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.