Ground State on the Dumbbell Graph

Abstract

We consider standing waves in the focusing nonlinear Schrödinger (NLS) equation on a dumbbell graph (two rings attached to a central line segment subject to the Kirchhoff boundary conditions at the junctions). In the limit of small L 2 norm, the ground state (the orbitally stable standing wave of the smallest energy at a fixed L 2 norm) is represented by a constant solution. However, when the L 2 norm is increased, this constant solution undertakes two bifurcations, where the first is the pitchfork (symmetry breaking) bifurcation and the second one is the symmetry preserving bifurcation. As a result of the first symmetry breaking bifurcation, the standing wave becomes more localized in one of the two rings. As a result of the second symmetry preserving bifurcation, the standing wave becomes localized in the central line segment. In the limit of large norm solutions, both standing waves are represented by a truncated solitary wave localized in either the ring or the central line segment. Both waves are stable local constrained minimizers of the energy for the fixed L 2 norm but the asymmetric wave supported in the ring has a smaller energy. The analytical results are confirmed by numerical approximations of the ground state on the dumbbell graph.

Quantum graphs and dimensional crossover: the honeycomb

“…a star graph) it has been proven that there is no ground state, irrespectively of the choice of µ ([2]). Starting from this negative result, the problem of ensuring (or excluding) the existence of ground states for the NLS on graphs gained some popularity in the community, and some general results were found, isolating a key topological condition ( [5]), studying in detail particular cases ( [14,26,27]), dealing with compact graphs ( [12,13]), introducing concentrated nonlinearities ( [15,28,29,30]), focusing on the more challenging L 2 -critical case (i.e. p = 6 [6]).…”

Nonlinear Dirac Equation on Graphs with Localized Nonlinearities: Bound States and Nonrelativistic Limit

“…We also mention some interesting results on the problem of the bound states on compact graphs. For a purely variational approach we recall, e.g., [23,19], whereas for a bifurcation approach we recall, e.g., [40].…”

An Overview on the Standing Waves of Nonlinear Schrödinger and Dirac Equations on Metric Graphs with Localized Nonlinearity

“…For the sake of completeness, we also remark that the NLSE on compact graphs (which, in particular, do not fulfill (H1)) has been studied, e.g., in [20,24,28,38]; while the case of one or higher-dimensional periodic graphs (which, in particular, do not fulfill (H2) as, for instance, in Figure 3) has been addressed, e.g., by [6,25,32,43].…”