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NLS bifurcations on the bowtie combinatorial graph and the dumbbell metric graph

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  • 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 [27]. The authors of that study found the ground state undergoes two bifurcations, first a symmetry-breaking, and the second which they call a symmetry-preserving bifurcation. We clarify the type of the symmetry-preserving bifurcation, showing it to be transcritical. We then reduce the question, and show that the phenomena described in that paper can be reproduced in a simple discrete self-trapping equation on a combinatorial graph of bowtie shape. This allows for complete analysis by parameterizing the full solution space. We then expand the question, and describe the bifurcations of all the standing waves of this system, which can be classified into three families, and of which there exists a countably infinite set.

    Mathematics Subject Classification: Primary: 35R02; Secondary: 35B32.


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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 1.2.  The dumbbell graph with its vertices and edges labeled

    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 2.1.  The bowtie combinatorial graph

    Figure 2.2.  Branches of stationary solutions to the bowtie-shaped DST system on the subspace $ \mathcal{S} _2 $

    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 3.2.  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

    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 4.1.  A graph that supports similar bifurcations

    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 5.1.  The phase plane of Equation (1.9), whose trajectories are level sets of the energy given by Equation (5.2)

    Figure 5.2.  The shooting function described in the text whose zeros correspond to nonlinear standing waves on the graph $ \Gamma $

    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 5.4.  Three views of a typical solution with two complete loops

    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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