Ground state on the dumbbell graph

TL;DR

This paper analyzes the behavior of ground states in the focusing nonlinear Schrödinger equation on a dumbbell graph, revealing bifurcations and localization phenomena as the $L^2$ norm varies, supported by analytical and numerical results.

Contribution

It provides a detailed bifurcation analysis of ground states on a dumbbell graph, including the identification of symmetry breaking and symmetry preserving bifurcations, and characterizes their localization properties.

Findings

Ground state is constant at small $L^2$ norm.

First bifurcation is a symmetry breaking pitchfork bifurcation.

Second bifurcation preserves symmetry, leading to localization in the central segment.

Abstract

We consider standing waves in the focusing nonlinear Schr\"odinger (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. Although the asymmetric wave supported in the ring is a ground state near the symmetry breaking bifurcation of the constant solution, it is the symmetric wave supported in the central line segment which becomes the ground state in the limit of large $L^2$ norm. The analytical results are confirmed by numerical approximations of the ground state on the dumbbell graph.