We consider the bifurcations of standing wave solutions to the nonlinear Schrödinger equation (NLS) posed on a quantum graph consisting of two loops connected by a single edge, the so-called dumbbell, recently studied in [
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Figure 1.1. The three most common bifurcations, after [39]. (a) Saddle-node, (b) Transcritical, (c) Pitchfork. Top row: coordinate $ a $ vs. parameter $ \Lambda $. Bottom row: power $ Q $ vs. $ \Lambda $
Figure 1.3. A numerically computed bifurcation diagram from Ref. [27]. The red $ \times $ symbols, added by this author, mark the bifurcation locations predicted by equation (3.3)
Figure 3.1. The first two members of the even family of eigenfunctions (a-b), odd family (c-d), and loop-localized family (e-f) of the linear eigenvalue problem (3.1) on the dumbbell graph, computed numerically, along with the associated eigenvaluess. In subfigure (f) the analytical value is obviously $ \lambda = 4 $, giving an indication of the accuracy of this computation
Figure 3.3. Numerical continuation of the PDE on the quantum graph. Comparison with Fig. 1.1 indicates that the loop-centered and constant solutions meet in a transcritical bifurcation. The computation indicates that the centered solution also undergoes saddle-node and pitchfork bifurcations
Figure 3.4. (a) Large-amplitude centered solution on the half-branch discovered in Ref. [27]. (b) Large-amplitude two-soliton solution. (c) Solution arising from symmetry-breaking of centered state. (d) Solution arising from symmetry-breaking of constant state. Subplot labels correspond to marked points in Figure 3.3
Figure 4.2. The analogy of Fig. 3.3 with $ L = 15 $ and $ L = 50 $. As $ L $ is increased, the angle with which the two branches of solution approach the transcritical bifurcation decreases, making it appear, locally, more like a pitchfork
Figure 5.3. Two views of a partial bifurcation diagram with $ L = 2 $. (a) Plotting $ Q $ the squared $ L^2 $ norm of the standing wave solutions. (b) Plotting the value $ q $ used in the shooting function. Colors of branches are consistent between the two panels and with Fig. 3.3
Figure 5.5. Bifurcation diagram for solutions with two complete loops. Plotted are solutions with $\left| {n_j} \right| \le 2$ and $ \left| m \right| \le2$. Color indicates type of solution on the edge $ \mathtt{e} _2 $. The dashed line shows the nonzero constant solution $ \Phi = \sqrt{-\Lambda/2} $
Figure 5.6. The standing waves at the six marked points in the bifurcation diagram of Fig. 5.5. (a) $ (0, 0, 2) $, (b) $ (1, 0, 2) $, (c) $ (1, 1, 2) $, (d) $ (1, \Lambda, 2) $, (e) $ (1, -1, 2) $, (f) $ (2, -1, 1) $. Note from (e) and (f) that reversing $ n_1 $ and $ n_3 $ is not equivalent to a symmetry operation since a half-period of the $ dn $-function has no symmetries. As $ \Lambda $ decreases, (b) bifurcates from (a), and then (c), (d), and (e-f) bifurcate from (b) in that order
Figure 5.7. (a) Solid curves: Partial bifurcation diagram on the lollipop subgraph. Dashed curves (red) indicate the maximum values of the quantized cnoidal solutions and the dash-dot curves (green) the maximum and minimum values of the quantized conoidal solutions on edge $ \mathtt{e} _3 $, with the regions between them shaded, alternately, for clarity. The marked points at intersections between the two families of curves indicate saddle node bifurcations of solutions with cnoidal or dnoidal solutions on the edge $ \mathtt{e} _3 $. (b) Partial bifurcation diagram on the dumbbell graph
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The three most common bifurcations, after [39]. (a) Saddle-node, (b) Transcritical, (c) Pitchfork. Top row: coordinate
The dumbbell graph with its vertices and edges labeled
A numerically computed bifurcation diagram from Ref. [27]. The red
The bowtie combinatorial graph
Branches of stationary solutions to the bowtie-shaped DST system on the subspace
The first two members of the even family of eigenfunctions (a-b), odd family (c-d), and loop-localized family (e-f) of the linear eigenvalue problem (3.1) on the dumbbell graph, computed numerically, along with the associated eigenvaluess. In subfigure (f) the analytical value is obviously
A pitchfork bifurcation may split into either (a) one branch with no bifurcations and one branch with a saddle node (b) a saddle-node and a transcritical bifurcation
Numerical continuation of the PDE on the quantum graph. Comparison with Fig. 1.1 indicates that the loop-centered and constant solutions meet in a transcritical bifurcation. The computation indicates that the centered solution also undergoes saddle-node and pitchfork bifurcations
(a) Large-amplitude centered solution on the half-branch discovered in Ref. [27]. (b) Large-amplitude two-soliton solution. (c) Solution arising from symmetry-breaking of centered state. (d) Solution arising from symmetry-breaking of constant state. Subplot labels correspond to marked points in Figure 3.3
A graph that supports similar bifurcations
The analogy of Fig. 3.3 with
The phase plane of Equation (1.9), whose trajectories are level sets of the energy given by Equation (5.2)
The shooting function described in the text whose zeros correspond to nonlinear standing waves on the graph
Two views of a partial bifurcation diagram with
Three views of a typical solution with two complete loops
Bifurcation diagram for solutions with two complete loops. Plotted are solutions with
The standing waves at the six marked points in the bifurcation diagram of Fig. 5.5. (a)
(a) Solid curves: Partial bifurcation diagram on the lollipop subgraph. Dashed curves (red) indicate the maximum values of the quantized cnoidal solutions and the dash-dot curves (green) the maximum and minimum values of the quantized conoidal solutions on edge