Gradient flow approach to the calculation of stationary states on nonlinear quantum graphs
Annales Henri Lebesgue, Volume 5 (2022), pp. 387-428.

Keywordsnormalized gradient flow, ground states, stationary states, quantum graphs, nonlinear Schrödinger equation

### Abstract

We introduce and implement a method to compute stationary states of nonlinear Schrödinger equations on metric graphs. Stationary states are obtained as local minimizers of the nonlinear Schrödinger energy at fixed mass. Our method is based on a normalized gradient flow for the energy (i.e. a gradient flow projected on a fixed mass sphere) adapted to the context of nonlinear quantum graphs. We first prove that, at the continuous level, the normalized gradient flow is well-posed, mass-preserving, energy diminishing and converges (at least locally) towards stationary states. We then establish the link between the continuous flow and its discretized version. We conclude by conducting a series of numerical experiments in model situations showing the good performance of the discrete flow to compute stationary states. Further experiments as well as detailed explanation of our numerical algorithm are given in a companion paper.

### References

[ABR20] Adami, Riccardo; Boni, Filippo; Ruighi, Alice Non-Kirchhoff Vertices and Nonlinear Schrödinger Ground States on Graphs, Mathematics, Volume 8 (2020) no. 4, 617 | DOI

[ACFN12a] Adami, Riccardo; Cacciapuoti, Claudio; Finco, Domenico; Noja, Diego On the structure of critical energy levels for the cubic focusing NLS on star graphs, J. Phys. A, Math. Theor., Volume 45 (2012) no. 19, 192001 | DOI | MR | Zbl

[ACFN12b] Adami, Riccardo; Cacciapuoti, Claudio; Finco, Domenico; Noja, Diego Stationary states of NLS on star graphs, Eur. Phys. Lett., Volume 100 (2012) no. 1, 10003 | DOI

[ACFN14] Adami, Riccardo; Cacciapuoti, Claudio; Finco, Domenico; Noja, Diego Variational properties and orbital stability of standing waves for NLS equation on a star graph, J. Differ. Equations, Volume 257 (2014) no. 10, pp. 3738-3777 | DOI | MR | Zbl

[ACFN16] Adami, Riccardo; Cacciapuoti, Claudio; Finco, Domenico; Noja, Diego Stable standing waves for a NLS on star graphs as local minimizers of the constrained energy, J. Differ. Equations, Volume 260 (2016) no. 10, pp. 7397-7415 | DOI | MR | Zbl

[AGHKH88] Albeverio, Sergio; Gesztesy, Friedrich; Høegh-Krohn, Raphael; Holden, Helge Solvable models in quantum mechanics, Texts and Monographs in Physics, Springer, 1988 | Zbl

[AN09] Adami, Riccardo; Noja, Diego Existence of dynamics for a 1D NLS equation perturbed with a generalized point defect, J. Phys. A, Math. Gen., Volume 42 (2009) no. 49, 495302 | DOI | MR | Zbl

[AN13] Adami, Riccardo; Noja, Diego Stability and symmetry-breaking bifurcation for the ground states of a NLS with a ${\delta }^{\prime }$ interaction, Commun. Math. Phys., Volume 318 (2013) no. 1, pp. 247-289 | DOI | MR | Zbl

[ANV13] Adami, Riccardo; Noja, Diego; Visciglia, Nicola Constrained energy minimization and ground states for NLS with point defects, Discrete Contin. Dyn. Syst., Volume 18 (2013) no. 5, pp. 1155-1188 | DOI | MR | Zbl

[AST15] 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

[AST16] 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

[AST17a] Adami, Riccardo; Serra, Enrico; Tilli, Paolo Negative energy ground states for the ${L}^{2}$-critical NLSE on metric graphs, Commun. Math. Phys., Volume 352 (2017) no. 1, pp. 387-406 | DOI | MR | Zbl

[AST17b] 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

[Bao07] Bao, Weizhu Ground states and dynamics of rotating Bose–Einstein condensates, Transport Phenomena and Kinetic Theory. Applications to gases, semiconductors, photons, and biological systems (Modeling and Simulation in Science, Engineering and Technology) (2007), pp. 215-255 | Zbl

[BD04] Bao, Weizhu; Du, Qiang Computing the ground state solution of Bose–Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput., Volume 25 (2004) no. 5, pp. 1674-1697 | DOI | MR | Zbl

[BDLC21a] Besse, Christophe; Duboscq, Romain; Le Coz, Stefan Grafidi. A Python library for the numerical simulation of quantum metric graphs with finite difference, 2021 (PLMlab repository, https://plmlab.math.cnrs.fr/cbesse/grafidi)

[BDLC21b] Besse, Christophe; Duboscq, Romain; Le Coz, Stefan Numerical Simulations on Nonlinear Quantum Graphs with the GraFiDi Library (2021) (https://arxiv.org/abs/2103.09650, to appear in SMAI - Journal of Computational Mathematics)

[BGRN15] De Bièvre, Stephan; Genoud, François; Rota Nodari, Simona Orbital stability: analysis meets geometry, Nonlinear optical and atomic systems (Lecture Notes in Mathematics), Volume 2146, Springer, 2015, pp. 147-273 | DOI | MR | Zbl

[BK13] Berkolaiko, Gregory; Kuchment, Peter Introduction to quantum graphs, Mathematical Surveys and Monographs, 186, American Mathematical Society, 2013 | MR | Zbl

[BRN19] De Bièvre, Stephan; Rota Nodari, Simona Orbital Stability via the Energy–Momentum Method: The Case of Higher Dimensional Symmetry Groups, Arch. Ration. Mech. Anal., Volume 231 (2019) no. 1, pp. 233-284 | DOI | MR | Zbl

[CL82] Cazenave, Thierry; Lions, Pierre-Louis Orbital stability of standing waves for some nonlinear Schrödinger equations, Commun. Math. Phys., Volume 85 (1982) no. 4, pp. 549-561 | DOI | MR | Zbl

[DST20] 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 | DOI | MR | Zbl

[DZ06] Dáger, René; Zuazua, Enrique Wave propagation, observation and control in $1\text{-}d$ flexible multi-structures, Mathématiques & Applications (Berlin), 50, Springer, 2006 | DOI | MR | Zbl

[FJ08] Fukuizumi, Reika; Jeanjean, Louis Stability of standing waves for a nonlinear Schrödinger equation with a repulsive Dirac delta potential, Discrete Contin. Dyn. Syst., Volume 21 (2008) no. 1, pp. 121-136 | DOI | Zbl

[FJ18] Faou, Erwan; Jézéquel, Tiphaine Convergence of a normalized gradient algorithm for computing ground states, IMA J. Numer. Anal., Volume 38 (2018) no. 1, pp. 360-376 | DOI | MR | Zbl

[FOO08] Fukuizumi, Reika; Ohta, Masahito; Ozawa, Tohru Nonlinear Schrödinger equation with a point defect, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 25 (2008) no. 5, pp. 837-845 | DOI | Numdam | MR | Zbl

[GHW04] Goodman, Roy H.; Holmes, Philip J.; Weinstein, Michael I. Strong NLS soliton-defect interactions, Physica D, Volume 192 (2004) no. 3-4, pp. 215-248 | DOI | MR | Zbl

[GLCT17] Gustafson, Stephen J.; Le Coz, Stefan; Tsai, Tai-Peng Stability of periodic waves of 1D cubic nonlinear Schrödinger equations, AMRX, Appl. Math. Res. Express (2017) no. 2, pp. 431-487 | DOI | MR | Zbl

[Goo19] 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

[Goo20] Goodman, Roy H. Quantum Graph Package, 2020 (GitHub repository, https://github.com/manroygood/Quantum-Graphs)

[GSS87] Grillakis, Manoussos G.; Shatah, Jalal; Strauss, Walter Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal., Volume 74 (1987) no. 1, pp. 160-197 | DOI | MR | Zbl

[GSS90] Grillakis, Manoussos G.; Shatah, Jalal; Strauss, Walter Stability theory of solitary waves in the presence of symmetry. II, J. Funct. Anal., Volume 94 (1990) no. 2, pp. 308-348 | DOI | MR | Zbl

[Hof19] Hofmann, Matthias An existence theory for nonlinear equations on metric graphs via energy methods (2019) (https://arxiv.org/abs/1909.07856)

[ILCR17] 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

[KMPX21] 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

[KPG19] 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

[LCFF + 08] 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 | MR | Zbl

[Lum80] Lumer, Günter Connecting of local operators and evolution equations on networks, Potential theory, Copenhagen 1979 (Proc. Colloq., Copenhagen, 1979) (Lecture Notes in Mathematics), Volume 787 (1980), pp. 219-234 | DOI | MR | Zbl

[Lun13] Lunardi, Alessandra Analytic semigroups and optimal regularity in parabolic problems, Modern Birkhäuser Classics, Springer, 2013 (reprint of the 1995 harback edition) | MR | Zbl

[MP16] Marzuola, Jeremy L.; Pelinovsky, Dmitry E. Ground State on the Dumbbell Graph, AMRX, Appl. Math. Res. Express (2016) no. 1, pp. 98-145 | DOI | MR | Zbl

[Nic85] Nicaise, Serge Some results on spectral theory over networks, applied to nerve impulse transmission, Orthogonal polynomials and applications (Bar-le-Duc, 1984) (Lecture Notes in Mathematics), Volume 1171, Springer, 1985, pp. 532-541 | DOI | MR | Zbl

[Noj14] 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 | DOI | MR | Zbl

[NP20] 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 | MR | Zbl

[NSMS11] Nakamura, Katsuhiro; Sobirov, Zarif A.; Matrasulov, Davron U.; Sawada, Shin-ichi Transport in simple networks described by an integrable discrete nonlinear Schrödinger equation, Phys. Rev. E, Volume 84 (2011) no. 2, 026609 | DOI

[PS17] Pelinovsky, Dmitry E.; Schneider, Guido Bifurcations of standing localized waves on periodic graphs, Ann. Henri Poincaré, Volume 18 (2017) no. 4, pp. 1185-1211 | DOI | MR | Zbl

[PSV21] Pierotti, Dario; Soave, Nicola; Verzini, Gianmaria Local minimizers in absence of ground states for the critical NLS energy on metric graphs, Proc. R. Soc. Edinb., Sect. A, Math., Volume 151 (2021) no. 2, pp. 705-733 | DOI | MR | Zbl

[QS19] Quittner, Pavol; Souplet, Philippe Superlinear parabolic problems. Blow-up, global existence and steady states, Birkhäuser Advanced Texts. Basler Lehrbücher, Birkhäuser; Springer, 2019 | DOI | MR | Zbl

[SBM + 16] Sobirov, Zarif A.; Babajanov, Doniyor; Matrasulov, Davron U.; Nakamura, Katsuhiro; Uecker, Hannes Sine–Gordon solitons in networks: Scattering and transmission at vertices, Eur. Phys. Lett., Volume 115 (2016) no. 5, 50002 | DOI

[SBMK18] Sabirov, Karimjon K.; Babajanov, Doniyor; Matrasulov, Davron U.; Kevrekidis, Panayotis G. Dynamics of Dirac solitons in networks, J. Phys. A, Math. Theor., Volume 51 (2018) no. 43, 435203 | DOI | MR

[SMS + 10] 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) no. 6, 066602 | DOI

[Wei85] Weinstein, Michael I. Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., Volume 16 (1985), pp. 472-491 | DOI | Zbl

[YSA + 20] 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) no. 6, 062208 | DOI | MR

[YSEM19a] Yusupov, Jambul R.; Sabirov, Karimjon K.; Ehrhardt, Matthias; Matrasulov, Davron U. Transparent nonlinear networks, Phys. Rev. E, Volume 100 (2019) no. 3, 032204 | DOI

[YSEM19b] 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