# American Institute of Mathematical Sciences

April  2019, 39(4): 2203-2232. doi: 10.3934/dcds.2019093

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

 Department of Mathematical Sciences, New Jersey Institute of Technology, University Heights, Newark, NJ 07102, USA

Received  June 2018 Revised  September 2018 Published  January 2019

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.

Citation: Roy H. Goodman. NLS bifurcations on the bowtie combinatorial graph and the dumbbell metric graph. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 2203-2232. doi: 10.3934/dcds.2019093
##### References:
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Phys., 55 (2014), 061901, 16pp. doi: 10.1063/1.4879497.  Google Scholar [11] C. Cacciapuoti, D. Finco and D. Noja, Topology-induced bifurcations for the nonlinear Schrödinger equation on the tadpole graph, Phys. Rev. E, 91 (2015), 013206, 8pp. doi: 10.1103/PhysRevE.91.013206.  Google Scholar [12] C. Cacciapuoti, D. Finco and D. Noja, Ground state and orbital stability for the NLS equation on a general starlike graph with potentials, Nonlinearity, 30 (2017), 3271-3303.  doi: 10.1088/1361-6544/aa7cc3.  Google Scholar [13] B. Delourme, S. Fliss, P. Joly and E. Vasilevskaya, Trapped modes in thin and infinite ladder like domains. Part 1: Existence results, Asymptotic Anal., 103 (2017), 103-134.  doi: 10.3233/ASY-171422.  Google Scholar [14] A. Dhooge, W. Govaerts, Y. A. Kuznetsov, H. G. E. Meijer and B. Sautois, New features of the software MatCont for bifurcation analysis of dynamical systems, Math. Comp. Model. 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I, Springer New York, 1985. doi: 10.1007/978-1-4612-5034-0.  Google Scholar [23] W. J. F. Govaerts, Numerical Methods for Bifurcations of Dynamical Equilibria, SIAM, 2000. doi: 10.1137/1.9780898719543.  Google Scholar [24] P. G. Kevrekidis, The Discrete Nonlinear Schrödinger Equation: Mathematical Analysis, Numerical Computations and Physical Perspectives, vol. 232 of Springer Tr. Mod. Phys., Springer, Berlin Heidelberg, 2009. doi: 10.1007/978-3-540-89199-4.  Google Scholar [25] E.-W. Kirr, Long time dynamics and coherent states in nonlinear wave equations, in Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science (eds. R. Melnik, R. Makarov and J. Belair), vol. 79 of Fields Inst. Commun., Springer, 2017, 59–88.  Google Scholar [26] P. Kuchment and O. Post, On the spectra of carbon nano-structures, Communications in Mathematical Physics, 275 (2007), 805-826.  doi: 10.1007/s00220-007-0316-1.  Google Scholar [27] J. L. Marzuola and D. E. Pelinovsky, Ground state on the dumbbell graph, Appl. Math. Res. Express, 2016 (2016), 98-145.  doi: 10.1093/amrx/abv011.  Google Scholar [28] J. L. Marzuola and D. E. Pelinovsky, Ground state on the dumbbell graph (v4), 2017. https://arXiv.org/abs/1509.04721. Google Scholar [29] J. L. Marzuola and M. I. Weinstein, Long time dynamics near the symmetry breaking bifurcation for nonlinear Schrödinger/Gross-Pitaevskii equations, Discrete Contin. Dyn. Syst., 28 (2010), 1505-1554.  doi: 10.3934/dcds.2010.28.1505.  Google Scholar [30] A. Nayfeh and B. Balachandran, Applied Nonlinear Dynamics: Analytical, Computational and Experimental Methods, Wiley Series in Nonlinear Science, Wiley, New York, 1995. doi: 10.1002/9783527617548.  Google Scholar [31] H. Niikuni, Schrödinger operators on a periodically broken zigzag carbon nanotube, P. Indian Acad. Sci.–Math. Sci., 127 (2017), 471-516.  doi: 10.1007/s12044-017-0342-7.  Google Scholar [32] D. Noja, S. Rolando and S. Secchi, Standing waves for the NLS on the double-bridge graph and a rational-irrational dichotomy, J. Differ. Equations, 266, (2019), 147-178. doi: 10.1016/j.jde.2018.07.038.  Google Scholar [33] D. Noja, D. E. Pelinovsky and G. Shaikhova, Bifurcations and stability of standing waves in the nonlinear Schrödinger equation on the tadpole graph, Nonlinearity, 28 (2015), 2343-2378.  doi: 10.1088/0951-7715/28/7/2343.  Google Scholar [34] D. E. Pelinovsky and T. V. Phan, Normal form for the symmetry-breaking bifurcation in the nonlinear Schrödinger equation, J. Diff. Eq., 253 (2012), 2796-2824.  doi: 10.1016/j.jde.2012.07.007.  Google Scholar [35] D. E. Pelinovsky and G. Schneider, Bifurcations of standing localized waves on periodic graphs, Ann. Henri Poincaré, 18 (2017), 1185–1211. doi: 10.1007/s00023-016-0536-z.  Google Scholar [36] The Mathworks, Inc., MATLAB Release 2018a, Natick, Massachusetts, United States. Google Scholar [37] Wolfram Research, Inc., Mathematica, Version 11.3, Champaign, IL, 2018. Google Scholar [38] J. Yang, Newton-conjugate-gradient methods for solitary wave computations, J. Comput. Phys., 228 (2009), 7007-7024.  doi: 10.1016/j.jcp.2009.06.012.  Google Scholar [39] J. Yang, Classification of solitary wave bifurcations in generalized nonlinear Schrödinger equations, Stud. Appl. Math., 129 (2012), 133-162.  doi: 10.1111/j.1467-9590.2012.00549.x.  Google Scholar [40] J. Yang, Personal communication, 2018. Google Scholar

show all references

##### References:
 [1] R. Adami, C. Cacciapuoti, D. Finco and D. Noja, Stationary states of NLS on star graphs, EPL–Europhys. Lett., 100 (2012). http://iopscience.iop.org/article/10.1209/0295-5075/100/10003/meta. doi: 10.1209/0295-5075/100/10003.  Google Scholar [2] R. Adami, E. Serra and P. Tilli, NLS ground states on graphs, Calc. Var., 54 (2014), 743-761.  doi: 10.1007/s00526-014-0804-z.  Google Scholar [3] R. Adami, E. Serra and P. Tilli, Lack of ground state for NLSE on bridge-type graphs, in Mathematical Technology of Networks (ed. D. Mugnolo), vol. 128 of Springer Proc. in Math. and Stat., Springer, 2015, 1–11. doi: 10.1007/978-3-319-16619-3_1.  Google Scholar [4] R. Adami, E. Serra and P. Tilli, Negative Energy Ground States for the $L^2$-Critical NLSE on Metric Graphs, Commun. Math. Phys., 352 (2017), 387-406.  doi: 10.1007/s00220-016-2797-2.  Google Scholar [5] R. Adami, E. Serra and P. Tilli, Threshold phenomena and existence results for NLS ground states on metric graphs, Journal of Functional Analysis, 271 (2016), 201-223.  doi: 10.1016/j.jfa.2016.04.004.  Google Scholar [6] R. Adami, E. Serra and P. Tilli, Nonlinear dynamics on branched structures and networks, Riv. Math. Univ. Parma (N.S.), 8 (2017), 109-159.   Google Scholar [7] G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Mathematical surveys and monographs, Amer. Math. Soc., 2013.  Google Scholar [8] G. Berkolaiko, An elementary introduction to quantum graphs, in Geometric and Computational Spectral Theory, vol. 700 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2017, 41–72. doi: 10.1090/conm/700/14182.  Google Scholar [9] G. Berkolaiko, Y. Latushkin and S. Sukhtaiev, Limits of quantum graph operators with shrinking edges, 2018. https://arXiv.org/abs/1806.00561. Google Scholar [10] J. Bolte and J. Kerner, Many-particle quantum graphs and Bose-Einstein condensation, J. Math. Phys., 55 (2014), 061901, 16pp. doi: 10.1063/1.4879497.  Google Scholar [11] C. Cacciapuoti, D. Finco and D. Noja, Topology-induced bifurcations for the nonlinear Schrödinger equation on the tadpole graph, Phys. Rev. E, 91 (2015), 013206, 8pp. doi: 10.1103/PhysRevE.91.013206.  Google Scholar [12] C. Cacciapuoti, D. Finco and D. Noja, Ground state and orbital stability for the NLS equation on a general starlike graph with potentials, Nonlinearity, 30 (2017), 3271-3303.  doi: 10.1088/1361-6544/aa7cc3.  Google Scholar [13] B. Delourme, S. Fliss, P. Joly and E. Vasilevskaya, Trapped modes in thin and infinite ladder like domains. Part 1: Existence results, Asymptotic Anal., 103 (2017), 103-134.  doi: 10.3233/ASY-171422.  Google Scholar [14] A. Dhooge, W. Govaerts, Y. A. Kuznetsov, H. G. E. Meijer and B. Sautois, New features of the software MatCont for bifurcation analysis of dynamical systems, Math. Comp. Model. Dyn., 14 (2008), 147-175.  doi: 10.1080/13873950701742754.  Google Scholar [15] A. Dhooge, W. Govaerts and Y. A. Kuznetsov, MATCONT: a MATLAB package for numerical bifurcation analysis of ODEs, ACM T. Math. Software, 29 (2003), 141-164.  doi: 10.1145/779359.779362.  Google Scholar [16] NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/, Release 1.0.15 of 2017-06-01, F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller and B. V. Saunders, eds. Google Scholar [17] J. C. Eilbeck and M. Johansson, The discrete nonlinear Schrödinger equation–20 years on, in Proceedings Of The Third Conference On Localization And Energy Transfer In Nonlinear Systems (eds. R. S. MacKay, L. Vázquez and M. P. Zorzano), World Scientific, Madrid, 2003, 44–67. https://www.worldscientific.com/doi/abs/10.1142/9789812704627_0003. doi: 10.1142/9789812704627_0003.  Google Scholar [18] J. C. Eilbeck, P. S. Lomdahl and A. C. Scott, The discrete self-trapping equation, Phys. D, 16 (1985), 318-338.  doi: 10.1016/0167-2789(85)90012-0.  Google Scholar [19] P. Glendinning, Stability, Instability and Chaos, An Introduction to the Theory of Nonlinear Differential Equations, Cambridge University Press, 1994.  doi: 10.1017/CBO9780511626296.  Google Scholar [20] S. Gnutzmann and D. Waltner, Stationary waves on nonlinear quantum graphs: General framework and canonical perturbation theory, Phys. Rev. E, 93 (2016), 032204, 19pp. doi: 10.1103/physreve.93.032204.  Google Scholar [21] S. Gnutzmann and D. Waltner, Stationary waves on nonlinear quantum graphs. Ⅱ. Application of canonical perturbation theory in basic graph structures, Phys. Rev. E, 94 (2016), 062216. https://journals.aps.org/pre/abstract/10.1103/PhysRevE.94.062216. doi: 10.1103/PhysRevE.94.062216.  Google Scholar [22] M. Golubitsky and D. Schaeffer, Singularities and Groups in Bifurcation Theory vol. I, Springer New York, 1985. doi: 10.1007/978-1-4612-5034-0.  Google Scholar [23] W. J. F. Govaerts, Numerical Methods for Bifurcations of Dynamical Equilibria, SIAM, 2000. doi: 10.1137/1.9780898719543.  Google Scholar [24] P. G. Kevrekidis, The Discrete Nonlinear Schrödinger Equation: Mathematical Analysis, Numerical Computations and Physical Perspectives, vol. 232 of Springer Tr. Mod. Phys., Springer, Berlin Heidelberg, 2009. doi: 10.1007/978-3-540-89199-4.  Google Scholar [25] E.-W. Kirr, Long time dynamics and coherent states in nonlinear wave equations, in Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science (eds. R. Melnik, R. Makarov and J. Belair), vol. 79 of Fields Inst. Commun., Springer, 2017, 59–88.  Google Scholar [26] P. Kuchment and O. Post, On the spectra of carbon nano-structures, Communications in Mathematical Physics, 275 (2007), 805-826.  doi: 10.1007/s00220-007-0316-1.  Google Scholar [27] J. L. Marzuola and D. E. Pelinovsky, Ground state on the dumbbell graph, Appl. Math. Res. Express, 2016 (2016), 98-145.  doi: 10.1093/amrx/abv011.  Google Scholar [28] J. L. Marzuola and D. E. Pelinovsky, Ground state on the dumbbell graph (v4), 2017. https://arXiv.org/abs/1509.04721. Google Scholar [29] J. L. Marzuola and M. I. Weinstein, Long time dynamics near the symmetry breaking bifurcation for nonlinear Schrödinger/Gross-Pitaevskii equations, Discrete Contin. Dyn. Syst., 28 (2010), 1505-1554.  doi: 10.3934/dcds.2010.28.1505.  Google Scholar [30] A. Nayfeh and B. Balachandran, Applied Nonlinear Dynamics: Analytical, Computational and Experimental Methods, Wiley Series in Nonlinear Science, Wiley, New York, 1995. doi: 10.1002/9783527617548.  Google Scholar [31] H. Niikuni, Schrödinger operators on a periodically broken zigzag carbon nanotube, P. Indian Acad. Sci.–Math. Sci., 127 (2017), 471-516.  doi: 10.1007/s12044-017-0342-7.  Google Scholar [32] D. Noja, S. Rolando and S. Secchi, Standing waves for the NLS on the double-bridge graph and a rational-irrational dichotomy, J. Differ. Equations, 266, (2019), 147-178. doi: 10.1016/j.jde.2018.07.038.  Google Scholar [33] D. Noja, D. E. Pelinovsky and G. Shaikhova, Bifurcations and stability of standing waves in the nonlinear Schrödinger equation on the tadpole graph, Nonlinearity, 28 (2015), 2343-2378.  doi: 10.1088/0951-7715/28/7/2343.  Google Scholar [34] D. E. Pelinovsky and T. V. Phan, Normal form for the symmetry-breaking bifurcation in the nonlinear Schrödinger equation, J. Diff. Eq., 253 (2012), 2796-2824.  doi: 10.1016/j.jde.2012.07.007.  Google Scholar [35] D. E. Pelinovsky and G. Schneider, Bifurcations of standing localized waves on periodic graphs, Ann. Henri Poincaré, 18 (2017), 1185–1211. doi: 10.1007/s00023-016-0536-z.  Google Scholar [36] The Mathworks, Inc., MATLAB Release 2018a, Natick, Massachusetts, United States. Google Scholar [37] Wolfram Research, Inc., Mathematica, Version 11.3, Champaign, IL, 2018. Google Scholar [38] J. Yang, Newton-conjugate-gradient methods for solitary wave computations, J. Comput. Phys., 228 (2009), 7007-7024.  doi: 10.1016/j.jcp.2009.06.012.  Google Scholar [39] J. Yang, Classification of solitary wave bifurcations in generalized nonlinear Schrödinger equations, Stud. Appl. Math., 129 (2012), 133-162.  doi: 10.1111/j.1467-9590.2012.00549.x.  Google Scholar [40] J. Yang, Personal communication, 2018. Google Scholar
]. (a) Saddle-node, (b) Transcritical, (c) Pitchfork. Top row: coordinate $a$ vs. parameter $\Lambda$. Bottom row: power $Q$ vs. $\Lambda$">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. $\Lambda$
The dumbbell graph with its vertices and edges labeled
]. The red $\times$ symbols, added by this author, mark the bifurcation locations predicted by equation (3.3)">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)
The bowtie combinatorial graph
Branches of stationary solutions to the bowtie-shaped DST system on the subspace $\mathcal{S} _2$
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
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
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 3.3.  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
]. (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 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
A graph that supports similar bifurcations
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 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
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 $\Gamma$
">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
Three views of a typical solution with two complete loops
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 $\Phi = \sqrt{-\Lambda/2}$
. (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 5.6.  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
(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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