ from $\psi_{\om}$ when not needed to avoid ambiguity): \begin{theorem} \label{prel} For any $\om > \f 4 {\gamma^2}$ there exists at least one minimizer of $S_\om$ among all functions on the Nehari manifold. Furthermore, the minimizer solves the stationary Schr\"odinger equation with defect: \be \label{stazz} H_\gamma \psi + \om \psi - \lambda | \psi |^{2 \mu} \psi \ = \ 0. \ee \end{theorem} The line of the proof is standard, except that: first, the functional space of reference $Q$ is larger than $H^1 (\erre)$; second, the problem is one-dimensional, so that one must cope with a lack of compactness when passing from weak convergence in $Q$ to strong convergence in $L^p$; third, the boundary condition to be reconstructed is non standard. A complete proof is in \cite{[AN2]}. An important point about Theorem \ref{prel} is that, in order to find the ground states, it suffices to determine which one among the solutions of \eqref{stazz} has least action. This can be made directly, as the solutions to equation \eqref{stazz} can be explicitly found. Moreover the minimum is constrained to a finite codimension (one in this case) manifold, an information which is important for stability issues. \subsection{Symmetry breaking} Equation \eqref{stazz} can be rephrased as follows: \be \label{reph} -\partial_x^2 \psi + \om \psi - \lambda | \psi |^{2 \mu} \psi \ = \ 0, \ee with $\psi \in H^2 (\erre^+) \oplus H^2(\erre^-)$ satisfying the boundary condition \eqref{bc}. The only solutions to \eqref{reph} that vanish at infinity are constructed by gluing together two pieces of a solitary wave for the NLS, namely \begin{equation*} \psi^{x_1, x_2}_{\om, \pm} (x) = \left\{ \begin{array}{cc} \pm \lambda^{-\f 1 {2\mu}} (\mu + 1)^{\f 1 {2\mu}} \om^{\f 1 {2\mu}} \cosh^{-\f 1 \mu} [\mu \sqrt \om ( x - x_1)], & x<0 \\ \lambda^{-\f 1 {2\mu}} (\mu + 1)^{\f 1 {2\mu}} \om^{\f 1 {2 \mu}} \cosh^{-\f 1 \mu} [\mu \sqrt \om ( x - x_2)], & x>0 \end{array} \right. \end{equation*} where the parameters $x_1$ and $x_2$ are to be adjusted so that \eqref{bc} is satisfied. Now, it is immediately seen by \eqref{som} that due to contribution of the point interaction energy, one has $$S_{\om} (\psi^{x_1, x_2}_{\om, -}) \ < \ S_\om (\psi^{x_1, x_2}_{\om, +})$$ so we can restrict the search for minimizers to the functions $\psi^{x_1, x_2}_{\om, -}$. For any such function, the boundary condition \eqref{bc} translates into the system \begin{equation} \label{tsystem} \left\{ \begin{array}{ccc} t_1^{2 \mu} - t_1^{2 \mu + 2} & = & t_2^{2 \mu} - t_2^{2 \mu + 2} \\ t_1^{-1} + t_2^{-1} & = & \gamma \sqrt \om \end{array} \right. , \ 0 \leq t_i = | \tanh (\mu \sqrt \om x_i)| \leq 1, \end{equation} whose solutions can be depicted as the intersection of the full and the dashed lines in Figure 2. \begin{figure} \label{fig1} \begin{center} %{}{}\scalebox{0.50}{\includegraphics{bif2-bis.pdf}} {}{}\scalebox{0.50}{\includegraphics{bif2-bis.eps}} \caption{The full lines represent the solutions to the first equation in \eqref{tsystem}: they consist of the line $0 \leq t_1 = t_2 \leq 1$, and of a curve, that is concave if $\mu$ is not too small. The dashed lines represent the solutions to the second equation of \eqref{tsystem}: they consist of a family of hyperbola parametrized by $\om$. } \end{center} \end{figure} One immediately finds that for $\f 4 {\gamma^2} < \om \leq \f 4 {\gamma^2} \f {\mu + 1} \mu$ the unique solution is given by $t_1 = t_2 = \f 2 {\gamma \sqrt \om}$, that corresponds to an antisymmetric stationary state $\psi_\om^{y, -y}$, where $$ y = x_1 = - x_2 = \f 1 {2 \mu \sqrt \om} \log \f {\gamma \sqrt \om + 2} {\gamma \sqrt \om - 2}. $$ At $ \om = \f 4 {\gamma^2} \f {\mu + 1} \mu$ two new solutions arise, giving birth to two new branches of stationary states that persist for $\om >\f 4 {\gamma^2} \f {\mu + 1} \mu$; they correspond to the couple of asymmetric stationary states $\psi_\om^{y_1, - y_2}, \, \psi_\om^{y_2, - y_1}$, with both $y_1$ and $y_2$ positive but, except in the cubic case $\mu = 1$, not in explicit form. A direct computation yields, for these values of $\omega$, $$ S_\om (\psi_\om^{y_1, -y_2}) \ < \ S_\om (\psi_\om^{y, - y})\ .$$ We conclude that with the growth of the frequency $\omega$ there exist {\em two} branches of asymmetric ground states which bifurcate from the branch of (anti)symmetric ones. We are then in the presence of a spontaneous symmetry breaking of the set of ground states. \subsection{Stability: a pitchfork bifurcation} The study of the stability for such a system can be made by applying the Grillakis-Shatah-Strauss theory (\cite{gss1}, \cite{gss2}). This theory provides sufficient conditions for the {\em{orbital stability}} of stationary states, which is stability ``up to the symmetries". Roughly speaking, the notion of orbital stability coincides with the ordinary Ljapunov stability {\em for orbits instead of states}, where orbits are to be understood with respect to a symmetry group. In our case the symmetry group is $U(1)$, corresponding to the well known phase invariance of the NLS, which persists in the presence of point perturbation too. So, a stationary state $\psi_\om$ is said to be {\em orbitally stable} if at any time a solution to \eqref{gpdefect} remains arbitrarily close {\em to the orbit} $\{ e^{i \theta} \psi_\om, \, \theta \in [0, 2 \pi) \}$, provided that it started sufficiently close to it. More rigorously, \begin{definition} \label{os} A stationary state $\psi_\om$ is called {\em orbitally stable} if for any $\ve > 0$ there exists a $\sigma > 0$ s.t. $$ \inf_{\theta \in [0, 2 \pi)} \| \psi_0 - e^{i \theta} \psi_\om \|_Q \leq \sigma \Rightarrow \sup_{t>0} \inf_{\theta \in [0, 2 \pi)} \| \psi_t - e^{i \theta} \psi_\om \|_Q \leq \ve, $$ where $\psi_t$ is the solution corresponding to the initial condition $\psi_0$. A stationary state is called {\em orbitally unstable} if it is not orbitally stable. \end{definition} The Grillakis-Shatah-Strauss theory (see \cite{gss1,gss2}) carries out a deep investigation of the orbital stability of stationary states of (infinite dimensional) hamiltonian systems with symmetries, generalizing previous work by the same authors and independently by Michael Weinstein (\cite{[W1],[W2],[W3]}).\par\noindent They succeed in giving sufficient conditions for stability and instability by studying second-order approximation of the action (linearization) around a stationary state, and carefully controlling the nonlinear remainders exploiting symmetries and conservation laws. In the present situation, as it is well known, one gets a hamiltonian system from NLS equation passing to real variables $(\eta, \rho)=({\rm Re} \psi, {\rm Im} \psi)$. We confine ourself to a brief operative summary of the method, and so we omit the (however important) connection with hamiltonian systems referring to the original literature for details.\par\noindent Neglecting higher order terms, one has for the action expanded around the stationary state $\psi_{\omega}$ (we omit other superscripts for simplicity) $$ S_\om (\psi_\om + \eta + i \rho) \cong S_\om (\psi_\om) + \f 1 2 \left( S^{\prime \prime}(\psi_\om) \left( \begin{array}{c} \eta \\ \rho \end{array} \right), \left( \begin{array}{c} \eta \\ \rho \end{array} \right) \right). $$ The Hessian operator $S_\om^{\prime \prime}(\psi_{\om})$ can be represented in matrix form as (we implicitly introduced the representation of a function $\eta + i \rho$ as the real vector function $(\eta, \rho)$) $$S^{\prime \prime}_\om (\psi_\om) \ : = \ \left( \begin{array}{cc} L_1 & 0 \\ 0 & L_2 %\left( \begin{array}{c} \eta_t \\ \rho_t \end{array} \right), $$ where $L_1$ and $L_2$ are two selfadjoint operators with $D(L_1)=D(L_2)=D(H_{\gamma})$ given by \begin{eqnarray*} L_1 & = & H_\gamma + \om - \lambda (2 \mu + 1) | \psi_\om |^{2 \mu} \\ L_2 & = & H_\gamma + \om - \lambda | \psi_\om |^{2 \mu} . \end{eqnarray*} Now, were $S^{\prime \prime}_\om (\psi_\om)$ a positive operator, the (linear) stability of $\psi_\om$ would be immediately established, as the situation would be analogous to what happens for a classical particle in a potential well. However this is cannot be the case. First of all, the operator $S_\om^{\prime \prime} (\psi_\om)$ is endowed with a non trivial kernel that consists of the linear span of $(0, \psi_\om)$, due to the symmetry. Second, recall that every ground state $\psi_\om$ is a minimizer only on the constraint provided by the Nehari manifold, which has codimension one. On the space orthogonal to the Nehari manifold, $S_\om^{\prime \prime} (\psi_\om)$ is surely negative, as $$ (\psi_\om, S_\om^{\prime \prime} \psi_\om) \ < \ 0. $$ It follows that there exists a cone on which $S_\om^{\prime \prime} (\psi_\om)$ is actually negative. \noindent Nevertheless, it is possible that the dynamical constraints given by the conservation laws prevent the wave function from further evolving far inside that cone, finally forcing the solution to remain close to the orbit of the ground state. The Grillakis-Shatah-Strauss theory establishes that this is the case if a certain number of conditions are satisfied. In its easiest version, such a set of conditions can be collected as follows \noindent $i)$ {\em Spectral conditions:} \begin{enumerate} \item ${\rm{Ker}} L_2 \ = \ {\rm{Span}} \{ \psi_\om \}$, \item $L_2 \geq 0$, \item ${\rm{Ker}} L_1 \ = \ \{ 0 \}$, \item $L_1$ has exactly {\em one} negative eigenvalue. \end{enumerate} \noindent $ii)$ {\em Vakhitov-Kolokolov's criterion} (\cite{vk}): $$\f {d \| \psi_\om \|_2}{d\om} > 0,$$ that, since $\f {dS_\om (\psi_\om)} {d \om} = \f 1 2 \| \psi_\om \|_2^2$, is equivalent to \be \label{vk} \f {d^2S_\om (\psi_\om)} {d \om^2} > 0. \ee In the case of interest, conditions $i)$ and $ii)$ are verified except for the the stationary states in the branch $\psi_\om^{y, -y}$ with $\omega>\f 4 {\gamma^2} \f {\mu + 1} \mu $, where more a more sophisticated version of conditions $i)$ and $ii)$ is needed, again provided by the Grillakis-Shatah-Strauss theory (see \cite{gss2}). The results on stability can be summed up as follows. \begin{theorem} For any $\mu > 0$ \begin{enumerate} \item If $\om < \f 4 {\gamma^2} \f {\mu + 1} \mu$, then the unique (up to a phase) ground state $\psi_\om^{y, -y}$ is orbitally stable. \item If $\om > \f 4 {\gamma^2} \f {\mu + 1} \mu$, then the stationary state $\psi_\om^{y, -y}$ is orbitally unstable. \end{enumerate} For $0 \leq \mu \leq 2$, $\om > \f 4 {\gamma^2} \f {\mu + 1} \mu$, the two ground states $\psi_\om^{y_1, -y_2}$, $\psi_\om^{y_2, -y_1}$ are orbitally stable. For $\mu > 2$ there exist $\om_1 > \f 4 {\gamma^2} \f {\mu + 1} \mu$, $\om_2 > \om_1$, such that, if $\f 4 {\gamma^2} \f {\mu + 1} \mu < \om < \om_1$ then $\psi_\om^{y_1, -y_2}$ and $\psi_\om^{y_2, -y_1}$ are orbitally stable; if $\om > \om_2$, then $\psi_\om^{y_1, -y_2}$ and $\psi_\om^{y_2, -y_1}$ are orbitally unstable. \end{theorem} The bifurcation diagrams for the system are portrayed in Figures 3 and 4. \begin{figure} \label{figA} \begin{center} %{}{}\scalebox{0.50}{\includegraphics{bifstab-bis.pdf}} {}{}\scalebox{0.50}{\includegraphics{bifstab-bis.eps}} \caption{Bifurcation diagram for $\mu \leq 2$. The full line denotes stable stationary states, while the dashed line represents unstable stationary states. Notice that ground states are always stable. } \end{center} \end{figure} \begin{figure} \label{figB} \begin{center} %{}{}\scalebox{0.50}{\includegraphics{bifuncertainright-bis.pdf}} {}{}\scalebox{0.50}{\includegraphics{bifuncertainright-bis.eps}} \caption{Bifurcation diagram for $\mu > 2$. We have no results for the interval $(\om_1, \om_2)$, but we conjecture that it is always possible to choose $\om_1 = \om_2$. } \end{center} \end{figure} \section{Proof of stability} The content of this section is technical. Here we we give a proof of the stability of all ground states in the case $\mu \leq 2$. Under such a restriction, every ground state satisfies the Vakhitov-Kolokolov's criterion. The proof we present differs from the one given in \cite{[AN2]}, as it does not use the Grillakis-Shatah-Strauss theory and so it does not refer to linearization. We decided to include in this report such a technical part in order to convey some information on the method of proofs and on the techniques employed. An analogous analysis is given for the case of a NLS with $\delta$ interaction in \cite{reika}, and both are inspired by \cite{fibich}.\par\noindent In order to proceed we need some preliminary definitions and results. First, the definition of orbital stability can be reformulated using the notion of {\em orbital} {\em neighbourhood}. \begin{definition} The set $$ U_\eta (\phi) : = \{ \psi \in Q, \, {\rm s.t.} \, \inf_{\theta \in [0, 2 \pi)} \| \psi - e^{i \theta} \phi \|_Q \leq \eta \}$$ is called the {\em orbital neighbourhood with radius $\eta$ of the function $\phi$}. \end{definition} It is convenient to introduce a function that associates to any frequency $\om > \f 4 {\gamma^2}$ the value of the minimum attained by $S_\om$ evaluated on functions in the Nehari manifold. Namely, \begin{eqnarray*} d & : & \left( \f 4 {\gamma^2}, + \infty \right) \to \erre \\ & & \om \mapsto \min \{ S_\om (\phi), \phi \in Q, I_\om (\phi) = 0 \}. \end{eqnarray*} It is then important to stress other points that we did not mention explicitly so far. \begin{remark} \label{tredue} \ \noindent \begin{enumerate} \item In the energy space $Q$ the following norm is defined: \begin{equation*} \begin{split} \| \phi \|_Q^2 \ : = & \ \| \phi \|_2^2 + \lim_{\ve \to 0+} \int_\ve^{+\infty} | \phi^\prime|^2 \, dx + \lim_{\ve \to 0+} \int^{-\ve}_{-\infty} | \phi^\prime|^2 \, dx \end{split} \end{equation*} \item For any $\theta \in [0, 2 \pi)$, any $\phi \in Q$, one has $S_\om (e^{i \theta} \phi) = S_\om (\phi)$. As a consequence, if $\psi_\om$ is a ground state, then all the functions $e^{i \theta} \psi_\om$ in its orbit are ground states too. In the proof of Theorem \eqref{teo:stability} we will make the phase explicit by denoting $$\psi_{\om}^{x_1, x_2, \theta} \ : = \ e^{i \theta} \psi_{\om}^{x_1, x_2, 0}.$$ \end{enumerate} \end{remark} The result we need to go through the proof are summarized in the following Proposition. Their proof is contained in \cite{[AN2]}. \begin{prop} \label{tretre} \ \noindent \begin{enumerate} \item For any function $\phi$ in the Nehari manifold one has $S_\om (\phi) = \widetilde S (\phi)$, where $\widetilde S$ is the functional defined by $$ \widetilde S (\phi) : = \f {\lambda \mu} {2 \mu + 2} \| \phi \|_{2 \mu + 2}^{2 \mu + 2}.$$ \item Any minimizer $\psi_\om$ of the functional $S_\om$ on the Nehari manifold minimizes also the functional $\widetilde S$ on the region $I_\om \leq 0$. \item Following Fibich and Wang (\cite{fibich}) we recall that the map \be \begin{split} \nonumber \widehat \om : Q \ \longrightarrow & \ \erre, \qquad \phi \ \mapsto \ d^{-1} \left( \f \lambda 2 \f {\mu} {\mu + 1} \| \phi \|_{2 \mu + 2}^{2 \mu + 2} \right) \end{split} \ee is well-defined. Notice that $\widehat \om$ maps a function $\phi$ into the frequency of a ground state having the same $L^{2 \mu + 2}$-norm as $\phi$. Such a ground state may not be unique, but the $L^{2 \mu + 2}$-norm always is. \item If $\psi_\om$ minimizes $S_\om$ on the Nehari manifold $I_\om = 0$, then $\psi_\om$ minimizes $S_\om$ on the set $ \{ \phi \in Q, \, \| \phi \|_{2 \mu + 2 } = \| \psi_\om \|_{2 \mu + 2} \}. $ \item For any $\om > 0$, the function $\chi_{[0, + \infty)} \psi_{\om}^{0,0}$ minimizes the functional $$ S_\om^0 (\phi) : = \f 1 2 \| \phi^\prime \|_2^2 + \f \om 2 \| \phi \|_2^2 - \f \lambda {2 \mu + 2} \| \phi \|_{2 \mu + 2}^{2 \mu + 2} $$ among the functions in $Q$ that satisfy $$ I_\om^0 (\phi) : = \| \phi^\prime \|_2^2 + \om \| \phi \|_2^2 - \lambda \| \phi \|_{2 \mu + 2}^{2 \mu + 2} \ = \ 0. $$ \item If $\mu \leq 2$, then any ground state satisfies the Vakhitov-Kolokolov's condition \eqref{vk}. %\item %Given $c>0$, %the set of the minima for the functional %$S_\om^0$ on the set % $$ %\{ \psi \in Q, \, \| \psi \|_{2 \mu + 2 } = c %$$ %reads %\be \label{minima0} %\{ e^{i \theta} \chi_+ \phi_0 , \ e^{i % \theta} \chi_- \phi_0, \, \theta \in [0, 2 \pi) \}. %\ee %\item Questo non lo capito %$$d^{\prime \prime} \ = \ $$ \end{enumerate} \end{prop} Now we can prove the \begin{theorem} \label{teo:stability} If $1 \leq \mu \leq 2$, then any ground state is stable. \end{theorem} \begin{proof} We specialize to the case with $\om > \f 4 {\gamma^2} \f {\mu + 1} \mu$, namely, beyond the frequency of bifurcation. In fact, for $\om < \f 4 {\gamma^2} \f {\mu + 1} \mu$, this proof can be easily adapted and one recovers essentially the argument given in \cite{reika}. Fix $\om_0 > \f 4 {\gamma^2} \f {\mu + 1} \mu$ and suppose that the stationary solution $e^{i \om_0 t} \psi_\om^{y_1, -y_2,0}$ is orbitally unstable. This means that there exists $\ve_0 > 0$ and a sequence $\varphi_k \in U_{\f 1 k} (\psi_\om^{y_1, -y_2,0})$ such that $$ \sup_{t \geq 0} \inf_{\theta \in [0, 2 \pi)} \| \varphi_k (t) - \psi^{y_1, -y_2,\theta}_{\om_0} \|_Q \geq \ve_0,$$ where $\varphi_k (t)$ is the solution to equation \eqref{gpdefect} with initial data $\varphi_k$. With no loss of generality, we assume \be \label{choiceps} \ve_0 \ \leq \ \inf_{\theta \in [0, 2 \pi)} \| \psi^{y_1, -y_2,0}_{\om_0} - \psi^{y_2, -y_1, \theta}_{\om_0} \|_Q \ = \ \| \psi^{y_1, -y_2,0}_{\om_0} - \psi^{y_2, -y_1,0}_{\om_0} \|_Q . \ee Let $t_k$ be the smallest positive time for which \be \label{smallest} \inf_{\theta \in [0, 2 \pi)} \| \varphi_k (t_k) - \psi^{y_1, -y_2,\theta}_{\om_0} \|_Q = \f {\ve_0} 2, \ee and let us use the notation $\xi_k \ = \ \varphi_k (t_k)$. By conservation laws, %\be \label{prefibich} %S_\om (\xi_k) \, \longrightarrow \, S_\om (\phi_0) %\ee %for any $\om$, in particular \be \label{particulare} \begin{split} S_{\om_0} (\xi_k) \, = \, & {\mathcal E} (\xi_k) + \f {\om_0} 2 \| \xi_k \|_2^2 \, = \, {\mathcal E} (\varphi_k) + \f {\om_0} 2 \| \varphi_k \|_2^2 \\ \, \longrightarrow \, & {\mathcal E} (\psi^{y_1, -y_2,0}_{\om_0}) + \f {\om_0} 2 \| \psi^{y_1, -y_2,0}_{\om_0} \|_2^2 \, = \, S_{\om_0} (\psi^{y_1, -y_2,0}_{\om_0}) \, = \, d (\om_0). \end{split} \ee where we used the fact that, by construction, the sequence $\varphi_k$ converges to $\psi^{y_1, -y_2,0}_{\om_0}$ strongly in $Q$, that implies the convergence of the energy and of the $L^2$-norm. Let us denote $\om_k = \widehat \om (\xi_k)$. We recall the following result from \cite{fibich}, used in \cite{reika} also: \be \label{fibich} S_{\om_k} (\xi_k) - S_{\om_k} (\phi_0) \ \geq \ \f 1 4 d^{\prime \prime} (\om_0) (\om_k - \om_0)^2 \ee where we denoted $\om_k \ = \ \om (\xi_k)$. The fact that the Vakhitov-Kolokolov's condition is satisfied (see (6)), together with \eqref{fibich} and \eqref{particulare}, implies $\om_k \rightarrow \om_0$, and therefore, by the definition of the function $\widehat \om$, we have \be \label{xil4} \| \xi_k \|_{2 \mu + 2} \ = \ \left[ \f {2 \mu + 2} {\lambda \mu} S_{\om_k} (\psi_{\om_k} ) \right]^\f 1 {2 \mu + 2} \ \longrightarrow \ \left[ \f {2 \mu + 2} {\lambda \mu} S_{\om_0} (\psi_{\om_0} ) \right]^\f 1 {2 \mu + 2} \ = \ \| \psi^{y_1, -y_2,0}_{\om_0}\|_{2 \mu + 2}. \ee \n We define the sequence $\zeta_k : = \f {\| \psi^{y_2, y_1,0}_{\om_0} \|_{\mu + 2}} {\| \xi_k \|_{\mu + 2}} \xi_k$. By \eqref{xil4}, \be \label{convzQ} \| \zeta_k - \xi_k \|_Q \ = \ \left| \f {\| \psi^{y_2, y_1,0}_{\om_0} \|_{\mu + 2}} {\| \xi_k \|_{\mu + 2}} - 1 \right| \| \xi_k \|_Q \ \longrightarrow \ 0. \ee As a consequence, $S_{\om_0} (\zeta_k) - S_{\om_0} (\xi_k) \rightarrow 0$, so $S_{\om_0} (\zeta_k) \rightarrow S_{\om_0} (\psi_{\om_0})$. For this reason, and as $\| \zeta_k \|_{2 \mu + 2} \ = \ \| \psi^{y_1, -y_2,0}_{\om_0} \|_{2 \mu + 2}$, point (4) in Proposition \ref{tretre} implies that $\{ \zeta_k \}$ is a minimizing sequence for the problem \be \nonumber \min \{ S_{\om_0} (\psi), \ \psi \in Q \backslash \{ 0 \}, \ \| \psi \|_{2 \mu + 2} = \| \psi^{y_2, -y_1,0}_{\om_0} \|_{2 \mu + 2} \}. \ee By Banach-Alaoglu theorem there exists a subsequence, whose elements we denote by $\zeta_k$ too, that converges weakly in $Q$ and therefore in $L^{2 \mu + 2}$. Let us call $\zeta_\infty$ its weak limit. First, notice that $\zeta_\infty \neq 0$. Indeed, were it zero, then weak convergence in $Q$ would imply $\zeta_\infty (0 \pm) \rightarrow 0$, and therefore $S_{\om_0} (\zeta_k) - S_{\om_0}^0 (\zeta_k) \rightarrow 0$, so $$ \lim_{k \to \infty} S_{\om_0}^0 (\zeta_k) \ = \ \lim_{k \to \infty} S_{\om_0} (\zeta_k) \ = \ \f \lambda 2 \f \mu {\mu + 1} \| \psi^{y_1, -y_2,0}_{\om_0} \|_{2 \mu + 2}^{2 \mu + 2}. $$ Then, employing the fact that $\phi_0 : = \chi_{[0, + \infty)} \psi_{\om_0}^{0,0} (0) \neq 0$, points (4) and (5) in Proposition \ref{tretre} yield \be \nonumber \f \lambda 2 \f \mu {\mu + 1} \| \psi^{y_1, -y_2,0}_{\om_0} \|_{2 \mu + 2}^{2 \mu + 2} \ = \ \lim S_{\om_0}^0 (\zeta_k) \ \geq \ S_{\om_0}^0 (\chi_+ \phi_0) \ > \ S_{\om_0} (\chi_+ \phi_0) \ \geq \ \f \lambda 2 \f \mu {\mu + 1} \| \psi^{y_1, -y_2, 0}_{\om_0} \|_{2 \mu + 2}^{2 \mu + 2}, \ee which is absurd. So it must be $\zeta_\infty \neq 0$. We claim that $I_{\om_0} (\zeta_\infty) \ = \ 0$. We proceed by contradiction. \n Suppose indeed that $I_\om (\zeta_\infty) < 0$. Then, by point (2) in Remark \ref{tredue} and points (1) and (2) in Proposition \ref{tretre} we know that the minimizers of the functional $\widetilde S$ on the region $I_{\om_0} \leq 0$ are given by $\psi_{\om_0}^{y_1, -y_2, \theta}$ and $\psi_{\om_0}^{y_2, -y_1, \theta}$, for all $\theta \in [0, 2 \pi)$. Furthermore, all such functions lie on the set $I_{\om_0} = 0$. As a consequence, recalling the definition of the functional $\widetilde S$, one obtains $$\| \zeta_\infty \|_{2 \mu + 2} \ = \ \left[ \f{2 (\mu + 1)} {\lambda \mu} \widetilde S (\zeta_\infty) \right]^{\f 1 {2 \mu + 2}} \ > \ \left[ \f{2 (\mu + 1)} {\lambda \mu} \widetilde S (\psi_{\om_0}^{y_1, -y_2, 0}) \right]^{\f 1 {2 \mu + 2}} \ = \ \| \psi_{\om_0}^{y_1, -y_2, 0} \|_{2 \mu + 2}.$$ But this is not possible, as $\zeta_\infty$ is the weak limit of functions having the same $L^{2 \mu +2}$-norm as $\psi^{y_1, -y_2,0}_{\om_0}$. \n On the other hand, suppose that $I_{\om_0} (\zeta_\infty) \ > \ 0$. By \eqref{particulare} and \eqref{xil4} \begin{equation*} \begin{split} & \lim_{k \to \infty} I_{\om_0} (\xi_k) \ = \ 2 \lim_{k \to \infty} S_{\om_0} (\xi_k) - \f {\lambda \mu} {\mu + 1} \lim_{k \to \infty} \| \xi_k \|_{2 \mu + 2}^{2 \mu + 2} \\ \ = \ & \ 2 S_{\om_0} (\psi^{y_1, -y_2,0}_{\om_0}) - \f {\lambda \mu} {\mu + 1} \| \psi^{y_1, -y_2,0}_{\om_0}\|_{2 \mu + 2}^{2 \mu + 2} \ = \ I_{\om_0} (\psi^{y_1, -y_2,0}_{\om_0}) \ = \ 0. \end{split} \end{equation*} Therefore, by \eqref{convzQ}, $$ \lim_{k \to \infty} I_{\om_0} (\zeta_k) \ = \ 0. $$ From the following inequality (see \cite{bl}) \begin{eqnarray} \| u_n \|_{p}^{p} - \| u_n - u_\infty \|_{p}^{p} - \| u_\infty \|_{p}^{p} & \longrightarrow & 0, \quad \forall \, 1 < p < \infty. \label{breli1} \end{eqnarray} one easily has $$I_{\om_0} (\zeta_k - \zeta_\infty) \longrightarrow - I_{\om_0} (\zeta_\infty) \ < \ 0.$$ As a consequence, eventually in $k$ we obtain $I_{\om_0} (\zeta_k - \zeta) < 0$ and then, using point (2) in Proposition \ref{tretre} \be \label{precontra} \| \zeta_k - \zeta_\infty \|_{2 \mu + 2} \ > \ \| \psi^{y_1, -y_2,0}_{\om_0} \|_{2 \mu + 2}. \ee But from \eqref{breli1}, and knowing that $\zeta_\infty \neq 0$, we have that the following inequality holds eventually in $k$ $$\| \zeta_k - \zeta_\infty \|_{2 \mu + 2} \ \leq \ \| \psi^{y_1, -y_2,0}_{\om_0} \|_{2 \mu + 2},$$ that contradicts \eqref{precontra}. We conclude that $I_{\om_0} (\zeta_\infty)$ cannot be strictly positive and, as we already proved that it cannot be negative, it must vanish. As a consequence, from point (2) in Proposition \ref{tretre} again, we get $\| \zeta_\infty \|_{2 \mu + 2} \geq \| \psi^{y_1, -y_2,0}_{\om_0} \|_{2 \mu + 2}$. But, since $\zeta_\infty$ is a weak limit, it must be \be \nonumber \| \zeta_\infty \|_{2 \mu + 2} \ = \ \| \psi^{y_1, -y_2,0}_{\om_0} \|_{2 \mu + 2}. \ee This fact has the following relevant consequences: \begin{itemize} \item Owing to \eqref{breli1}, the sequence $\{ \zeta_n \}$ converges strongly to $\zeta_\infty$ in the topology of $L^{2 \mu + 2}$. \item The sequence $\{ \zeta_k \}$ converges to $\zeta_\infty$ in the strong topology of $Q$. Indeed, by the convergence of $S_{\om_0} (\zeta_k)$ to $S_{\om_0} (\zeta_\infty)$, the weak convergence in $Q$, and the strong convergence of $\{ \zeta_k \}$ in $L^{2 \mu + 2}$, we have \be \label{qomega} \| \zeta_k^\prime \|^2 + {\om_0} \| \zeta_k \|^2 \ \longrightarrow \ \| \zeta_\infty^\prime \|^2 + {\om_0} \| \zeta_\infty \|^2. \ee So the convergence is strong in the space $Q$ endowed with the norm given by \eqref{qomega}, that is equivalent to the usual $Q$-norm. \item The sequence $\{ \xi_k \}$ also converges to $\zeta_\infty$ in the strong topology of $Q$. Indeed, applying \eqref{convzQ}, we have \be \label{poscontra} \| \xi_k - \zeta_\infty \|_Q \ \leq \ \| \xi_k - \zeta_k \|_Q + \| \zeta_k - \zeta_\infty \|_Q \ \longrightarrow \ 0. \ee \item The function $\zeta_\infty$ minimizes $S_{\om_0}$ with the constraint $I_{\om_0} \ = \ 0$, so, either $\zeta_\infty \ = \ \psi^{y_1, -y_2,\theta}_{\om_0}$ or $\zeta_\infty \ = \ \psi^{y_2, -y_1, \theta}_{\om_0}$ for some value of $\theta$ in $[0, 2 \pi)$. \end{itemize} Let us suppose that $\zeta_\infty \ = \ \psi^{y_1, -y_2,\theta}_{\om_0}$, for a certain value of $\theta$. By \eqref{poscontra} we obtain $\xi_k \to \psi^{y_1, -y_2,\theta}_{\om_0}$ strongly in $Q$, that contradicts inequality \eqref{smallest}, and thus the assumption of the orbital instability of the stationary state $\psi^{y_1, - y_2, 0}_{\om_0} $ proves false. On the other hand, consider the case with $\zeta_\infty \ = \ \psi^{y_2, -y_1, \theta}_{\om_0}$ for some value of $\theta$. By \eqref{smallest} there exists a sequence $\theta_k$ such that \be \label{subsub} \| \xi_k - \psi^{y_1, -y_2, \theta_k}_{\om_0} \|_Q \ \leq \ \f 2 3 \ve_0. \ee Using elementary triangular identity, \eqref{choiceps} and \eqref{subsub}, we obtain, for any $\theta \in [0, 2 \pi)$, \be \nonumber \| \xi_k - \psi^{y_2, -y_1, \theta}_{\om_0} \|_Q \ \geq \ \| \psi^{y_1, -y_2, \theta_k}_{\om_0} - \psi^{y_2, -y_1, \theta}_{\om_0} \|_Q - \| \psi^{y_1, -y_2, \theta_k}_{\om_0} - \xi_k \|_Q \ \geq \ \f {\ve_{0}} 3. \ee This contradicts \eqref{poscontra}, % which implies $\xi_k \to %\psi^{y_2, -y_1, \theta}_{\om_0}$ strongly in $Q$. %This is absurd, so the proof is complete. \end{proof} \section{Perspectives} The interplay between nonlinearity and defect is, in our opinion, a promising and worth developing field. In particular, already in simple models it makes highly non trivial behaviour emerge. An enlightening example has been supplied by means of the $\delta^\prime$ defect, in which the occurrence of a pitchfork bifurcation with symmetry breaking has been proved for the family of nonlinear ground states. Such results have to be considered as the first achievements of our research project. Many non trivial variations on the theme could be given by studying the the entire family of one-dimensional defects (a four parameters family, see \cite{abd}) and thus investigate the effect of various self-adjoint boundary conditions, in particular, of those that give rise to {\em two} bound states. We expect that, in the nonlinear problem, each of the two linear modes could be deformed into nonlinear modes for any frequency greater than the energy of the corresponding linear mode. Think, for instance, of a point interaction that, roughly speaking, is the sum of a $\delta$ and a $\delta$-prime defect {\em at the same point}. It exhibits two bound states, one of which is even (as the ground state for a Dirac's delta), while the other is odd (as the ground state for a delta prime). A number of question then arises: how do the corresponding nonlinear mode interact? Does it exist a third family of stationary (possibly ground) states that does not preserve any parity symmetry? However, all these steps are only preliminary to the problem of studying the evolution of a soliton that meets an impurity. It remains completely open the problem of defining analogous models in higher dimension. We recall that in dimension two and three, the only point interaction is the delta interaction, and in dimension higher than three there are no point perturbations of the laplacian. For instance, in the three dimensional case a bare power nonlinearity seems to be too strong to be added to a Dirac's delta potential; so a different type of nonlinearity with a moderated behaviour at infinity should be considered. Conversely, in space dimension two the na{\"{\i}}f power nonlinearity could be not necessarily in conflict with the domain of a delta interaction, but up to now no rigorous result exists. Another related topic is given by quantum graphs (\cite{kuchment1, kuchment2, ks} for the relevant definitions and analysis in the linear case). Also in the relatively simple case of a NLS on a star graph, the richer structure provides a larger number of nonlinear stationary states, for example two stationary states for a three edge star graph with a delta vertex, both attractive and repulsive, and the number increases with the number of edges (see \cite{[ACFN2]}). In this respect, besides the determination of the ground state, it is an open interesting problem the analysis of stability of excited states, here explicitly known. 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