Numerical Simulations on Nonlinear Quantum Graphs with the GraFiDi Library
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 t...
Saved in:
| Published in: | SMAI Journal of Computational Mathematics Vol. 8; pp. 1 - 47 |
|---|---|
| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Société de Mathématiques Appliquées et Industrielles (SMAI)
08.04.2022
|
| Subjects: | |
| ISSN: | 2426-8399, 2426-8399 |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Abstract | 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. |
|---|---|
| AbstractList | 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. |
| Author | Besse, Christophe Duboscq, Romain Le Coz, Stefan |
| Author_xml | – sequence: 1 givenname: Christophe surname: Besse fullname: Besse, Christophe – sequence: 2 givenname: Romain surname: Duboscq fullname: Duboscq, Romain – sequence: 3 givenname: Stefan surname: Le Coz fullname: Le Coz, Stefan |
| BackLink | https://hal.science/hal-03164639$$DView record in HAL |
| BookMark | eNptkEtLAzEUhYMoWGtX_oFsRabm5SRZlmpbZaiIug6ZTMKkzKMkM4r_3hmrIOLqPjjnHu53Bo6btrEAXGA0vxGIXMda-2Rn6jkXR2BCGEkTQaU8_tWfglmMO4QQkYTylEzAw7avbfBGV_DZ132lO982EbYN3LZN5RurA3zqddP1NVwHvS8jfPddCbvSjvPK33qY-Tzo8HEOTpyuop191yl4Xd29LDdJ9ri-Xy6yxBAsRaIdEdYUjFPnjGaOcExdQXCa89QS6RgizDiLac6Y4xpLmReGUcEdLozglE7B5eFuqSu1D74eslWrvdosMjXuEMUpS6l8w4MWH7QmtDEG65Tx3dePXdC-UhipkZ0a2amBneJi8Fz98fyE_Kf-BLdQdHM |
| CitedBy_id | crossref_primary_10_1088_1361_6544_ad1535 crossref_primary_10_1016_j_jfa_2024_110760 crossref_primary_10_1088_1361_6544_ad5133 crossref_primary_10_1038_s41467_022_34073_3 crossref_primary_10_1007_s00526_023_02497_4 crossref_primary_10_1088_1361_6544_ac7505 crossref_primary_10_1137_23M1627729 crossref_primary_10_1112_plms_70025 crossref_primary_10_1016_j_aim_2024_109633 crossref_primary_10_1016_j_jmaa_2023_127672 crossref_primary_10_5802_smai_jcm_118 crossref_primary_10_1515_math_2024_0129 |
| Cites_doi | 10.1016/j.jde.2017.12.025 10.4171/PM/1970 10.1016/j.jfa.2016.04.004 10.1093/amrx/abv011 10.1016/j.physleta.2013.02.011 10.1103/PhysRevE.100.032204 10.1137/19M127447X 10.1137/15M1029606 10.1209/0295-5075/100/10003 10.1007/s00032-018-0288-y 10.1103/PhysRevE.91.023209 10.1103/PhysRevA.84.053618 10.1016/j.jde.2020.09.010 10.1103/PhysRevE.101.062208 10.1007/s00220-013-1699-9 10.1103/PhysRevLett.100.140402 10.1016/j.physleta.2019.04.059 10.1016/j.jcp.2017.04.040 10.1016/j.jmaa.2016.10.060 10.2140/apde.2014.7.903 10.1103/physreve.93.032204 10.1038/s41586-020-2649-2 10.1088/1751-8113/42/41/415305 10.1098/rsta.2013.0002 10.1088/0951-7715/28/7/2343 10.3934/dcds.2018030 10.1007/s00526-018-1461-4 10.1103/PhysRevE.91.013206 10.1007/s00526-020-01832-3 10.1016/0021-9991(81)90052-8 10.1103/PhysRevE.81.066602 10.1201/9780203902196 10.1137/17M1121974 10.1016/j.jmaa.2020.124297 10.1016/j.anihpc.2020.11.003 10.1016/j.anihpc.2013.09.003 10.1109/MCSE.2007.55 10.1137/19M1246146 10.1017/prm.2020.36 10.1137/S0036142901396521 10.1038/s41592-019-0686-2 10.1093/amrx/abx004 10.2478/caim-2019-0016 10.1007/s00030-016-0417-7 10.1007/s00023-016-0536-z 10.1007/s00498-021-00285-z 10.1016/j.physd.2007.12.004 10.1007/s00526-014-0804-z 10.25080/TCWV9851 10.2140/apde.2019.12.1597 10.1112/jlms.12361 10.1137/0705041 10.1016/j.aim.2020.107352 10.3934/dcds.2019093 10.1137/0723033 10.1007/s00030-019-0576-4 |
| ContentType | Journal Article |
| Copyright | Distributed under a Creative Commons Attribution 4.0 International License |
| Copyright_xml | – notice: Distributed under a Creative Commons Attribution 4.0 International License |
| DBID | AAYXX CITATION 1XC VOOES |
| DOI | 10.5802/smai-jcm.78 |
| DatabaseName | CrossRef Hyper Article en Ligne (HAL) Hyper Article en Ligne (HAL) (Open Access) |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | CrossRef |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Mathematics |
| EISSN | 2426-8399 |
| EndPage | 47 |
| ExternalDocumentID | oai:HAL:hal-03164639v1 10_5802_smai_jcm_78 |
| GroupedDBID | AAYXX AEXTA ALMA_UNASSIGNED_HOLDINGS ARCSS CITATION GROUPED_DOAJ M~E 1XC VOOES |
| ID | FETCH-LOGICAL-c2198-af28ecd473ffca4f2713fd216b76e29f4024cfe13b44f7a199bdc4387f1dc8733 |
| ISSN | 2426-8399 |
| IngestDate | Sat Oct 25 11:16:24 EDT 2025 Sat Oct 25 05:39:37 EDT 2025 Tue Nov 18 22:33:24 EST 2025 |
| IsDoiOpenAccess | true |
| IsOpenAccess | true |
| IsPeerReviewed | true |
| IsScholarly | true |
| Keywords | Finite Differences Ground states Python Library Nonlinear Schrodinger equation Quantum Graphs |
| Language | English |
| License | Distributed under a Creative Commons Attribution 4.0 International License: http://creativecommons.org/licenses/by/4.0 |
| LinkModel | OpenURL |
| MergedId | FETCHMERGED-LOGICAL-c2198-af28ecd473ffca4f2713fd216b76e29f4024cfe13b44f7a199bdc4387f1dc8733 |
| ORCID | 0000-0002-0086-8710 0000-0002-3739-3640 0000-0003-0680-0175 |
| OpenAccessLink | https://hal.science/hal-03164639 |
| PageCount | 47 |
| ParticipantIDs | hal_primary_oai_HAL_hal_03164639v1 crossref_citationtrail_10_5802_smai_jcm_78 crossref_primary_10_5802_smai_jcm_78 |
| PublicationCentury | 2000 |
| PublicationDate | 2022-04-08 |
| PublicationDateYYYYMMDD | 2022-04-08 |
| PublicationDate_xml | – month: 04 year: 2022 text: 2022-04-08 day: 08 |
| PublicationDecade | 2020 |
| PublicationTitle | SMAI Journal of Computational Mathematics |
| PublicationYear | 2022 |
| Publisher | Société de Mathématiques Appliquées et Industrielles (SMAI) |
| Publisher_xml | – name: Société de Mathématiques Appliquées et Industrielles (SMAI) |
| References | Ali Mehmeti, Felix (key2025101714505957745_10) 1994; 80 Adami, Riccardo (key2025101714505957745_8) 2017; 8 Adami, Riccardo (key2025101714505957745_5) 2019; 12 Yusupov, Jambul R. (key2025101714505957745_64) 2019; 383 Besse, Christophe (key2025101714505957745_18) 2021 Pankov, Alexander (key2025101714505957745_51) 2018; 38 Adami, Riccardo (key2025101714505957745_6) 2015; 54 Besse, Christophe (key2025101714505957745_16) 2004; 42 Banica, Valeria (key2025101714505957745_13) 2014; 7 Adami, Riccardo (key2025101714505957745_2) 2014; 31 Cacciapuoti, Claudio (key2025101714505957745_20) 2018; 86 Kairzhan, Adilbek (key2025101714505957745_42) 2019; 18 Gustafson, Stephen (key2025101714505957745_35) 2017; 2 Sobirov, Zarif A. (key2025101714505957745_56) 2010; 81 Tokuno, Akiyuki (key2025101714505957745_58) 2008; 100 Noja, Diego (key2025101714505957745_50) 2020; 59 Delfour, Michel C. (key2025101714505957745_23) 1981; 44 Berkolaiko, Gregory (key2025101714505957745_15) 2021; 38 Uecker, Hannes (key2025101714505957745_59) 2015; 91 Goodman, Roy H. (key2025101714505957745_34) 2020 Adami, Riccardo (key2025101714505957745_9) 2019; 58 (key2025101714505957745_11) 2001; 219 Weideman, J. A. C. (key2025101714505957745_61) 1986; 23 Kairzhan, Adilbek (key2025101714505957745_41) 2021; 271 Mehmeti, Felix Ali (key2025101714505957745_47) 2017; 448 Gilg, Steffen (key2025101714505957745_31) 2016; 23 Marzuola, Jeremy L. (key2025101714505957745_45) 2016; 2016 Adami, Riccardo (key2025101714505957745_7) 2016; 271 Sobirov, Zarif A. (key2025101714505957745_55) 2017; 8 Ianni, Isabella (key2025101714505957745_40) 2017; 49 Hung, Nguyen Viet (key2025101714505957745_38) 2011; 84 Antoine, Xavier (key2025101714505957745_12) 2017; 343 Yusupov, Jambul R. (key2025101714505957745_62) 2020; 101 Mehmeti, Felix Ali (key2025101714505957745_46) 2015; 72 Bhandari, Kuntal (key2025101714505957745_19) 2021; 33 Besse, Christophe (key2025101714505957745_17) Pierotti, Dario (key2025101714505957745_53) 2020 Dovetta, Simone (key2025101714505957745_28) 2020; 374 Harris, Charles R. (key2025101714505957745_37) 2020; 585 Dovetta, Simone (key2025101714505957745_24) 2018; 264 Dovetta, Simone (key2025101714505957745_27) 2020; 102 Dovetta, Simone (key2025101714505957745_25) 2019; 26 Danaila, Ionut (key2025101714505957745_22) 2017; 39 Dovetta, Simone (key2025101714505957745_26) 2020; 52 Yusupov, Jambul R. (key2025101714505957745_63) 2019; 100 Goodman, Roy H. (key2025101714505957745_33) 2019; 39 Pelinovsky, Dmitry (key2025101714505957745_52) 2017; 18 Adami, Riccardo (key2025101714505957745_4) 2019; 10 Exner, Pavel (key2025101714505957745_29) 2009; 42 Exner, Pavel (key2025101714505957745_30) 2013; 322 Adami, Riccardo (key2025101714505957745_1) 2012; 100 Noja, Diego (key2025101714505957745_48) 2014; 372 Sabirov, Karimjon K. (key2025101714505957745_54) 2013; 377 Cacciapuoti, Claudio (key2025101714505957745_21) 2015; 91 Adami, Riccardo (key2025101714505957745_3) 2018; 39 Kurata, Kazuhiro (key2025101714505957745_43) 2020; 491 Berkolaiko, Gregory (key2025101714505957745_14) 2013; 186 Le Coz, Stefan (key2025101714505957745_44) 2008; 237 Gnutzmann, Sven (key2025101714505957745_32) 2016; 93 Hunter, John D. (key2025101714505957745_39) 2007; 9 Strang, Gilbert (key2025101714505957745_57) 1968; 5 Virtanen, Pauli (key2025101714505957745_60) 2020; 17 Hagberg, Aric A. (key2025101714505957745_36) 2008 Noja, Diego (key2025101714505957745_49) 2015; 28 |
| References_xml | – volume: 264 start-page: 4806 issue: 7 year: 2018 ident: key2025101714505957745_24 article-title: Existence of infinitely many stationary solutions of the <span class="mathjax-formula" data-tex="${L}^2$"><math xmlns="http://www.w3.org/1998/Math/MathML">L 2 </math></span>-subcritical and critical NLSE on compact metric graphs publication-title: J. Differ. Equations doi: 10.1016/j.jde.2017.12.025 – volume: 72 start-page: 309 issue: 4 year: 2015 ident: key2025101714505957745_46 article-title: Dispersive effects and high frequency behaviour for the Schrödinger equation in star-shaped networks publication-title: Port. Math. doi: 10.4171/PM/1970 – volume: 271 start-page: 201 issue: 1 year: 2016 ident: key2025101714505957745_7 article-title: Threshold phenomena and existence results for NLS ground states on metric graphs publication-title: J. Funct. Anal. doi: 10.1016/j.jfa.2016.04.004 – volume: 2016 start-page: 98 issue: 1 year: 2016 ident: key2025101714505957745_45 article-title: Ground State on the Dumbbell Graph publication-title: AMRX, Appl. Math. Res. Express doi: 10.1093/amrx/abv011 – volume: 377 start-page: 860 issue: 12 year: 2013 ident: key2025101714505957745_54 article-title: Stationary nonlinear Schrödinger equation on simplest graphs publication-title: Phys. Lett., A doi: 10.1016/j.physleta.2013.02.011 – volume: 100 start-page: 032204 year: 2019 ident: key2025101714505957745_63 article-title: Transparent nonlinear networks publication-title: Phys. Rev. E doi: 10.1103/PhysRevE.100.032204 – volume: 52 start-page: 2874 issue: 3 year: 2020 ident: key2025101714505957745_26 article-title: Peaked and Low Action Solutions of NLS Equations on Graphs with Terminal Edges publication-title: SIAM J. Math. Anal. doi: 10.1137/19M127447X – volume: 49 start-page: 1060 issue: 2 year: 2017 ident: key2025101714505957745_40 article-title: On the Cauchy problem and the black solitons of a singularly perturbed Gross-Pitaevskii equation publication-title: SIAM J. Math. Anal. doi: 10.1137/15M1029606 – volume: 100 start-page: 10003 issue: 1 year: 2012 ident: key2025101714505957745_1 article-title: Stationary states of NLS on star graphs publication-title: Eur. Phys. Lett. doi: 10.1209/0295-5075/100/10003 – volume: 86 start-page: 305 issue: 2 year: 2018 ident: key2025101714505957745_20 article-title: Variational and stability properties of constant solutions to the NLS equation on compact metric graphs publication-title: Milan J. Math. doi: 10.1007/s00032-018-0288-y – volume: 91 start-page: 023209 year: 2015 ident: key2025101714505957745_59 article-title: Soliton transport in tubular networks: Transmission at vertices in the shrinking limit publication-title: Phys. Rev. E doi: 10.1103/PhysRevE.91.023209 – volume: 84 start-page: 053618 year: 2011 ident: key2025101714505957745_38 article-title: Symmetric and asymmetric solitons trapped in <span class="mathjax-formula" data-tex="$\mathsf {H}$"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="sans-serif">H</mi></math></span>-shaped potentials publication-title: Phys. Rev. A doi: 10.1103/PhysRevA.84.053618 – volume: 271 start-page: 719 year: 2021 ident: key2025101714505957745_41 article-title: Standing waves on a flower graph publication-title: J. Differ. Equations doi: 10.1016/j.jde.2020.09.010 – volume: 101 start-page: 062208 year: 2020 ident: key2025101714505957745_62 article-title: Dirac particles in transparent quantum graphs: Tunable transport of relativistic quasiparticles in branched structures publication-title: Phys. Rev. E doi: 10.1103/PhysRevE.101.062208 – year: 2021 ident: key2025101714505957745_18 article-title: Grafidi – volume: 322 start-page: 207 issue: 1 year: 2013 ident: key2025101714505957745_30 article-title: A general approximation of quantum graph vertex couplings by scaled Schrödinger operators on thin branched manifolds publication-title: Commun. Math. Phys. doi: 10.1007/s00220-013-1699-9 – volume: 100 start-page: 140402 year: 2008 ident: key2025101714505957745_58 article-title: Dynamics of One-Dimensional Bose Liquids: Andreev-Like Reflection at <span class="mathjax-formula" data-tex="${Y}$"><math xmlns="http://www.w3.org/1998/Math/MathML">Y</math></span> Junctions and the Absence of the Aharonov-Bohm Effect publication-title: Phys. Rev. Lett. doi: 10.1103/PhysRevLett.100.140402 – volume: 383 start-page: 2382 issue: 20 year: 2019 ident: key2025101714505957745_64 article-title: Transparent quantum graphs publication-title: Phys. Lett., A doi: 10.1016/j.physleta.2019.04.059 – volume: 343 start-page: 92 year: 2017 ident: key2025101714505957745_12 article-title: Efficient spectral computation of the stationary states of rotating Bose–Einstein condensates by preconditioned nonlinear conjugate gradient methods publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2017.04.040 – volume: 448 start-page: 262 issue: 1 year: 2017 ident: key2025101714505957745_47 article-title: Dispersive effects for the Schrödinger equation on the tadpole graph publication-title: J. Math. Anal. Appl. doi: 10.1016/j.jmaa.2016.10.060 – volume: 7 start-page: 903 issue: 4 year: 2014 ident: key2025101714505957745_13 article-title: Dispersion for the Schrödinger equation on the line with multiple Dirac delta potentials and on delta trees publication-title: Anal. PDE doi: 10.2140/apde.2014.7.903 – volume: 93 issue: 3 year: 2016 ident: key2025101714505957745_32 article-title: Stationary waves on nonlinear quantum graphs: general framework and canonical perturbation theory publication-title: Phys. Rev. E doi: 10.1103/physreve.93.032204 – year: 2020 ident: key2025101714505957745_34 article-title: Quantum Graph Package publication-title: <a href="https://github.com/manroygood/Quantum-Graphs" target="_blank">https://github.com/manroygood/Quantum-Graphs</a> – volume: 585 start-page: 357 issue: 7825 year: 2020 ident: key2025101714505957745_37 article-title: Array programming with NumPy publication-title: Nature doi: 10.1038/s41586-020-2649-2 – volume: 42 issue: 41 year: 2009 ident: key2025101714505957745_29 article-title: Approximation of quantum graph vertex couplings by scaled Schrödinger operators on thin branched manifolds publication-title: J. Phys. A, Math. Gen. doi: 10.1088/1751-8113/42/41/415305 – volume: 372 issue: 2007 year: 2014 ident: key2025101714505957745_48 article-title: Nonlinear Schrödinger equation on graphs: recent results and open problems publication-title: Philos. Trans. R. Soc. Lond., Ser. A doi: 10.1098/rsta.2013.0002 – volume: 28 start-page: 2343 issue: 7 year: 2015 ident: key2025101714505957745_49 article-title: Bifurcations and stability of standing waves in the nonlinear Schrödinger equation on the tadpole graph publication-title: Nonlinearity doi: 10.1088/0951-7715/28/7/2343 – volume: 38 start-page: 697 issue: 2 year: 2018 ident: key2025101714505957745_51 article-title: Nonlinear Schrödinger equations on periodic metric graphs publication-title: Discrete Contin. Dyn. Syst. doi: 10.3934/dcds.2018030 – volume: 39 start-page: 181 issue: 2 year: 2018 ident: key2025101714505957745_3 article-title: One-dimensional versions of three-dimensional system: ground states for the NLS on the spatial grid publication-title: Rend. Mat. Appl. (7) – volume: 8 start-page: 109 issue: 1 year: 2017 ident: key2025101714505957745_8 article-title: Nonlinear dynamics on branched structures and networks publication-title: Riv. Math. Univ. Parma (N.S.) – volume: 58 issue: 1 year: 2019 ident: key2025101714505957745_9 article-title: Multiple positive bound states for the subcritical NLS equation on metric graphs publication-title: Calc. Var. Partial Differ. Equ. doi: 10.1007/s00526-018-1461-4 – volume: 91 issue: 1 year: 2015 ident: key2025101714505957745_21 article-title: Topology-induced bifurcations for the nonlinear Schrödinger equation on the tadpole graph publication-title: Phys. Rev. E (3) doi: 10.1103/PhysRevE.91.013206 – volume: 59 issue: 5 year: 2020 ident: key2025101714505957745_50 article-title: Standing waves of the quintic NLS equation on the tadpole graph publication-title: Calc. Var. Partial Differ. Equ. doi: 10.1007/s00526-020-01832-3 – volume: 80 year: 1994 ident: key2025101714505957745_10 publication-title: Nonlinear waves in networks – volume: 44 start-page: 277 issue: 2 year: 1981 ident: key2025101714505957745_23 article-title: Finite-difference solutions of a nonlinear Schrödinger equation publication-title: J. Comput. Phys. doi: 10.1016/0021-9991(81)90052-8 – volume: 81 start-page: 066602 year: 2010 ident: key2025101714505957745_56 article-title: Integrable nonlinear Schrödinger equation on simple networks: Connection formula at vertices publication-title: Phys. Rev. E doi: 10.1103/PhysRevE.81.066602 – volume: 219 start-page: xviii+248 year: 2001 ident: key2025101714505957745_11 publication-title: Partial differential equations on multistructures doi: 10.1201/9780203902196 – volume: 39 start-page: B1102 issue: 6 year: 2017 ident: key2025101714505957745_22 article-title: Computation of ground states of the Gross–Pitaevskii functional via Riemannian optimization publication-title: SIAM J. Sci. Comput. doi: 10.1137/17M1121974 – volume: 491 start-page: 124297 issue: 1 year: 2020 ident: key2025101714505957745_43 article-title: Least energy solutions to semi-linear elliptic problems on metric graphs publication-title: Journal of Mathematical Analysis and Applications doi: 10.1016/j.jmaa.2020.124297 – volume: 38 start-page: 1295 issue: 5 year: 2021 ident: key2025101714505957745_15 article-title: Edge-localized states on quantum graphs in the limit of large mass publication-title: Ann. Inst. Henri Poincaré, Anal. Non Linéaire doi: 10.1016/j.anihpc.2020.11.003 – volume: 31 start-page: 1289 issue: 6 year: 2014 ident: key2025101714505957745_2 article-title: Constrained energy minimization and orbital stability for the NLS equation on a star graph publication-title: Ann. Inst. Henri Poincaré, Anal. Non Linéaire doi: 10.1016/j.anihpc.2013.09.003 – volume: 9 start-page: 90 issue: 3 year: 2007 ident: key2025101714505957745_39 article-title: Matplotlib: A 2D graphics environment publication-title: Computing in Science & Engineering doi: 10.1109/MCSE.2007.55 – volume: 18 start-page: 1723 issue: 4 year: 2019 ident: key2025101714505957745_42 article-title: Drift of spectrally stable shifted states on star graphs publication-title: SIAM J. Appl. Dyn. Syst. doi: 10.1137/19M1246146 – start-page: 1 year: 2020 ident: key2025101714505957745_53 article-title: Local minimizers in absence of ground states for the critical NLS energy on metric graphs publication-title: Proceedings of the Royal Society of Edinburgh: Section A Mathematics doi: 10.1017/prm.2020.36 – volume: 186 year: 2013 ident: key2025101714505957745_14 publication-title: Introduction to quantum graphs – volume: 8 start-page: 29 issue: 1 year: 2017 ident: key2025101714505957745_55 article-title: Nonlinear standing waves on planar branched systems: shrinking into metric graph publication-title: Nanosystems: Physics, Chemistry, Mathematics – ident: key2025101714505957745_17 publication-title: Gradient Flow Approach to the Calculation of Ground States on Nonlinear Quantum Graphs – volume: 42 start-page: 934 issue: 3 year: 2004 ident: key2025101714505957745_16 article-title: A relaxation scheme for the nonlinear Schrödinger equation publication-title: SIAM J. Numer. Anal. doi: 10.1137/S0036142901396521 – volume: 17 start-page: 261 year: 2020 ident: key2025101714505957745_60 article-title: SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python publication-title: Nature Methods doi: 10.1038/s41592-019-0686-2 – volume: 2 start-page: 431 year: 2017 ident: key2025101714505957745_35 article-title: Stability of periodic waves of 1D cubic nonlinear Schrödinger equations publication-title: AMRX, Appl. Math. Res. Express doi: 10.1093/amrx/abx004 – volume: 10 start-page: 109 issue: 1 year: 2019 ident: key2025101714505957745_4 article-title: Quantum graphs and dimensional crossover: the honeycomb publication-title: Commun. Appl. Ind. Math. doi: 10.2478/caim-2019-0016 – volume: 23 issue: 6 year: 2016 ident: key2025101714505957745_31 article-title: Validity of the NLS approximation for periodic quantum graphs publication-title: NoDEA, Nonlinear Differ. Equ. Appl. doi: 10.1007/s00030-016-0417-7 – volume: 18 start-page: 1185 issue: 4 year: 2017 ident: key2025101714505957745_52 article-title: Bifurcations of standing localized waves on periodic graphs publication-title: Ann. Henri Poincaré doi: 10.1007/s00023-016-0536-z – volume: 33 start-page: 413 issue: 3 year: 2021 ident: key2025101714505957745_19 article-title: Boundary null-controllability of 1-D coupled parabolic systems with Kirchhoff-type conditions publication-title: Math. Control Signals Syst. doi: 10.1007/s00498-021-00285-z – volume: 237 start-page: 1103 issue: 8 year: 2008 ident: key2025101714505957745_44 article-title: Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential publication-title: Physica D doi: 10.1016/j.physd.2007.12.004 – volume: 54 start-page: 743 issue: 1 year: 2015 ident: key2025101714505957745_6 article-title: NLS ground states on graphs publication-title: Calc. Var. Partial Differ. Equ. doi: 10.1007/s00526-014-0804-z – start-page: 11 year: 2008 ident: key2025101714505957745_36 article-title: Exploring network structure, dynamics, and function using NetworkX publication-title: Proceedings of the 7th Python in Science Conference (SciPy2008) doi: 10.25080/TCWV9851 – volume: 12 start-page: 1597 issue: 6 year: 2019 ident: key2025101714505957745_5 article-title: Dimensional crossover with a continuum of critical exponents for NLS on doubly periodic metric graphs publication-title: Anal. PDE doi: 10.2140/apde.2019.12.1597 – volume: 102 start-page: 1223 issue: 3 year: 2020 ident: key2025101714505957745_27 article-title: NLS ground states on metric trees: existence results and open questions publication-title: J. Lond. Math. Soc. doi: 10.1112/jlms.12361 – volume: 5 start-page: 506 year: 1968 ident: key2025101714505957745_57 article-title: On the construction and comparison of difference schemes publication-title: SIAM J. Numer. Anal. doi: 10.1137/0705041 – volume: 374 year: 2020 ident: key2025101714505957745_28 article-title: Uniqueness and non-uniqueness of prescribed mass NLS ground states on metric graphs publication-title: Adv. Math. doi: 10.1016/j.aim.2020.107352 – volume: 39 start-page: 2203 issue: 4 year: 2019 ident: key2025101714505957745_33 article-title: NLS bifurcations on the bowtie combinatorial graph and the dumbbell metric graph publication-title: Discrete Contin. Dyn. Syst. doi: 10.3934/dcds.2019093 – volume: 23 start-page: 485 issue: 3 year: 1986 ident: key2025101714505957745_61 article-title: Split-step methods for the solution of the nonlinear Schrödinger equation publication-title: SIAM J. Numer. Anal. doi: 10.1137/0723033 – volume: 26 issue: 5 year: 2019 ident: key2025101714505957745_25 article-title: Mass-constrained ground states of the stationary NLSE on periodic metric graphs publication-title: NoDEA, Nonlinear Differ. Equ. Appl. doi: 10.1007/s00030-019-0576-4 |
| SSID | ssj0002923762 |
| Score | 2.3475196 |
| Snippet | Nonlinear quantum graphs are metric graphs equipped with a nonlinear Schrödinger equation. Whereas in the last ten years they have known considerable... |
| SourceID | hal crossref |
| SourceType | Open Access Repository Enrichment Source Index Database |
| StartPage | 1 |
| SubjectTerms | Analysis of PDEs Mathematics Nonlinear Sciences Numerical Analysis Pattern Formation and Solitons |
| Title | Numerical Simulations on Nonlinear Quantum Graphs with the GraFiDi Library |
| URI | https://hal.science/hal-03164639 |
| Volume | 8 |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| journalDatabaseRights | – providerCode: PRVHPJ databaseName: ROAD: Directory of Open Access Scholarly Resources customDbUrl: eissn: 2426-8399 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0002923762 issn: 2426-8399 databaseCode: M~E dateStart: 20150101 isFulltext: true titleUrlDefault: https://road.issn.org providerName: ISSN International Centre |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV1NbxMxELVo4cAFgQBRvmShnlgt1GtnbR8joBREIqQWqbeV12uLrZpNSZqq4sBvZyZ2nE2VQzlwsbKWvVl5nmbG4_EbQvZLd2ANtyZ3pdO5MNAYAY9mwBo8VbJ1HYpNyPFYnZ7q7zGqNF-WE5Bdp66v9cV_FTX0gbDx6uw_iDu9FDrgNwgdWhA7tLcS_HgRDmHOs-N2skipbl02DqwYZoaZnGBqJtlnZKuO99vQAYXnw_Zjm8XLDH3H9Xg0_JL1vNdQDWIVSRwl8td11B1JyTfpC9ZOcz2d218hsXti2l5WELz3d0w-8xG3MSQBu1nMZOlpUTT7OXheQRe6LX1R9aqe6mQ9GxxIOG9q94FassXO4cPyMzt5F2r_bHJo37BtKeMQ9jo4vcLJFUyupNohdws50KgKR3_WgblCY6IQnj6lTw73OnH--96fb3gyOz9XgfilY3LykDyIMqHDgIRH5I7rHpOvCQW0hwI67WhCAY0ooAEFFFFAQYw0ooBGFDwhPw4_nXw4ymPZjNyC-VG58YVythGSe2-N8IVk3DcFK2tZukJ7AW6Z9Y7xWggvDdO6bqzgSnrWWCU5f0p2u2nnnhEqFHdNUfK6briAnalmpjwwpSwcEkcO5B55u1qCykZOeSxtcl5tWe89sp8GXwQqle3D3sBaphFIf340_FZhHxigUoBLfcWe3-5dL8j9NT5fkt3L2cK9Ivfs1WU7n71eyv0vc4B8PA |
| linkProvider | ISSN International Centre |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Numerical+Simulations+on+Nonlinear+Quantum+Graphs+with+the+GraFiDi+Library&rft.jtitle=SMAI+Journal+of+Computational+Mathematics&rft.au=Besse%2C+Christophe&rft.au=Duboscq%2C+Romain&rft.au=Le+Coz%2C+Stefan&rft.date=2022-04-08&rft.issn=2426-8399&rft.eissn=2426-8399&rft.volume=8&rft.spage=1&rft.epage=47&rft_id=info:doi/10.5802%2Fsmai-jcm.78&rft.externalDBID=n%2Fa&rft.externalDocID=10_5802_smai_jcm_78 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=2426-8399&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=2426-8399&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=2426-8399&client=summon |