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Remarks on the Schrödinger-Lohe model
Department of Mathematics, Chung-Ang University, Seoul 156-756, Republic of Korea |
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.
References:
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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. |
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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. |
| [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. |
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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. |
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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. |
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M. A. Lohe, Quantum synchronization over quantum networks, J. Phys. A, 43 (2010), 465301, 20 pp.
doi: 10.1088/1751-8113/43/46/465301. |
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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. |
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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. |
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. |
| [2] |
R. Bellman, Stability Theory of Differential Equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953. |
| [3] |
S. Blanes, F. Casas, J. 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. |
| [4] |
S. Blanes, F. Casas, J. 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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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