Confinement and orbital stability of solitons of the NLS equation on metric graphs

Abstract

We study the behavior of soliton states for the subcritical, time-dependent focusing NLS equation on a large family of non-compact metric graphs with Kirchhoff boundary conditions. This family is characterized by a topological assumption (``Assumption H'' in the literature) which rules out the existence of a ground state for all members of the class, with a single exception: the bubble-tower metric graph. We present two main results. First, we show that if the initial datum is close (in the energy norm) to a soliton placed on a single half-line of the graph and sufficiently far from the nearest vertex, then the corresponding solution remains confined to the same half-line for all times, and close to the soliton, up to a remainder that stays small in the energy norm. As a nontrivial application, this yields reflection of a slow soliton upon collision with the compact core of the graph, a phenomenon that first we prove and then we further investigate numerically. Second, for the exceptional case of bubble-tower graphs, we prove that the ground state (which exists only in this case) is orbitally stable. We emphasize that this example does not allow an immediate application of the Cazenave--Lions orbital stability argument, which requires a suitable modification. Finally, we discuss how the ideas and methods developed here may extend beyond the class of metric graphs with Kirchhoff boundary conditions and satisfying Assumption H. In particular, we extend the results to the meaningful case of the line in the presence of a smooth potential or a delta interaction.

Figures (9)

• Figure 1: Bubble tower graph
• Figure 2: On the left we plot the profile of the real soliton $\phi_{\mu}$ on the real line. On the right we have a bubble-tower graph $\mathcal{G}$, with two bubbles. The top bubble has perimeter $l_1$ and the bottom one has perimeter $l_2$, with $l_1<l_2$. Define the intervals $I_1,I_2,I_3 \subset \mathbf{R}$ as in the left plot and identify the portions $u, v_a,v_b, w_a,w_b$ of the soliton $\phi_{\mu}$. The function $\Phi_{\mu} \in H^1(\mathcal{G})$ is defined as follows. We place the profile of $u$ on the top bubble (with the maximum located at a distance $l_1/2$ from the vertex). The profiles of $v_a$ and $v_b$ are placed on the left and right edges of second bubble. The portions $w_a$, $w_b$ are placed on the half-lines. Gluing these profiles together we obtain the profile of $\Phi_{\mu}$. The arrows in the right picture denote the directions along which $\Phi_{\mu}$ decreases. The same procedure can be followed in the case of any number of bubbles.
• Figure 3: N half-lines and a pendant all emanating from the same vertex. The number of half-lines is $N \geq 3$ and the length of the pendant is $l >0$.
• Figure 4: Snapshots of the amplitude $|u(t)|^2$ of the solution to the NLS equation \ref{['NLS_eq']} with $p=5$ on the star graph at six different times. The initial datum corresponds to an approximated soliton placed halfway along one edge of the graph, and with velocity $v=-0.08$ towards the vertex. We observe the soliton approaching the vertex of the graph and hitting it at approximately $t=144$. It is then reflected completely.
• Figure 5: We plot the amplitude $|u(x,t)|^2$ as a function of space and time, for the same solution as in Figure \ref{['sub_graphs']}. In the plots, the position of the vertex of the star graph corresponds to the origin of the vertical axis. In Figure (A) we display the edge that initially contains the soliton. We observe how the soliton proceeds with uniform motion until the collision time, at which it is reflected. It then proceeds backwards, still with uniform motion. In Figure (B) we display one of the edges of the graph which are initially empty, i.e. which initially contain only the tail of the soliton (for convenience, we display only one tenth of the total length of the edge). We observe how the soliton is unable to access this edge, and how its maximal amplitude is approximately $1.4\times10^{-2}$ at the collision time $t^* = 144$. A similar result holds for the other empty edge.

Theorems & Definitions (36)

• Proposition 2.1: Gagliardo-Nirenberg inequality
• Proposition 2.2
• Definition 2.1
• Proposition 2.3
• proof
• Proposition 2.4
• Definition 2.2
• Proposition 2.5: adamiCFN17
• Definition 3.1
• Proposition 3.1: adami_2