Numerical Simulations on Nonlinear Quantum Graphs with the GraFiDi Library

Besse, Christophe; Duboscq, Romain; Le Coz, Stefan

The SMAI Journal of computational mathematics, Tome 8 (2022), pp. 1-47.

Nonlinear quantum graphs are metric graphs equipped with a nonlinear Schrödinger equation. Whereas in the last ten years they have known considerable developments on the theoretical side, their study from the numerical point of view remains in its early stages. The goal of this paper is to present the Grafidi library, a Python library which has been developed with the numerical simulation of nonlinear Schrödinger equations on graphs in mind. We will show how, with the help of the Grafidi library, one can implement the popular normalized gradient flow and nonlinear conjugate gradient flow methods to compute ground states of a nonlinear quantum graph. We will also simulate the dynamics of the nonlinear Schrödinger equation with a Crank-Nicolson relaxation scheme and a Strang splitting scheme. Finally, in a series of numerical experiments on various types of graphs, we will compare the outcome of our numerical calculations for ground states with the existing theoretical results, thereby illustrating the versatility and efficiency of our implementations in the framework of the Grafidi library.

Classification : 35R02, 65N06, 35Q55
Mots clés : Quantum Graphs, Python Library, Nonlinear Schrodinger equation, Finite Differences, Ground states

[1] Adami, Riccardo; Cacciapuoti, Claudio; Finco, Domenico; Noja, Diego Stationary states of NLS on star graphs, Eur. Phys. Lett., Volume 100 (2012) no. 1, p. 10003 | DOI

[2] Adami, Riccardo; Cacciapuoti, Claudio; Finco, Domenico; Noja, Diego Constrained energy minimization and orbital stability for the NLS equation on a star graph, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 31 (2014) no. 6, pp. 1289-1310 | DOI | Numdam | MR | Zbl

[3] Adami, Riccardo; Dovetta, Simone One-dimensional versions of three-dimensional system: ground states for the NLS on the spatial grid, Rend. Mat. Appl. (7), Volume 39 (2018) no. 2, pp. 181-194 | MR | Zbl

[4] Adami, Riccardo; Dovetta, Simone; Ruighi, Alice Quantum graphs and dimensional crossover: the honeycomb, Commun. Appl. Ind. Math., Volume 10 (2019) no. 1, pp. 109-122 | DOI | MR | Zbl

[5] Adami, Riccardo; Dovetta, Simone; Serra, Enrico; Tilli, Paolo Dimensional crossover with a continuum of critical exponents for NLS on doubly periodic metric graphs, Anal. PDE, Volume 12 (2019) no. 6, pp. 1597-1612 | DOI | MR | Zbl

[6] Adami, Riccardo; Serra, Enrico; Tilli, Paolo NLS ground states on graphs, Calc. Var. Partial Differ. Equ., Volume 54 (2015) no. 1, pp. 743-761 | DOI | MR | Zbl

[7] Adami, Riccardo; Serra, Enrico; Tilli, Paolo Threshold phenomena and existence results for NLS ground states on metric graphs, J. Funct. Anal., Volume 271 (2016) no. 1, pp. 201-223 | DOI | MR | Zbl

[8] Adami, Riccardo; Serra, Enrico; Tilli, Paolo Nonlinear dynamics on branched structures and networks, Riv. Math. Univ. Parma (N.S.), Volume 8 (2017) no. 1, pp. 109-159 | MR | Zbl

[9] Adami, Riccardo; Serra, Enrico; Tilli, Paolo Multiple positive bound states for the subcritical NLS equation on metric graphs, Calc. Var. Partial Differ. Equ., Volume 58 (2019) no. 1, 5, 16 pages | DOI | MR | Zbl

[10] Ali Mehmeti, Felix Nonlinear waves in networks, Mathematical Research, 80, Akademie-Verlag, 1994, 171 pages | MR | Zbl

[11] Partial differential equations on multistructures, Lecture Notes in Pure and Applied Mathematics, 219 (2001), p. xviii+248 | DOI

[12] Antoine, Xavier; Levitt, Antoine; Tang, Qinglin Efficient spectral computation of the stationary states of rotating Bose–Einstein condensates by preconditioned nonlinear conjugate gradient methods, J. Comput. Phys., Volume 343 (2017), pp. 92-109 | DOI | MR | Zbl

[13] Banica, Valeria; Ignat, Liviu I. Dispersion for the Schrödinger equation on the line with multiple Dirac delta potentials and on delta trees, Anal. PDE, Volume 7 (2014) no. 4, pp. 903-927 | DOI | Zbl

[14] Berkolaiko, Gregory; Kuchment, Peter Introduction to quantum graphs, Mathematical Surveys and Monographs, 186, American Mathematical Society, 2013, xiv+270 pages

[15] Berkolaiko, Gregory; Marzuola, Jeremy L.; Pelinovsky, Dmitry E. Edge-localized states on quantum graphs in the limit of large mass, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 38 (2021) no. 5, pp. 1295-1335 | DOI | MR | Zbl

[16] Besse, Christophe A relaxation scheme for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., Volume 42 (2004) no. 3, pp. 934-952 | DOI | Zbl

[17] Besse, Christophe; Duboscq, Romain; Le Coz, Stefan Gradient Flow Approach to the Calculation of Ground States on Nonlinear Quantum Graphs (https://hal.archives-ouvertes.fr/hal-02798328, https://arxiv.org/abs/2006.04404, to appear in Ann. Henri Lebesgue)

[18] Besse, Christophe; Duboscq, Romain; Le Coz, Stefan Grafidi, 2021 (https://plmlab.math.cnrs.fr/cbesse/grafidi)

[19] Bhandari, Kuntal; Boyer, Franck; Hernández-Santamaría, Víctor Boundary null-controllability of 1-D coupled parabolic systems with Kirchhoff-type conditions, Math. Control Signals Syst., Volume 33 (2021) no. 3, pp. 413-471 | DOI | MR | Zbl

[20] Cacciapuoti, Claudio; Dovetta, Simone; Serra, Enrico Variational and stability properties of constant solutions to the NLS equation on compact metric graphs, Milan J. Math., Volume 86 (2018) no. 2, pp. 305-327 | DOI | MR | Zbl

[21] Cacciapuoti, Claudio; Finco, Domenico; Noja, Diego Topology-induced bifurcations for the nonlinear Schrödinger equation on the tadpole graph, Phys. Rev. E (3), Volume 91 (2015) no. 1, 013206, 8 pages | DOI

[22] Danaila, Ionut; Protas, Bartosz Computation of ground states of the Gross–Pitaevskii functional via Riemannian optimization, SIAM J. Sci. Comput., Volume 39 (2017) no. 6, p. B1102-B1129 | DOI | MR | Zbl

[23] Delfour, Michel C.; Fortin, Michel; Payr, G. Finite-difference solutions of a nonlinear Schrödinger equation, J. Comput. Phys., Volume 44 (1981) no. 2, pp. 277-288 | DOI

[24] Dovetta, Simone Existence of infinitely many stationary solutions of the $L^2$-subcritical and critical NLSE on compact metric graphs, J. Differ. Equations, Volume 264 (2018) no. 7, pp. 4806-4821 | DOI | MR | Zbl

[25] Dovetta, Simone Mass-constrained ground states of the stationary NLSE on periodic metric graphs, NoDEA, Nonlinear Differ. Equ. Appl., Volume 26 (2019) no. 5, 30, 30 pages | DOI | MR | Zbl

[26] Dovetta, Simone; Ghimenti, Marco; Micheletti, Anna Maria; Pistoia, Angela Peaked and Low Action Solutions of NLS Equations on Graphs with Terminal Edges, SIAM J. Math. Anal., Volume 52 (2020) no. 3, pp. 2874-2894 | DOI | MR | Zbl

[27] Dovetta, Simone; Serra, Enrico; Tilli, Paolo NLS ground states on metric trees: existence results and open questions, J. Lond. Math. Soc., Volume 102 (2020) no. 3, pp. 1223-1240 | DOI | MR | Zbl

[28] Dovetta, Simone; Serra, Enrico; Tilli, Paolo Uniqueness and non-uniqueness of prescribed mass NLS ground states on metric graphs, Adv. Math., Volume 374 (2020), 107352, 41 pages | DOI | MR | Zbl

[29] Exner, Pavel; Post, Olaf Approximation of quantum graph vertex couplings by scaled Schrödinger operators on thin branched manifolds, J. Phys. A, Math. Gen., Volume 42 (2009) no. 41, 415305, 22 pages | DOI | Zbl

[30] Exner, Pavel; Post, Olaf A general approximation of quantum graph vertex couplings by scaled Schrödinger operators on thin branched manifolds, Commun. Math. Phys., Volume 322 (2013) no. 1, pp. 207-227 | DOI | Zbl

[31] Gilg, Steffen; Pelinovsky, Dmitry; Schneider, Guido Validity of the NLS approximation for periodic quantum graphs, NoDEA, Nonlinear Differ. Equ. Appl., Volume 23 (2016) no. 6, 63, 30 pages | DOI | MR | Zbl

[32] Gnutzmann, Sven; Waltner, Daniel Stationary waves on nonlinear quantum graphs: general framework and canonical perturbation theory, Phys. Rev. E, Volume 93 (2016) no. 3, 032204, 19 pages | DOI | MR

[33] Goodman, Roy H. NLS bifurcations on the bowtie combinatorial graph and the dumbbell metric graph, Discrete Contin. Dyn. Syst., Volume 39 (2019) no. 4, pp. 2203-2232 | DOI | MR | Zbl

[34] Goodman, Roy H. Quantum Graph Package, https://github.com/manroygood/Quantum-Graphs, 2020

[35] Gustafson, Stephen; Le Coz, Stefan; Tsai, Tai-Peng Stability of periodic waves of 1D cubic nonlinear Schrödinger equations, AMRX, Appl. Math. Res. Express, Volume 2 (2017), pp. 431-487 | DOI | Zbl

[36] Hagberg, Aric A.; Schult, Daniel A.; Swart, Pieter J. Exploring network structure, dynamics, and function using NetworkX, Proceedings of the 7th Python in Science Conference (SciPy2008) (2008), pp. 11-15

[37] Harris, Charles R.; Millman, K. Jarrod; van der Walt, Stéfan J.; Gommers, Ralf; Virtanen, Pauli; Cournapeau, David; Wieser, Eric; Taylor, Julian; Berg, Sebastian; Smith, Nathaniel J.; Kern, Robert; Picus, Matti; Hoyer, Stephan; van Kerkwijk, Marten H.; Brett, Matthew; Haldane, Allan; del Río, Jaime Fernández; Wiebe, Mark; Peterson, Pearu; Gérard-Marchant, Pierre; Sheppard, Kevin; Reddy, Tyler; Weckesser, Warren; Abbasi, Hameer; Gohlke, Christoph; Oliphant, Travis E. Array programming with NumPy, Nature, Volume 585 (2020) no. 7825, pp. 357-362 | DOI

[38] Hung, Nguyen Viet; Trippenbach, Marek; Malomed, Boris A. Symmetric and asymmetric solitons trapped in $\mathsf{H}$-shaped potentials, Phys. Rev. A, Volume 84 (2011), p. 053618 | DOI

[39] Hunter, John D. Matplotlib: A 2D graphics environment, Computing in Science & Engineering, Volume 9 (2007) no. 3, pp. 90-95 | DOI

[40] Ianni, Isabella; Le Coz, Stefan; Royer, Julien On the Cauchy problem and the black solitons of a singularly perturbed Gross-Pitaevskii equation, SIAM J. Math. Anal., Volume 49 (2017) no. 2, pp. 1060-1099 | DOI | MR | Zbl

[41] Kairzhan, Adilbek; Marangell, Robert; Pelinovsky, Dmitry E.; Xiao, Ke Liang Standing waves on a flower graph, J. Differ. Equations, Volume 271 (2021), pp. 719-763 | DOI | MR | Zbl

[42] Kairzhan, Adilbek; Pelinovsky, Dmitry E.; Goodman, Roy H. Drift of spectrally stable shifted states on star graphs, SIAM J. Appl. Dyn. Syst., Volume 18 (2019) no. 4, pp. 1723-1755 | DOI | MR | Zbl

[43] Kurata, Kazuhiro; Shibata, Masataka Least energy solutions to semi-linear elliptic problems on metric graphs, Journal of Mathematical Analysis and Applications, Volume 491 (2020) no. 1, p. 124297 | DOI | MR | Zbl

[44] Le Coz, Stefan; Fukuizumi, Reika; Fibich, Gadi; Ksherim, Baruch; Sivan, Yonatan Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential, Physica D, Volume 237 (2008) no. 8, pp. 1103-1128 | DOI | Zbl

[45] Marzuola, Jeremy L.; Pelinovsky, Dmitry E. Ground State on the Dumbbell Graph, AMRX, Appl. Math. Res. Express, Volume 2016 (2016) no. 1, pp. 98-145 | DOI | MR | Zbl

[46] Mehmeti, Felix Ali; Ammari, Kaïs; Nicaise, Serge Dispersive effects and high frequency behaviour for the Schrödinger equation in star-shaped networks, Port. Math., Volume 72 (2015) no. 4, pp. 309-355 | DOI | Zbl

[47] Mehmeti, Felix Ali; Ammari, Kaïs; Nicaise, Serge Dispersive effects for the Schrödinger equation on the tadpole graph, J. Math. Anal. Appl., Volume 448 (2017) no. 1, pp. 262-280 | DOI | Zbl

[48] Noja, Diego Nonlinear Schrödinger equation on graphs: recent results and open problems, Philos. Trans. R. Soc. Lond., Ser. A, Volume 372 (2014) no. 2007, 20130002, 20 pages | DOI | Zbl

[49] Noja, Diego; Pelinovsky, Dmitry; Shaikhova, Gaukhar Bifurcations and stability of standing waves in the nonlinear Schrödinger equation on the tadpole graph, Nonlinearity, Volume 28 (2015) no. 7, pp. 2343-2378 | DOI | Zbl

[50] Noja, Diego; Pelinovsky, Dmitry E. Standing waves of the quintic NLS equation on the tadpole graph, Calc. Var. Partial Differ. Equ., Volume 59 (2020) no. 5, 173, 30 pages | DOI | MR | Zbl

[51] Pankov, Alexander Nonlinear Schrödinger equations on periodic metric graphs, Discrete Contin. Dyn. Syst., Volume 38 (2018) no. 2, pp. 697-714 | DOI | MR | Zbl

[52] Pelinovsky, Dmitry; Schneider, Guido Bifurcations of standing localized waves on periodic graphs, Ann. Henri Poincaré, Volume 18 (2017) no. 4, pp. 1185-1211 | DOI | MR | Zbl

[53] Pierotti, Dario; Soave, Nicola; Verzini, Gianmaria Local minimizers in absence of ground states for the critical NLS energy on metric graphs, Proceedings of the Royal Society of Edinburgh: Section A Mathematics (2020), pp. 1-29 | DOI

[54] Sabirov, Karimjon K.; Sobirov, Zarif A.; Babajanov, Doniyor; Matrasulov, Davron U. Stationary nonlinear Schrödinger equation on simplest graphs, Phys. Lett., A, Volume 377 (2013) no. 12, pp. 860-865 | DOI

[55] Sobirov, Zarif A.; Babajanov, Doniyor; Matrasulov, Davron U. Nonlinear standing waves on planar branched systems: shrinking into metric graph, Nanosystems: Physics, Chemistry, Mathematics, Volume 8 (2017) no. 1, p. 29

[56] Sobirov, Zarif A.; Matrasulov, Davron U.; Sabirov, Karimjon K.; Sawada, Shin-ichi; Nakamura, Katsuhiro Integrable nonlinear Schrödinger equation on simple networks: Connection formula at vertices, Phys. Rev. E, Volume 81 (2010), p. 066602 | DOI

[57] Strang, Gilbert On the construction and comparison of difference schemes, SIAM J. Numer. Anal., Volume 5 (1968), pp. 506-517 | DOI | MR | Zbl

[58] Tokuno, Akiyuki; Oshikawa, Masaki; Demler, Eugene Dynamics of One-Dimensional Bose Liquids: Andreev-Like Reflection at $Y$ Junctions and the Absence of the Aharonov-Bohm Effect, Phys. Rev. Lett., Volume 100 (2008), p. 140402 | DOI

[59] Uecker, Hannes; Grieser, Daniel; Sobirov, Zarif; Babajanov, Doniyor; Matrasulov, Davron U. Soliton transport in tubular networks: Transmission at vertices in the shrinking limit, Phys. Rev. E, Volume 91 (2015), p. 023209 | DOI | MR

[60] Virtanen, Pauli; Gommers, Ralf; Oliphant, Travis E.; Haberland, Matt; Reddy, Tyler; Cournapeau, David; Burovski, Evgeni; Peterson, Pearu; Weckesser, Warren; Bright, Jonathan; van der Walt, Stéfan J.; Brett, Matthew; Wilson, Joshua; Millman, K. Jarrod; Mayorov, Nikolay; Nelson, Andrew R. J.; Jones, Eric; Kern, Robert; Larson, Eric; Carey, C. J.; Polat, İlhan; Feng, Yu; Moore, Eric W.; Van der Plas, Jake; Laxalde, Denis; Perktold, Josef; Cimrman, Robert; Henriksen, Ian; Quintero, E. A.; Harris, Charles R.; Archibald, Anne M.; Ribeiro, Antônio H.; Pedregosa, Fabian; van Mulbregt, Paul; SciPy 1.0 Contributors SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python, Nature Methods, Volume 17 (2020), pp. 261-272 | DOI

[61] Weideman, J. A. C.; Herbst, B.M Split-step methods for the solution of the nonlinear Schrödinger equation, SIAM J. Numer. Anal., Volume 23 (1986) no. 3, pp. 485-507 | DOI | Zbl

[62] Yusupov, Jambul R.; Sabirov, Karimjon K.; Asadov, Q. U.; Ehrhardt, Matthias; Matrasulov, Davron U. Dirac particles in transparent quantum graphs: Tunable transport of relativistic quasiparticles in branched structures, Phys. Rev. E, Volume 101 (2020), p. 062208 | DOI | MR

[63] Yusupov, Jambul R.; Sabirov, Karimjon K.; Ehrhardt, Matthias; Matrasulov, Davron U. Transparent nonlinear networks, Phys. Rev. E, Volume 100 (2019), p. 032204 | DOI

[64] Yusupov, Jambul R.; Sabirov, Karimjon K.; Ehrhardt, Matthias; Matrasulov, Davron U. Transparent quantum graphs, Phys. Lett., A, Volume 383 (2019) no. 20, pp. 2382-2388 | DOI | MR