December  2019, 14(4): 759-769. doi: 10.3934/nhm.2019030

Remarks on the Schrödinger-Lohe model

Department of Mathematics, Chung-Ang University, Seoul 156-756, Republic of Korea

* Corresponding author: Hyungjin Huh

Received  October 2018 Revised  July 2019 Published  October 2019

Fund Project: This research was supported by LG Yonam Foundation (of Korea) and Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (2017R1D1A1B03028308)

We study the Schrödinger-Lohe model. Making use of the principal fundamental matrix $ Y $ of linear ODEs with variable coefficients, the coupled nonlinear Schrödinger-Lohe system is transformed into the decoupled linear Schrödinger equations. The boundedness of $ Y $ is shown for the case of complete synchronization. We also study the cases where the principal fundamental matrices can be derived explicitly.

Citation: Hyungjin Huh. Remarks on the Schrödinger-Lohe model. Networks & Heterogeneous Media, 2019, 14 (4) : 759-769. doi: 10.3934/nhm.2019030
References:
[1]

P. Antonelli and P. Marcati, A model of synchronization over quantum networks, J. Phys. A, 50 (2017), 315101, 19 pp. doi: 10.1088/1751-8121/aa79c9.  Google Scholar

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R. Bellman, Stability Theory of Differential Equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953.  Google Scholar

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S. BlanesF. CasasJ. A. Oteo and J. Ros, Magnus and Fer expansions for matrix differential equations: The convergence problem, J. Phys. A, 31 (1998), 259-268.  doi: 10.1088/0305-4470/31/1/023.  Google Scholar

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S.-H. Choi, J. Cho and S.-Y. Ha, Practical quantum synchronization for the Schrödinger-Lohe system, J. Phys. A, 49 (2016), 205203, 17 pp. doi: 10.1088/1751-8113/49/20/205203.  Google Scholar

[6]

S.-H. Choi and S.-Y. Ha, Quantum synchronization of the Schrödinger-Lohe model, J. Phys. A, 47 (2014), 355104, 16 pp. doi: 10.1088/1751-8113/47/35/355104.  Google Scholar

[7]

H. Huh and S.-Y. Ha, Dynamical system approach to synchronization of the coupled Schrödinger-Lohe system, Quart. Appl. Math., 75 (2017), 555-579.  doi: 10.1090/qam/1465.  Google Scholar

[8]

H. Huh, S.-Y. Ha and D. Kim, Emergent behaviors of the Schrödinger-Lohe model on cooperative-competitive networks, J. Differential Equations, 263 (2017), 8295–8321. doi: 10.1016/j.jde.2017.08.050.  Google Scholar

[9]

H. Huh, S.-Y. Ha and D. Kim, Asymptotic behavior and stability for the Schrödinger-Lohe model, J. Math. Phys., 59 (2018), 102701, 21 pp. doi: 10.1063/1.5041463.  Google Scholar

[10]

M. A. Lohe, Quantum synchronization over quantum networks, J. Phys. A, 43 (2010), 465301, 20 pp. doi: 10.1088/1751-8113/43/46/465301.  Google Scholar

[11]

M. A. Lohe, Non-Abelian Kuramoto model and synchronization, J. Phys. A, 42 (2009), 395101, 25 pp. doi: 10.1088/1751-8113/42/39/395101.  Google Scholar

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W. Magnus, On the exponential solution of differential equations for a linear operator, Comm. Pure Appl. Math., 7 (1954), 649-673.  doi: 10.1002/cpa.3160070404.  Google Scholar

show all references

References:
[1]

P. Antonelli and P. Marcati, A model of synchronization over quantum networks, J. Phys. A, 50 (2017), 315101, 19 pp. doi: 10.1088/1751-8121/aa79c9.  Google Scholar

[2]

R. Bellman, Stability Theory of Differential Equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953.  Google Scholar

[3]

S. BlanesF. CasasJ. A. Oteo and J. Ros, Magnus and Fer expansions for matrix differential equations: The convergence problem, J. Phys. A, 31 (1998), 259-268.  doi: 10.1088/0305-4470/31/1/023.  Google Scholar

[4]

S. BlanesF. CasasJ. A. Oteo and J. Ros, The Magnus expansion and some of its applications, Phys. Rep., 470 (2009), 151-238.  doi: 10.1016/j.physrep.2008.11.001.  Google Scholar

[5]

S.-H. Choi, J. Cho and S.-Y. Ha, Practical quantum synchronization for the Schrödinger-Lohe system, J. Phys. A, 49 (2016), 205203, 17 pp. doi: 10.1088/1751-8113/49/20/205203.  Google Scholar

[6]

S.-H. Choi and S.-Y. Ha, Quantum synchronization of the Schrödinger-Lohe model, J. Phys. A, 47 (2014), 355104, 16 pp. doi: 10.1088/1751-8113/47/35/355104.  Google Scholar

[7]

H. Huh and S.-Y. Ha, Dynamical system approach to synchronization of the coupled Schrödinger-Lohe system, Quart. Appl. Math., 75 (2017), 555-579.  doi: 10.1090/qam/1465.  Google Scholar

[8]

H. Huh, S.-Y. Ha and D. Kim, Emergent behaviors of the Schrödinger-Lohe model on cooperative-competitive networks, J. Differential Equations, 263 (2017), 8295–8321. doi: 10.1016/j.jde.2017.08.050.  Google Scholar

[9]

H. Huh, S.-Y. Ha and D. Kim, Asymptotic behavior and stability for the Schrödinger-Lohe model, J. Math. Phys., 59 (2018), 102701, 21 pp. doi: 10.1063/1.5041463.  Google Scholar

[10]

M. A. Lohe, Quantum synchronization over quantum networks, J. Phys. A, 43 (2010), 465301, 20 pp. doi: 10.1088/1751-8113/43/46/465301.  Google Scholar

[11]

M. A. Lohe, Non-Abelian Kuramoto model and synchronization, J. Phys. A, 42 (2009), 395101, 25 pp. doi: 10.1088/1751-8113/42/39/395101.  Google Scholar

[12]

W. Magnus, On the exponential solution of differential equations for a linear operator, Comm. Pure Appl. Math., 7 (1954), 649-673.  doi: 10.1002/cpa.3160070404.  Google Scholar

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