0$, we will be looking for $\phi\in H_D^1(\clG)$ such that \begin{equation}\label{eq:minimization} M(\phi)=m,\quad E(\phi)=\min\left\{E(\psi):\psi\in H_D^1(\clG),\,M(\psi)=m\right\}. \end{equation} The theoretical existence of minimizers for the problem~\eqref{eq:minimization} has attracted a lot of attention in the past decade and we will not attempt to give an exhaustive overview of the existing literature. Some examples have already been shortly mentioned in Section~\ref{sec:introduction}. In what follows, we give a few more details on the case of star graphs with two or more edges, and on the topological assumption preventing the existence of ground states. \subsubsection{Star graphs with two or more edges}\label{sec:delta-ground-states} One of the simplest nontrivial graph is given by two semi-infinite half-lines connected at a vertex, with $\delta$ type condition on the vertex. In this case, the operator $H$ is equivalent to the second order derivative on $\R$ with point interaction at $0$. In this setting, existence and stability of standing waves for a focusing power-type nonlinearity was treated by Fukuizumi and co.~\cite{FuJe08,FuOhOz08,LeFuFiKsSi08}, using techniques based on Grillakis--Shatah--Strauss stability theory (see~\cite{GrShSt87,GrShSt90} for the original papers and~\cite{DeGeRo15,DeRo19} for recent developments). Various generalizations have been obtained, in particular for a generic point interaction~\cite{AdNo09,AdNo13,AdNoVi13} ($\delta$ or $\delta'$ boundary conditions) or in the case of non-vanishing boundary conditions at infinity~\cite{IaLeRo17}. In particular, the following results have been obtained in~\cite{AdNoVi13}. \begin{prop} Assume that $\clG$ is formed by two semi-infinite edges $\{ e_1,e_2\}$ connected at the vertex $v$. Let $H:D(H)\subset L^2(\clG)\to L^2(\clG)$ be the operator $-\partial_{xx}$ with one of the following conditions to be satisfied at the vertex. \begin{itemize} \item Attractive $\delta$ conditions: \[ \varphi_{e_1}(v)=\varphi_{e_2}(v),\quad \varphi_{e_1}'(v)+\varphi_{e_2}'(v)=\alpha\varphi(v),\;\alpha<0. \] \item Attractive $\delta'$ conditions: \[ \varphi_{e_1}(v)-\varphi_{e_2}(v)=\beta\varphi_{e_2}'(v),\;\beta<0,\quad \varphi_{e_1}'(v)+\varphi_{e_2}'(v)=0. \] \item Dipole conditions: \[ \varphi_{e_1}(v)+\tau\varphi_{e_2}(v)=0,\quad \varphi_{e_1}'(v)+\tau\varphi_{e_2}'(v)=0,\quad \tau\in\R. \] \end{itemize} Define for $\varphi\in H^1_D(\clG)$ the energy \[ E(\varphi)=Q(\varphi)-\frac{1}{p+1}\norm{\varphi}_{L^{p+1}}^{p+1}, \] where $1

0$ there exists up to phase shift and translation a unique minimizer to \[ \min\left\{E(\varphi):\varphi\in H^1_D(\clG),\,M(\varphi)=m\right\}. \] \end{prop} A detailed review of these results as well as announcement of new results can be found in~\cite{AdBoRu20}. The minimizer is in fact explicitly known, and we use its explicit form in Section~\ref{sec:2edges} to compare the outcome of our numerical experiences with the theoretical ground states. \subsubsection{General non-compact graphs with Kirchhoff condition} The existence of ground states with prescribed mass for the focusing nonlinear Schr\"odinger equation on non-compact graphs $\clG$ equipped with Kirchhoff boundary conditions is linked to the topology of the graph. Actually, a topological hypothesis, usually referred to as Assumption~\ref{ass:H} can prevent a graph from having ground states for every value of the mass (see~\cite{AdSeTi17b} for a review). For the sake of clarity, we recall that a \textsl{trail} in a graph is a path made of adjacent edges, in which every edge is run through exactly once. In a trail, vertices can be run through more than once. The Assumption~\ref{ass:H} has many formulations (see~\cite{AdSeTi17b}) but we give here only the following one. %\begin{enonce}{Assumption}[Assumption~(H)]\label{ass:H} %Every $x\in \clG$ lies in a trail that contains two half-lines. %\end{enonce} \setcounter{assuml}{7} \begin{assuml}\label{ass:H} Every $x\in \clG$ lies in a trail that contains two half-lines. \end{assuml} Under Assumption~\ref{ass:H}, no {global} minimizer exists, unless $\clG$ is (up to symmetries) isomorphic to $\R$ {(note that this assumption does not prevent the existence of local minimizers)}. Let us consider for example a general $N$-edges star-graph $\clG$ (see Figure~\ref{fig:N_star_graph}). The $N$ star-graph with $N>2$ verifies Assumption~\ref{ass:H}, so there are no ground states in this case without adding more constraints. Another example satisfying Assumption~\ref{ass:H} is the triple bridge $\clB_3$ (represented in Figure~\ref{fig:three_bridge}). \begin{figure}[htbp!] \centering \begin{tikzpicture} \node[left] at (-4,0) {$\infty$}; \node[right] at (4,0) {$\infty$}; \draw[dashed] (-4,0) to (-2.5,0); \draw (-2.5,0) to (2.5,0); \draw[dashed] (2.5,0) to (4,0); \draw (-.75,0) to [bend right=45] (.75,0); \draw (-.75,0) to [bend left=45] (.75,0); \node at (-.75,0) {$\bullet$}; \node at (.75,0) {$\bullet$}; \end{tikzpicture} \caption{The $3$-bridge $\clB_3$}\label{fig:three_bridge} \end{figure} When we are searching to obtain ground states, we consider graphs violating Assumption~\ref{ass:H}, for example the signpost graph or a line with a tower of bubbles (Figure~\ref{fig:signpost_bubbles}). \begin{figure}[htbp!] \centering \begin{tabular}{cc} \begin{tikzpicture} \node[left] at (-2,0) {$\infty$}; \node[right] at (2,0) {$\infty$}; \draw (-2,0) -- (0,0) -- (0,1) -- (0,0) -- (2,0); \node at (0,0) {$\bullet$}; \node at (0,1) {$\bullet$}; \draw (0,1.25) circle (0.25); \end{tikzpicture}& \begin{tikzpicture} \node[left] at (-2,0) {$\infty$}; \node[right] at (2,0) {$\infty$}; \draw (-2,0) -- (0,0) -- (2,0); \node at (0,0) {$\bullet$}; \draw (0,0.5) circle (0.5); \node at (0,1) {$\bullet$}; \draw (0,1.25) circle (0.25); \end{tikzpicture}\\ \end{tabular} \caption{Line with a signpost graph (left) and with a tower of bubbles (right).}\label{fig:signpost_bubbles} \end{figure} \section{Continuous normalized gradient flow}\label{sec:cngf} We want here to show that, when the standing wave profile $\phi$ is a strict local minimizer for the energy on fixed mass, the corresponding continuous normalized gradient flow (i.e. the gradient flow of the energy projected on the mass constraint) converges towards $\phi$. The continuous normalized gradient flow is defined by \begin{equation}\label{eq:CNGF} \partial_t\psi=-E'(\psi)+\dual{E'(\psi)}{\frac{\psi}{\norm{\psi}_{L^2}}}\frac{\psi}{\norm{\psi}_{L^2}},\quad \psi(t=0)=\psi_0, \end{equation} where $\psi=\psi(t,\cdot)$. It is the projection of the usual gradient flow \[ \partial_t\psi=-E'(\psi) \] on the $L^2$ sphere \[ \clS_{\psi_0}=\left\{ u\in H^1_D(\clG) :\norm{u}_{L^2}=\norm{\psi_0}_{L^2}\right\}. \] Let $\phi\in H^1_D(\clG)$ be a standing wave profile solution of~\eqref{eq:snls}. We define the linearized action operator $L_+$ around $\phi$ by \begin{equation}\label{eq:L_+} \begin{aligned} L_+:D(H)\subset L^2(\clG)&\to L^2(\clG),\\ u&\mapsto Hu+\omega u-f'(\phi)u. \end{aligned} \end{equation} We will assume that the bound state $\phi$ is a strict local minimizer of the energy on fixed $L^2$-norm, which translates for $L_+$ into the following assumption. \begin{enonce}{Assumption}\label{ass:coercivity} There exists $\kappa>0$ such that for any $\varphi\in D(H)$ verifying \begin{align*} \scalar{\varphi}{\phi}_{L^2} &=0, \\ \intertext{we have} \scalar{L_+\varphi}{\varphi}_{L^2}&\geq \kappa\norm{\varphi}_{H^1}^2. \end{align*} \end{enonce} Since the pioneering work of Weinstein~\cite{We85}, this assumption is well known to hold (if one removes translations and phase shifts) in the classical case of Schr\"odinger equations on $\R^d$ with subcritical power-nonlinearities ($f(\varphi)=|\varphi|^{p-1}\varphi$, $1

0$ and $C>0$ such that for every $\psi_0\in H^1_D(\clG)$ such that
\[
\norm{\psi_0-\phi}_{H^1}<\eps
\]
the unique solution $\psi\in\clC([0,T),H^1_D(\clG))$ of~\eqref{eq:CNGF} is global (i.e. $T=\infty$) and converges to $\phi$ exponentially fast: for every $t\in[0,\infty)$ we have
\[
\norm{\psi(t)-\phi}_{H^1}