Ground State on the Dumbbell Graph

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 …