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NLS, graphs, ground state
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\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
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\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. Nothing is known for the a star graph with more general vertex conditions, 
for example the boundary condition of $\delta^\prime$ type. 




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\end{document} 
  
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