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J.L. Marzuola and D.E. Pelinovsky

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Ground states on the dumbbell graph

**Abstract:**

We consider standing waves in the focusing nonlinear Schrodinger (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.

**Keywords**:

Nonlinear Schrodinger equation; quantum graphs, standing wave solutions,
existence and stability, Jacobi elliptic functions.