1 Introduction↩︎

The study of the NLS equation on metric graphs has experienced substantial growth over the last decade. A rigorous analysis was probably initiated by [1], where the behavior of fast solitons of the cubic NLS equation after collision with the vertex of a three-edge star graph under several boundary conditions was investigated, in analogy with the seminal study on the line in [2]. In the subsequent years, research in the area has focused almost exclusively on the stationary NLS equation, describing the amplitude of bound states, i.e. periodic localized solutions of the NLS equation. This has led to a large body of results on the existence and properties of ground states (corresponding to several definitions and obtained by several methods) and on the existence of various classes of standing waves. In some cases, their orbital stability and instability has been established, essentially as a by-product of the variational or spectral analysis of the stationary problem and its linearization. These results rely on the methods originating from the seminal works of Cazenave and Lions [3] and of Grillakis, Shatah and Strauss [4], [5], together with later developments (see, e.g., [6] and references therein for a review). In this paper we firstly provide results on the time behavior of untrapped, soliton-like solutions of the NLS equation on a broad and well-defined class of non-compact metric graphs with finitely many edges, whose importance was highlighted in [7], [8]. The graphs we consider are characterized by the property that every point lies on a trail (i.e., a sequence of distinct edges) containing two half-lines. This topological assumption is called Assumption H in [7],[8]; it is shown there that Assumption H implies the generic absence of an NLS ground state (defined as a minimizer of the NLS energy at fixed \(L^2\) mass), with a single exception: the so-called tower of bubbles, for which the ground state exists and can be exhibited explicitly. This class is therefore large enough to contain both the generic no-ground-state regime and the unique exceptional case with a ground state and it offers a natural and robust setting. At the same time, as we will discuss in the last section, the confinement argument can be extended well beyond Assumption H and it reveals a more fundamental threshold phenomenon. We now come to the description of our results.

First, in Proposition 9, without distinguishing between the two cases, we show that if an initial datum is supported on a single half-line of the graph and is close in the energy norm to a truncated soliton, then along the NLS flow it remains confined to the same half-line and stays close (up to a suitable rephasing and translation) to the truncated soliton for all times. This statement resembles orbital stability, except that the standard definition compares solutions with an actual solitary/standing wave, whereas here the solution is compared with a reference function (a soliton on the half-line plus a small error on the rest of the graph), to which no genuine solitary wave corresponds. The idea is that a soliton placed on a half-line sufficiently far from the compact core (so that the truncation error is small in the energy norm) still effectively behaves as a configuration that is stable for the NLS dynamics, in an appropriate and rigorous sense. A further piece of information contained in Proposition 9 is that the initial datum can be chosen so that the corresponding soliton-like solution never comes closer than a fixed a priori distance to the compact core of the metric graph (see Proposition 9 for the precise formulation).

For comparison, a related result was recently proved for the KdV equation on the half-line with Dirichlet boundary condition (see [9] and the subsequent [10], [11] the latter treating also different b.c.). Note that a half-line is a very special metric graph, and the Dirichlet condition presumably prevents the existence of an exact solitary wave, as remarked by the authors of those papers. Their strategy uses an adaptation of the ideas in [12], based on the introduction of certain almost-conserved quantities and their monotonicity properties. Here, instead, we consider the NLS equation in the substantially more general setting of metric graphs. Our proofs exploit a version of concentration–compactness adapted to non-compact metric graphs with finitely many edges (in particular, covering all graphs satisfying Assumption H), developed in [13], combined with a contradiction argument. Roughly speaking, if the claim were false along a sequence of initial data, the corresponding evolutions would produce a minimizing sequence. Such a minimizing sequence could either converge (which is ruled out by the absence of ground states) or be runaway, i.e.escape to infinity along a single half-line carrying all the mass. The runaway scenario is also excluded, since it would imply a false inequality, yielding a contradiction. In the exceptional tower of bubbles case, where mass and energy alone are not sufficient, we introduce an additional functional \(F\) (see 16 ) that is almost conserved along minimizing sequences; this allows us to conclude, again by contradiction.

A relevant example of initial datum satisfying the hypotheses of Proposition 9 is that of a slow soliton placed far from the vertex and moving toward it, discussed in Subsection 4.2. In this case, the main result implies that the solution remains on the same half-line and stays close to a soliton in the energy norm, if the velocity initially present is below a critical velocity: slow solitons are reflected, as shown in Proposition 10. This behavior is reminiscent of the so-called quantum reflection of matter-wave solitons, well known in the physics of Bose–Einstein condensates and matter wave solitons (see, e.g., [14]–[16]). Beyond the rigorous result above, the quantum-reflection phenomenon is further investigated numerically in Section 6. In particular, we show that the kinetic energy of the incoming soliton attains its maximum at the collision time, a markedly non-classical behavior.
We also stress that analogous results on confinement, orbital stability and slow-soliton reflection hold for the more standard case of the subcritical NLS on the line with a repulsive potential; this is proved in the final part of the paper (see Proposition 12), where we also compare our approach with the existing literature on the subject. This model, and the other discussed in Section 6, show that confinement and reflection of slow solitons have to be interpreted as a general threshold energy phenomenon and they are not restricted to metric graphs in the class \(H\). We find that the latter remains however a convenient setting, useful to exhibit the relevant behaviors in both the situations of absence or presence of ground state.

Our second main result concerns the tower of bubbles (see the definition in Subsection 3.1 and Figure 1). This is the only subfamily within class H that admits an NLS ground state. The general expectation is that ground states are orbitally stable, following the celebrated argument of Cazenave and Lions [3]. Here, however, one must take into account that a key assumption in [3] is the relative compactness of all minimizing sequences in the energy space. This fails for tower of bubbles graphs, because there exist minimizing sequences that escape along a single half-line (runaway sequences in the terminology above; see [13] and Subsection 2.5), preventing convergence. To overcome this difficulty and prove orbital stability of the ground state (see Proposition 11), we again exploit the functional \(F\) introduced above, showing that instability of the ground state would contradict the continuity of the map \(t\mapsto F(\xi(t))\) for \(t\in\mathbf{R}\), where \(\xi\) denotes the solution of the time-dependent NLS equation.

The paper is organized as follows. In Section 2 we collect preliminary notation and background on metric graphs, introduce several constructions used later, and recall variational properties of minimizing sequences and of the NLS equation on metric graphs. Most of this material is known but scattered in the literature; here we present it in a compact and unified way. In Section 3 we focus on the class of non-compact metric graphs under consideration, including its definition and basic properties. In Section 4 we prove the first main result on confinement of solitons, stated as Proposition 9; the dynamics of slow solitons is also discussed there, see Proposition 10. In Section 5 we prove orbital stability of ground states for tower-of-bubbles graphs, stated as Proposition 11. In Section 6 we discuss how the hypotheses of the presented results can be weakened in various directions and we provide some open problems. Finally, we present numerical simulations of collision and reflection of a slow soliton at the vertex of a three-edge star graph, illustrating the results of Section 4. The simulations use the package QGLAB recently developed in [17].

2 Preliminaries↩︎

2.1 Quantum graphs↩︎

We begin by introducing some tools which are necessary to deal with the NLS equation on metric graphs.
We consider a metric graph \(\mathcal{G} = (V,E)\), where \(V\) is the set of vertices and \(E\) is the set of edges. We will always consider finite graphs, so that the cardinalities \(|V|\) and \(|E|\) are finite. Each edge \(e \in E\) has an associated length \(L_e \in (0,+\infty]\) and it is identified with the interval \(I_e = [0,L_e]\), if \(L_e\) is finite, or with the interval \(I_e=[0,+\infty)\), if \(L_e\) is infinite. An edge of finite length is called bounded edge, while an edge of infinite length is called half-line. If the graph \(\mathcal{G}\) possesses a half-line, we say that \(\mathcal{G}\) is non-compact; we say \(\mathcal{G}\) is compact if it has no half-lines. To every bounded edge we associate two vertices, and to every half-line we associate a single vertex. On each interval \(I_e\) we fix a coordinate \(x\), such that the points \(x=0\) and \(x = L_e\) correspond to vertices if \(L_e< +\infty\); if instead \(L_e = +\infty\), the point \(x = 0\) corresponds to the unique vertex associated to the half-line \(e\). When a vertex \(\underline{v} \in V\) belongs to an edge \(e \in E\), we write \(\underline{v} \in e\). We denote by \(\{e \prec \underline{v}\}\) the set of edges connecting the vertex \(\underline{v} \in V\). In the following, we will denote points on the graph with \(\underline{x} =(e,x)\), where \(e \in E\) identifies the edge and \(x \in I_e\) the coordinate on the corresponding edge. Given two points \(\underline{x}\) and \(\underline{y}\) on \(\mathcal{G}\), we define the distance \(d(\underline{x},\underline{y})\) as infimum of the length of the paths connecting the two points. With this distance, the couple \((\mathcal{G},d)\) is a locally compact metric space. We call trail a finite sequence of consecutive edges \(e \in E\) in which no edge is repeated (a vertex, instead, may be repeated, for example in presence of loops). A path is a trail in which no vertex is repeated. We say that a graph is connected if for every pair of vertices \(\underline{v_1},\underline{v_2} \in V\), there exists a path connecting the two. If a graph is not connected, we say it is disconnected.
A function \(\psi: \mathcal{G} \to \mathbf{C}\) can be regarded as a family of functions \(\{\psi_e\}_{e \in E}\) with \(\psi_e: I_e \to \mathbf{C}\). In particular, if \(\underline{x} = (e,x)\), then \[\psi(\underline{x}) = \psi_e(x).\] At convenience, we will also use the notation \((\psi)_e = \psi_e\) for \(e \in E\). The spaces \(L^q(\mathcal{G})\), \(1\leq q\leq \infty\), are made of functions \(\psi\) on \(\mathcal{G}\) such that \(\psi_e \in L^q(I_e)\) for every \(e \in E\) and we define \[||\psi||^q_{L^q(\mathcal{G})} = \sum_{e \in E} ||\psi_e||^q_{L^q(I_e)}, \;1 \leq q < \infty \quad \text{and} \quad ||\psi||_{L^{\infty}(\mathcal{G})} = \max_{e \in E} ||\psi_e||_{L^{\infty}(I_e)}\] The space \(L^2(\mathcal{G})\) forms a Hilbert space, with the scalar product defined in the natural way. We denote by \(C(\mathcal{G})\) the set of functions which are continuous on \(\mathcal{G}\), and we introduce the spaces \[H^1(\mathcal{G}) = \{ \psi \in C(\mathcal{G}) \;\text{s.t.} \;\psi_e \in H^1(I_e), \forall e \in E\}\] equipped with the norm \[||\psi||^2_{H^1(\mathcal{G})} = \sum_{e \in E}||\psi_e||^2_{H^1(I_e)}\] and \[H^2(\mathcal{G}) = \{ \psi \in C(\mathcal{G}) \;\text{s.t.} \;\psi_e \in H^2(I_e), \forall e \in E\}\] equipped with the norm \[||\psi||^2_{H^2(\mathcal{G})} = \sum_{e \in E}||\psi_e||^2_{H^2(I_e)}.\] We recall the Gagliardo-Nirenberg inequality for metric graphs [8], which we shall use in the case of connected non-compact graphs. Notice that the constants \(C,C'>0\) appearing in the statement do not depend on the structure of the non-compact graph.

2.2 Sub-graphs↩︎

In the following we will consider the restriction of a function \(u \in H^1(\mathcal{G})\) to a sub-graph of \(\mathcal{G}\). A sub-graph \(\mathcal{B} = (E',V')\) will be formed from a subset of the vertices and edges of \(\mathcal{G}\), and will be possibly disconnected. The vertex subset \(V'\) will include only the vertices connected to an edge of the subset \(E'\). For \(u \in H^1(\mathcal{G})\), we use the notation \[||u||^2_{H^1(\mathcal{B})} = \sum_{e \in E'} ||u||^2_{H^1(I_e)},\] and similarly for the other \(L^q\)-norms, with \(1\leq q<\infty\). We also make use of the following construction. Suppose \(\mathcal{G} = (V,E)\) is a connected graph and \(\mathcal{T} =(e_1, e_2, ..., e_n)\) is a trail of \(\mathcal{G}\), with \(n \in \mathbf{N}\). Suppose that \(e_1\) and \(e_n\) are half-lines. Call \(\underline{v_1}\) the vertex associated to \(e_1\). Given \(u \in H^1(\mathcal{G})\), our goal is to define a continuous function \(u|_{\mathbf{R}} \in H^1(\mathbf{R})\), by restricting \(u\) to the trail \(\mathcal{T}\). We proceed as follows. To the edge \(e_1\) of the trail, we associate the interval \(J_{1} = (-\infty,0]\). To the edge \(e_2\) we associate the interval \(J_2 = [0,L_{e_2}]\), where \(L_{e_2}\geq 0\) is the length of \(e_2\). We proceed by consecutive edges in this way, until the entire trail has been covered, except for the half-line \(e_n\). We associate to \(e_n\) the interval \(J_{n} = [l,+\infty]\), where \(l \geq 0\) is the total length of the edges \(e_2,...,e_{n-1}\). If \(u \in H^1(\mathcal{G})\), we define the function \(u|_{\mathbf{R}}: \mathbf{R} \to \mathbf{C}\) as follows. For \(x \in J_{1}\), we set \(u|_{\mathbf{R}}(x) = u_{e_1}(-x)\). If the edge \(e_2\) is outgoing from \(\underline{v_1}\), we set \(u|_{\mathbf{R}}(x) = u_{e_2}(x)\), for \(x \in J_{e_2}\). If the edge \(e_2\) is incoming in \(\underline{v_1}\), we set \(u|_{\mathbf{R}}(x) = u_{e_2}(L_{e_2}-x)\). We proceed in a similar way, until we reach the half-line \(e_n\). For \(x \in J_{n}\), we set \(u|_{\mathbf{R}}(x) = u_{e_n}(x-l)\). We obtain in this way a continuous function \(u|_{\mathbf{R}} \in H^1(\mathbf{R})\) such that \[||u|_{\mathbf{R}}||^2_{H^1(\mathbf{R})} = \sum_{e \in \mathcal{T}}||u_e||^2_{H^1(I_e)},\] and that \[||u|_{\mathbf{R}}||^q_{L^q(\mathbf{R})} = \sum_{e \in \mathcal{T}}||u_e||^q_{L^q(I_e)}\] for any \(1 \leq q < \infty\). If \(\mathcal{B}_{\mathcal{T}}\) represents the graph obtained by removing from \(\mathcal{G}\) the edges in the trail \(\mathcal{T}\), and the resulting disconnected vertices, we write \[||u||^2_{H^1(\mathcal{G})} = ||u||^2_{H^1(\mathcal{B}_{\mathcal{T}})} + ||u|_{\mathbf{R}}||^2_{H^1(\mathbf{R})},\] and similarly for the other \(L^q\)-norms, with \(1 \leq q < \infty\).

2.3 NLS equation on graphs↩︎

In this section we recall some properties of the NLS equation on metric graphs. For a more complete treatment we refer to [13].
The NLS equation on a graph \(\mathcal{G} = (V,E)\) reads \[i\partial_t\psi = H\psi -|\psi|^{p-2}\psi \label{NLS95eq}\tag{1}\] In the following, the power \(p\) will be a given number in the interval \((2,6)\), and we refer to this regime as subcritical. The operator \(H\) in (1 ) acts as \[(H \psi)_e = -\psi''_e, \qquad \forall e \in E.\] The domain of \(H\) is defined by means of the Kirchhoff condition at the vertices: \[D(H) = \{\psi \in H^2(\mathcal{G}), \;\text{s.t.} \;\sum_{e \prec \underline{v}}\partial_o\psi_e(\underline{v}) = 0 \quad \forall \underline{v} \in V\}, \label{domain}\tag{2}\] where \(\partial_o\) denotes the outgoing derivative from the vertex, and it corresponds to \(\partial_x\) or to \(-\partial_x\) according to the orientation of the edge. In particular, the operator \((H,D(H))\) is self-adjoint on \(L^2(\mathcal{G})\) and the quadratic form associated with \(H\) is \[Q[\psi] = \frac{1}{2}||\psi'||^2_{L^2(\mathcal{G})}, \quad \text{with domain} \quad \mathcal{D}(Q) = H^1(\mathcal{G}).\] Finally, the nonlinearity in equation (1 ) is interpreted edge by edge, i.e. \((|\psi|^{p-2}\psi)_e = |\psi_e|^{p-2}\psi_e, \;\forall e \in E.\)
The equation in (1 ) admits two conserved quantities, namely the mass \[M(\psi) = ||\psi||^2_{L^2(\mathcal{G})} \label{mass}\tag{3}\] and the energy \[E(\psi,\mathcal{G}) = \frac{1}{2} ||\psi'||_{L^2(\mathcal{G})}^2 - \frac{1}{p}||\psi||^p_{L^p(\mathcal{G})} \label{energy}\tag{4}\] The well-posedness of the Cauchy problem associated to equation (1 ) has been proven in [13], by applying a Banach fixed point theorem. The result can be summarized as follows.

2.4 Ground states on non-compact graphs↩︎

Given a non-compact graph \(\mathcal{G}\) and a mass \(\mu >0\), we introduce the following set of functions with prescribed mass \[H^1_{\mu}(\mathcal{G}) = \{ \psi \in H^1(\mathcal{G}), \quad M(\psi) = \mu\}. \label{eq:32H195mu}\tag{5}\] We then consider the following minimization problem \[\mathcal{E}_{\mathcal{G}}(\mu) := \inf_{\psi \in H^1_{\mu}(\mathcal{G})}E(\psi,\mathcal{G}). \label{minimization95problem}\tag{6}\] In the subcritical regime, the Gagliardo-Nirenberg inequality (?? ) implies that, for any \(\mu >0\), the energy in (4 ) is bounded from below in the set \(H^1_{\mu}(\mathcal{G})\), i.e. \(\mathcal{E}_{\mathcal{G}}(\mu)> -\infty\) for any \(\mu >0\). We give the following definition.

A familiar example of metric graph, trivial but however of great importance in the following, is the real line \(\mathcal{G}= \mathbf{R}\). It can be considered as composed by two half-lines joined at a vertex, where the Kirchhoff condition reduces to continuity of the function and its derivative. In this case, the set of ground states is well known and explicit: given a mass \(\mu >0\) the set of ground states is the set \[\label{Omu} \mathcal{O}_{\mu} = \{ e^{i\theta}\phi_{\mu}(\cdot-x_0), \; \text{with} \; (\theta, x_0) \in \mathbf{R}^2\}\tag{7}\] where \[\phi_{\mu}(x) = C_p \;\mu^{\frac{2}{6-p}}\mathop{\mathrm{sech}}^{\frac{2}{p-2}}(c_p \mu^{\frac{p-2}{6-p}}x), \label{eq:32soliton}\tag{8}\] with the constants \(C_p\) and \(c_p\) depending only on \(p\) (see [18]). The function \(\phi_{\mu}\) is the unique positive and even ground state.
In the case of connected non-compact graphs, NLS ground states do not always exist. This fact has been investigated in [7], [19], and it turns out that the existence of ground states is related to the topology of graphs, as we will see in Section 3. When they exist, ground states are real and positive. For now, we state the following proposition from [7], of which we report the proof.

2.5 Concentration-compactness principle↩︎

The concentration-compactness principle is a variational tool that has been introduced by P.L. Lions [20] to describe the possible behaviors of sequences with prescribed \(L^2\)-norm in \(\mathbf{R}^d\), with \(d \in \mathbf{N}\). In the case of non-compact graphs, this principle has been suitably adapted in [13], and can be summarized as follows.
Let \(\mathcal{G}\) be a connected non-compact graph. For any function \(\psi \in L^2(\mathcal{G})\) and \(t \geq 0\) we define the concentration function \[\rho(\psi,t) = \sup_{\underline{y}\in \mathcal{G}}||\psi||^2_{L^2(B(\underline{y},t))},\] where \(B(\underline{y},t)\) represents the open set of points on \(\mathcal{G}\) whose distance from \(\underline{y}\) is less than \(t\geq0\). Given a sequence \(\{\psi_n\}\subset L^2(\mathcal{G})\) such that \(||\psi_n||^2_{L^2(\mathcal{G})} \to \mu >0\), the concentration parameter \(\tau\in [0,\mu]\) is \[\tau = \lim_{t \to +\infty}\liminf_{n \to \infty}\rho(\psi_n,t)\] We can now state the following proposition (see [13] Section 3 and in particular Lemma 3.7).

In the next sections, we will apply the concentration-compactness principle to a particular class of sequences, called minimizing sequences, which are defined as follows.

Using arguments based on energy estimates (see [3], [13], [21]), it can be proven that for minimizing sequences the vanishing and dichotomy behaviors are prohibited on any connected non-compact graph. The only possible behaviors are convergence or runaway. If a minimizing sequence is runaway, it escapes toward infinity along a single half-line, in a concentrated manner. On the other hand, if a minimizing sequence is convergent, then the limit function provides a ground state for the system, as explained in the following proposition.

3 The topological assumption H↩︎

We are interested in a family of graphs which is characterized by a topological property, called assumption H. This property has been introduced for the first time in [7] (see also [19]) in relation to the problem of existence of ground states on connected non-compact graphs. We recall it in the following definition.

In other words, assumption H tells us that, given any point \(\underline{x}\) in \(\mathcal{G}\), it is possible to reconstruct a copy of the real line \(\mathbf{R}\) that passes from \(\underline{x}\), by collecting consecutive edges of \(\mathcal{G}\). This property allows one to connect the minimization problem in (6 ) to the one on the real line, as explained in the following proposition.

An interesting sub-family of graphs that satisfy assumption H is made of bubble tower graphs. These graphs consist in two half-lines attached to a single chain of bubbles, as depicted in Figure 1. The number of bubbles is arbitrary (but greater than zero) and the size of each bubble is also arbitrary. If a bubble is made of two edges, they must have the same length.

Together with the real line, bubble tower graphs are the only graphs that satisfy assumption H and admit ground states, as stated in the following proposition.

We refer to [7], [19] for the proof and an extended explanation.

3.1 Bubble tower graphs↩︎

In this section we show that, for any mass \(\mu>0\), bubble tower graphs admit a unique (up to phase shifts) ground state.
Suppose \(\mathcal{G}\) is a bubble tower graph. Thanks to the particular symmetry of \(\mathcal{G}\), we can place the positive and even ground state \(\phi_{\mu}\) in (8 ) on the graph, obtaining a positive function \(\Phi_{\mu} \in H^1_{\mu}(\mathcal{G})\) (we refer to the Figure 2 for the actual procedure, see also [7]). The function \(\Phi_{\mu}\) has the same energy and mass of the original function \(\phi_{\mu}\), i.e. \[M(\Phi_{\mu}) = \mu \quad \text{and} \quad E(\Phi_{\mu},\mathcal{G}) = E(\phi_{\mu},\mathbf{R}),\] and provides the unique positive ground state of \(\mathcal{G}\), as stated in the following proposition.

4 Soliton confinement and reflection of slow solitons↩︎

4.1 Soliton confinement and stability↩︎

In this section we state precisely and prove the first of the main results of this paper.

Before proving proposition ([main95prop]), we introduce the following weaker statement.

4.2 On the dynamics of slow solitons↩︎

In this subsection we investigate, by means of Proposition [main95prop], the dynamics of slow solitons on graphs that satisfy assumption H (assuming \(\mathcal{G} \neq \mathbf{R}\)). More precisely, we shall consider an approximate soliton, initially supported on a half-line \(e_*\) of \(\mathcal{G}\), that runs toward the nearest vertex of the graph with velocity \(v >0\), or that escapes from it with velocity \(v<0\). In both cases, we show that when the velocity \(v \in \mathbf{R}\) is small enough, the slow soliton preserves its shape in time and it remains confined on the half-line \(e_*\).
Consider a graph \(\mathcal{G}\) as in Proposition [main95prop], and call \(e_*\) one of its half-lines. Let \(\chi \in C^{\infty}(\mathbf{R}_+)\) be a smooth cut-off such that \(\chi(x) =0\) on \((0,1)\) and \(\chi(x) = 1\) on \((2,+\infty)\). Consider \(\psi_{(x_0,v)} \in H^1(\mathcal{G})\) of the form \[(\psi_{(x_0,v)})_e(x) = \begin{cases} e^{-i\frac{v}{2}x}\chi(x)\phi_{\mu}(x-x_0) \qquad\text{if} \quad e=e_*\\ 0 \qquad \qquad \qquad \qquad \;\;\qquad \text{if} \quad e \neq e_*. \end{cases} \label{slow95soliton}\tag{17}\] where \(x_0 \in \mathbf{R_+}\) and \(v \in \mathbf{R}\). The function \(\psi_{(x_0,v)}\) represents the initial datum of an approximate soliton supported on \(e_*\) and located at \(x_0 >0\), running towards the nearest vertex of the graph when \(v>0\) and escaping from it when \(v<0\). We have the following proposition.

5 Orbital stability of ground states in bubble tower graphs↩︎

In this section we first recall the notion of orbital stability on a metric graph and then prove the orbital stability of the ground state orbit on bubble tower graphs. The main point is that the standard Cazenave–Lions argument does not apply directly, because on bubble tower graphs one has non-compact minimizing sequences escaping along a half-line. We therefore isolate the precise compactness hypothesis needed in the classical argument and then modify the proof accordingly.

5.1 The argument of Cazenave-Lions on non-compact graphs↩︎

For the NLS equation 1 on a connected non-compact graph \(\mathcal{G}\), the point of the Cazenave–Lions argument for orbital stability is not simply the existence of a ground state at mass \(\mu\), but the compactness of every minimizing sequence at that mass. In the present setting the relevant hypothesis is the following.

• for some \(\mu>0\), every minimizing sequence with asymptotic mass \(\mu\) (see Definition 2) is relatively compact in \(H^1(\mathcal{G})\).

This is the compactness assumption behind Case 2 on p. 552 of [3], suitably adapted to the present setting. In particular, (HP) implies the existence of a ground state at mass \(\mu\), but the converse is false in general. Bubble tower graphs provide exactly such a situation. They do admit a ground state at every mass \(\mu>0\), but they do not satisfy (HP). Indeed, by Proposition 6 one has \(\mathcal{E}_{\mathcal{G}}(\mu)=E(\phi_\mu,\mathbf{R})\). As in the proof of Proposition 3, one can construct a sequence \(\{u_n\}\subset H^1_\mu(\mathcal{G})\) supported on a single half-line and escaping to infinity, while asymptotically reconstructing the soliton \(\phi_\mu\) on the line. This is a minimizing sequence which is runaway, hence not relatively compact in \(H^1(\mathcal{G})\). Therefore the classical Cazenave–Lions argument cannot be applied directly to bubble tower graphs. In the next subsection we adapt the argument and prove Proposition 11.

5.2 Proof of orbital stability↩︎

6 Extensions and some further problems↩︎

In this section we collect some further results and remarks on the scope of our findings and we also formulate a few open problems. Moreover, we report numerical simulations of the collision of a slow NLS soliton with the compact core of the simplest example of a metric graph satisfying Assumption H.

6.1 Generalizations and possible extensions: the line↩︎

We want to extend our results to the relevant case of the NLS on the line with an inhomogeneity, not provided by the compact core of the metric graph, but by an external potential.

Consider the focusing NLS equation in the presence of an external repulsive potential \(V(x)\): \[i\partial_tu = -\partial_x^2 u - |u|^{p-2}u +V(x)u. \label{eq:32NLS95potential}\tag{22}\] We treat two prototypical cases:
i) the potential \(V\) is continuous non-negative and vanishing at infinity: \(V(x)\geq0\) for all \(x \in \mathbf{R}\), such that \(V(x) \to 0\) as \(|x| \to \infty\) and \(V\) is different from the zero function;
ii) zero-range potential \(V(x) = g\delta(x)\), with \(g >0\), also called delta potential or, more properly, a point interaction.
In both cases it is well known that the Schrödinger operator \(H=-\Delta+V\), suitably interpreted, is self-adjoint and that the subcritical NLS equation enjoys well posedness in \(H^1(\mathbf{R})\). Moreover the NLS equation conserves the mass \(M(u) = ||u||^2_{L^2(\mathbf{R})}\) and the energy \[E_V(u) := \frac{1}{2}||u'||^{2}_{L^2(\mathbf{R})}-\frac{1}{p}||u||^{p}_{L^p(\mathbf{R})} + \frac{1}{2}\int_{\mathbf{R}}V(x)|u(x)|^2dx,\] or \[E_g(u) := \frac{1}{2}||u'||^{2}_{L^2(\mathbf{R})}-\frac{1}{p}||u||^{p}_{L^p(\mathbf{R})} + \frac{g}{2}|u(0)|^2,\] These facts are well known for the regular potential, while for the delta potential one can see for example [13], [22], [23] and references therein. For convenience, in this section we denote the free energy as \[E_0(u) = \frac{1}{2}||u'||^{2}_{L^2(\mathbf{R})}-\frac{1}{p}||u||^{p}_{L^p(\mathbf{R})},\] so that \(E_V(u)= E_0(u)+\frac{1}{2}\int_{\mathbf{R}}V(x)|u(x)|^2dx\) for any \(u \in H^1(\mathbf{R})\), and similarly for \(E_g\).
Similarly to 5 , we define for any \(\mu >0\) the set \(H^1_{\mu}(\mathbf{R}):= \{u \in H^1(\mathbf{R}), \;M(u) = \mu\}\), and we consider the following minimization problem: \[\inf_{u \in H_{\mu}^1(\mathbf{R})}E_V(u).\] Using the repulsive nature of the potential, and its decay at infinity, we obtain the following simple lemma, the proof of which we provide for completeness.

We can now state the following proposition, regarding the confinement of approximate solitons by means of the external potential. While the proof does not make use of essentially new ingredients with respect to the already treated case of metric graphs, we feel that it is useful to give it in detail, being the setting and scope a little different.

As in the case of graphs, it seems convenient to first focus on the following weaker statement.

Let us now prove Proposition 12.

Analogously to the case of metric graphs, reflection of slow solitons still holds. We state the result, avoiding the repetition of the proof, which in this case does not present any relevant difference with the one of Proposition 10. We preliminarily define the asymptotic approximated soliton: \[(\psi_{(x_0,v)})(x) = \begin{cases} e^{-i\frac{v}{2}x}\chi(x)\phi_{\mu}(x-x_0) \qquad\text{if} \quad x\in \mathbb{R}_+\\ 0 \qquad \qquad \qquad \qquad \;\;\qquad \text{if} \quad x\in \mathbb{R}_- \end{cases} \label{slow95soliton2}\tag{35}\] where \(x_0 \in \mathbf{R_+}\) and \(v \in \mathbf{R}\). Then, \(\chi \in C^{\infty}(\mathbf{R}_+)\) is a smooth cut-off such that \(\chi(x) =0\) on \((0,1)\) and \(\chi(x) = 1\) on \((2,+\infty)\).

6.2 Generalizations and possible extensions: non trivial metric graphs↩︎

We now come back to the case of non trivial metric graphs, discussing the generality of the results here presented. First, we note that Proposition [main95prop] holds, in fact, for a wider class of graphs than those characterized by Assumption H. Indeed, as noticed in Remark 2), the proof of Proposition [main95prop] it is sufficient to assume that the following two conditions hold: 1) the connected graph \(\mathcal{G}\) has at least two half-lines; 2) the infimum \(\mathcal{E}_{\mathcal{G}}(\mu)\) of the energy at mass \(\mu\) is not below the energy \(E(\phi_{\mu},\mathbf{R})\) of the soliton on the line (see also Remark 2). Of course one could consider metric graphs with non-negative delta vertices or non-negative potentials, as we did with the line, obtaining not surprising generalizations of those results, under suitable hypotheses. More interestingly, we show that there exist examples of metric graphs with Kirchhoff boundary conditions and no external potentials that satisfy conditions 1) and 2), but violate Assumption H. As we will see, when 1) and 2) hold, ground states may or may not exist. This indicates that, while Assumption H remains a convenient framework, the confinement or reflection behavior is not tied to topology and it is better understood as a threshold-energy mechanism.

Consider a graph \(\mathcal{G}\) consisting of \(N\) half-lines, with \(N \ge 3\), and a pendant edge of length \(l>0\), all attached to a single vertex (see Fig. 3). Because of the pendant edge, Assumption H is violated. By [8] there exists a critical mass \(\mu^*>0\) such that \(\mathcal{G}\) admits ground states at mass \(\mu\) if and only if \(\mu\ge \mu^*\). Therefore, for any \(\mu<\mu^*\) these graphs have no ground states. Thus, \(\mathcal{E}_{\mathcal{G}}(\mu) = E(\phi_{\mu},\mathbf{R})\) for all \(\mu <\mu^*\) and Proposition [main95prop] applies.

Next, if \(\mu\ge \mu^*\), then ground states exist. Since the map \(\mu\mapsto \mathcal{E}_{\mathcal{G}}(\mu)\) is continuous (see [8], Theorem 3.1), we have \[\mathcal{E}_{\mathcal{G}}(\mu^*) = \lim_{\mu\to (\mu^*)^-}\mathcal{E}_{\mathcal{G}}(\mu) = \lim_{\mu\to (\mu^*)^-}E(\phi_{\mu},\mathbf{R}) = E(\phi_{\mu^*},\mathbf{R}).\] We conclude that \(\mathcal{G}\) admits ground states at mass \(\mu^*\) with \(\mathcal{E}_{\mathcal{G}}(\mu^*)=E(\phi_{\mu^*},\mathbf{R})\). This example shows concretely that the confinement argument extends beyond Assumption H, even in the presence of ground states.

We conclude with some open problems. In Section 5 we discussed how the Cazenave–Lions argument for orbital stability of ground states on non-compact graphs relies on hypothesis (HP), namely the relative compactness of all minimizing sequences (with prescribed asymptotic mass). We also showed that bubble-tower graphs violate (HP) for every mass \(\mu>0\). In fact, any graph \(\mathcal{G}\) for which \(\mathcal{E}_{\mathcal{G}}(\mu)=E(\phi_{\mu},\mathbf{R})\) holds violates (HP) at mass \(\mu\), as discussed in Remark 5. Therefore, if any of these graphs admit ground states at that mass, their orbital stability cannot be proved by the Cazenave–Lions argument. Notice that graphs of this kind exist even outside the class H, as the example in the previous paragraph shows. For such graphs, it is natural to ask whether orbital stability of the set of ground states can be proved by the same strategy as in Proposition 11. We believe the answer is affirmative if the set of ground states at mass \(\mu^*\) consists of finitely many orbits (see [32] for a discussion of uniqueness of ground states on this type of graphs). In this respect, we mention [33], where the authors study stability of ground states for the NLS equation with general nonlinearities on the line. When the set of ground states contains multiple orbits, they prove orbital stability of each orbit by a method similar to that of Proposition 11. We expect that their approach can be adapted to non-compact graphs with multiple ground states, but we leave this conjecture for future work.

Finally, a natural extension and refinement of Proposition [main95prop] is to show that the parameters \(c\) and \(\theta\) are regular (e.g.continuous, or even differentiable) functions of time, in the spirit of deriving modulation equations (see, e.g., [34], or some of the paper cited in Remark 6.2, such as [27],[28] and reference therein). If this is possible, several interesting questions arise. For instance, one may ask whether the approximate solitons in Proposition [main95prop] drift farther and farther away from the compact core as time evolves, i.e.whether \(c(t)\to+\infty\) as \(t\to+\infty\) at some rate, or whether can stay bounded or oscillate. In this direction, we recall the work of Cavalcante and Muñoz [10] on asymptotic stability of KdV solitons on the half-line. We partially address these questions with the aid of numerical simulations in the next section and we plan to discuss more thoroughly the problem from a rigorous perspective in a subsequent work.

In the same spirit, one may ask whether there exist non-compact graphs \(\mathcal{G}\neq\mathbf{R}\) for which slow incoming solitons exhibit a genuinely different qualitative behavior, for instance capture, transmission or escape of moving solitons. In the integrable case \(p=4\), related reflection phenomena on the half-line with several boundary conditions are discussed in [35], and for NLS in \(\mathbf{R}^n\) see the already cited [28].

6.3 Numerical simulations↩︎

Using the QGLAB package developed by R. Goodman, G. Conte and J. Marzuola [17] we have performed some numerical simulations in order to illustrate and to test quantitatively the results in Section 4.2. In particular, we simulate the dynamics of a slow soliton colliding against the vertex of a star graph with three half-lines.
In the simulation the star graph is approximated by three segments of length \(L=50\), all emanating from a single vertex. We choose the power \(p = 5\) for the nonlinearity in 1 , and we check that for other nonlinearities similar results are obtained (in particular for the integrable case \(p=4\)). To build the initial datum, we consider the following moving soliton on the line [18] \[\phi(x) = e^{-ixv_0}\Bigl(\frac{p}{2}\Bigr)^{\frac{1}{p-2}} \frac{1}{\cosh^{\frac{2}{p-2}}(\frac{p-2}{2}x)}, \qquad \text{with} \quad p = 5. \label{num95soliton}\tag{36}\] The simulation is started with \(u_0(x)=\phi(x-L/2)\) on one edge of the graph, and \(u_0(x)=\phi(x+L/2)\) on the other edges. The incoming velocity is chosen to be \(v_0=-0.08\); the total time of simulation is set to \(t=300\). In Figure 4 we report some snapshots of the modulus of the solution \(|u(t)|^2\) at different times. The collision happens approximately at \(t^*=144\); after that the soliton is completely reflected. Figure 5 displays the evolution of the slow soliton on single half-lines. We observe how the vertex repels the soliton, allowing only a minimal overlap. In Figure 6 (A) we report the ratio \(p(t)/p(0)\) over time, where \[p(t) = \Im \int_{\mathcal{G}} u'\overline{u} dx\] is the linear momentum on \(\mathcal{G}\). We observe that, before and after the collision, the momentum is approximately conserved in modulus, with a transient time around \(t = 144\). On the other hand, its sign flips once the collision has happened. In Figure 6 (B) we report the ratio \(K(t)/K(0)\) over time, where \[K(t) = \frac{1}{2}||u'||^2_{L^2(\mathcal{G})}\] is the kinetic energy. This quantity also has a transient time around \(t=144\), but the value before and after the collision is approximately conserved, as in an elastic collision. It interesting to notice that the kinetic energy increases as the collision happens, and it reaches its maximal value at the collision time. Thus, the reflection of the soliton takes place without a classical turning point. This behavior is reminiscent of a quantum reflection [14], [15], which consists in the non classical reflection of slow solitons by an attractive and rapidly varying potential. We remark that, in the present case, the reflection takes place in the absence of an external potential and it is due to the vertex interaction, but the qualitative behavior we observe is analogous..
Finally we checked that the total energy (4 ) is conserved with accuracy \(\bigl|\frac{E(t)-E(0)}{E(0)}\bigr| < 8 \cdot 10^{-3}\) and the total mass (3 ) is conserved with accuracy \(\bigl|\frac{M(t)-M(0)}{M(0)}\bigr| < 1 \cdot 10^{-3}\), over the whole simulation.
The results of the numerical simulations seem to indicate that the compact part of the graph has a net repulsive effect on the slow soliton (we mention in this respect the analysis of instability of half-solitons on star graphs given in [36]). In the following, we want to investigate the spatial range of this effect. In order to do this, we consider an approximate soliton located on one half-line of the star graph, at distance \(x_0>0\) from the vertex, and with zero incoming velocity. Our goal is to study how the dynamics associated to \(u_0\) changes as we increase \(x_0>0\).
We start the simulation with the following initial datum: \(u_0=\phi(x-x_0)\) on one edge of the star graph and \(u_0(x) = \phi(x+x_0)\) on the other two edges, where \(\phi\) is given in 36 , with \(v=0\). The results of the simulations are reported in Figure 7. We observe a clear repulsive effect for the case \(x_0=L/10\), which pushes the soliton away from the vertex. This effect is present also for \(x_0=L/9\) and \(x_0=L/8\), but with milder intensity. Thus, the repulsive effect seems to diminish as we move away from the vertex. In the case \(x_0=L/2\), the repulsive effect appears to be so low that the solution remains in its initial position for the whole time interval of the simulation (this static behavior is observed even if we run the simulation until \(t=600\)). If the value of \(x_0>0\) is increased further so as to approach the free end of the edge (where Dirichlet conditions are applied), boundary effects may start to appear. We believe that these effects can be observed in Figure 8, where the evolution associated to \(x_0 = 0.875L\) and \(x_0=0.9L\) is reported. We observe that the presence of the Dirichlet boundary has a repulsive effect as well, which pushes the soliton towards the center of the edge, and whose intensity increases as we move closer to the free end of the edge. The presence of boundary effects is indicative of the fact that the infinite length approximation, which replaces the half-lines of the graph with finite length edges, is effective provided that the amplitude \(|u(t)|^2\) of the solution always remains small in the extremal regions of the compact graph. Otherwise, finite-size effects may appear. In the numerical simulations reported so far, it seems reasonable to assume that, as long as the distance between the soliton and the vertex remains smaller than half of the lentgh of the edge, boundary effect are negligible. Finally, we report in Figure 9 the evolution associated with \(x_0=0.875L\) as before, but this time with a nonzero incoming velocity \(v=-0.1\). In this case we believe boundary effect are still present, but less apparent, probably due to the fact that the solution spends a relatively short time in the vicinity of the boundary. In particular, we observe that the soliton proceeds in uniform motion towards the vertex until the collision time, at which it is reflected, in agreement with the results in Section 4.2.

Acknowledgments.↩︎

The authors are grateful to Roy Goodman for various comments and discussions. The second author acknowledges the support of the Next Generation EU - Prin 2022 project "Singular Interactions and Effective Models in Mathematical Physics- 2022CHELC7" and of Gruppo Nazionale di Fisica Matematica (GNFM-INdAM).

References↩︎