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Existence of the normalized solutions to the nonlocal elliptic system with partial confinement
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 |
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 [
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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[7] |
G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Mathematical surveys and monographs, Amer. Math. Soc., 2013. |
[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. |
[9] |
G. Berkolaiko, Y. Latushkin and S. Sukhtaiev, Limits of quantum graph operators with shrinking edges, 2018. https://arXiv.org/abs/1806.00561. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[19] |
P. Glendinning, Stability, Instability and Chaos, An Introduction to the Theory of Nonlinear Differential Equations, Cambridge University Press, 1994.
doi: 10.1017/CBO9780511626296.![]() ![]() ![]() |
[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. |
[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. |
[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. |
[23] |
W. J. F. Govaerts, Numerical Methods for Bifurcations of Dynamical Equilibria, SIAM, 2000.
doi: 10.1137/1.9780898719543. |
[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. |
[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. |
[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. |
[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. |
[28] |
J. L. Marzuola and D. E. Pelinovsky, Ground state on the dumbbell graph (v4), 2017. https://arXiv.org/abs/1509.04721. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[36] |
The Mathworks, Inc., MATLAB Release 2018a, Natick, Massachusetts, United States. |
[37] |
Wolfram Research, Inc., Mathematica, Version 11.3, Champaign, IL, 2018. |
[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. |
[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. |
[40] |
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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[7] |
G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Mathematical surveys and monographs, Amer. Math. Soc., 2013. |
[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. |
[9] |
G. Berkolaiko, Y. Latushkin and S. Sukhtaiev, Limits of quantum graph operators with shrinking edges, 2018. https://arXiv.org/abs/1806.00561. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[19] |
P. Glendinning, Stability, Instability and Chaos, An Introduction to the Theory of Nonlinear Differential Equations, Cambridge University Press, 1994.
doi: 10.1017/CBO9780511626296.![]() ![]() ![]() |
[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. |
[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. |
[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. |
[23] |
W. J. F. Govaerts, Numerical Methods for Bifurcations of Dynamical Equilibria, SIAM, 2000.
doi: 10.1137/1.9780898719543. |
[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. |
[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. |
[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. |
[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. |
[28] |
J. L. Marzuola and D. E. Pelinovsky, Ground state on the dumbbell graph (v4), 2017. https://arXiv.org/abs/1509.04721. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[36] |
The Mathworks, Inc., MATLAB Release 2018a, Natick, Massachusetts, United States. |
[37] |
Wolfram Research, Inc., Mathematica, Version 11.3, Champaign, IL, 2018. |
[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. |
[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. |
[40] |


















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