Two-point correlation function and its applications to the Schrödinger-Lohe type models
Authors:
Seung-Yeal Ha, Gyuyoung Hwang and Dohyun Kim
Journal:
Quart. Appl. Math. 80 (2022), 669-699
MSC (2020):
Primary 34C15; Secondary 34D06, 35Q55
DOI:
https://doi.org/10.1090/qam/1623
Published electronically:
May 10, 2022
MathSciNet review:
4489001
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Abstract: We study the asymptotic emergent dynamics and the continuum limit for the Schrödinger-Lohe (SL) model and semi-discrete SL model. For the SL model, emergent dynamics has been mostly studied for systems with identical potentials in literature. In this paper, we further extend emergent dynamics and stability estimate for the SL model with nonidentical potentials. To achieve this, we use two-point correlation functions defined as an inner product between wave functions. For the semi-discrete SL model, we provide a global unique solvability and a sufficient framework for the smooth transition from the semi-discrete SL model to the SL model in any finite-time interval, as the mesh size tends to zero. Our convergence estimate depends on the uniform-in-$h$ Strichartz estimate and the uniform-stability of the SL models with respect to initial data.
References
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- Yoshiki Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, International Symposium on Mathematical Problems in Theoretical Physics (Kyoto Univ., Kyoto, 1975) Lecture Notes in Phys., vol. 39, Springer, Berlin, 1975, pp. 420–422. MR 0676492
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- M. A. Lohe, Non-abelian Kuramoto models and synchronization, J. Phys. A 42 (2009), no. 39, 395101, 25. MR 2539317, DOI 10.1088/1751-8113/42/39/395101
- Renato E. Mirollo and Steven H. Strogatz, Synchronization of pulse-coupled biological oscillators, SIAM J. Appl. Math. 50 (1990), no. 6, 1645–1662. MR 1080514, DOI 10.1137/0150098
- Arkady Pikovsky, Michael Rosenblum, and Jürgen Kurths, Synchronization, Cambridge Nonlinear Science Series, vol. 12, Cambridge University Press, Cambridge, 2001. A universal concept in nonlinear sciences. MR 1869044, DOI 10.1017/CBO9780511755743
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References
- P. Antonelli and P. Marcati, A model of synchronization over quantum networks, J. Phys. A 50 (2017), no. 31, 315101, 19. MR 3673479, DOI 10.1088/1751-8121/aa79c9
- Weizhu Bao, Seung-Yeal Ha, Dohyun Kim, and Qinglin Tang, Collective synchronization of the multi-component Gross-Pitaevskii-Lohe system, Phys. D 400 (2019), 132158, 30. MR 3989155, DOI 10.1016/j.physd.2019.132158
- J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature 211 (1966), 562–564.
- Thierry Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, vol. 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. MR 2002047, DOI 10.1090/cln/010
- Dongpyo Chi, Sun-Ho Choi, and Seung-Yeal Ha, Emergent behaviors of a holonomic particle system on a sphere, J. Math. Phys. 55 (2014), no. 5, 052703, 18. MR 3390625, DOI 10.1063/1.4878117
- Sun-Ho Choi and Seung-Yeal Ha, Quantum synchronization of the Schrödinger-Lohe model, J. Phys. A 47 (2014), no. 35, 355104, 16. MR 3254872, DOI 10.1088/1751-8113/47/35/355104
- Seung-Yeal Ha, Shi Jin, and Doheon Kim, Convergence of a first-order consensus-based global optimization algorithm, Math. Models Methods Appl. Sci. 30 (2020), no. 12, 2417–2444. MR 4179193, DOI 10.1142/S0218202520500463
- Seung-Yeal Ha, Hwa Kil Kim, and Sang Woo Ryoo, Emergence of phase-locked states for the Kuramoto model in a large coupling regime, Commun. Math. Sci. 14 (2016), no. 4, 1073–1091. MR 3491817, DOI 10.4310/CMS.2016.v14.n4.a10
- Younghun Hong and Changhun Yang, Uniform Strichartz estimates on the lattice, Discrete Contin. Dyn. Syst. 39 (2019), no. 6, 3239–3264. MR 3959428, DOI 10.3934/dcds.2019134
- Younghun Hong and Changhun Yang, Strong convergence for discrete nonlinear Schrödinger equations in the continuum limit, SIAM J. Math. Anal. 51 (2019), no. 2, 1297–1320. MR 3939333, DOI 10.1137/18M120703X
- Hyungjin Huh and Seung-Yeal Ha, Dynamical system approach to synchronization of the coupled Schrödinger-Lohe system, Quart. Appl. Math. 75 (2017), no. 3, 555–579. MR 3636169, DOI 10.1090/qam/1465
- Hyungjin Huh, Seung-Yeal Ha, and Dohyun Kim, Asymptotic behavior and stability for the Schrödinger-Lohe model, J. Math. Phys. 59 (2018), no. 10, 102701, 21. MR 3869160, DOI 10.1063/1.5041463
- Hyungjin Huh, Seung-Yeal Ha, and Dohyun Kim, Emergent behaviors of the Schrödinger-Lohe model on cooperative- competitive networks, J. Differential Equations 263 (2017), no. 12, 8295–8321. MR 3710686, DOI 10.1016/j.jde. 2017.08.050
- Liviu I. Ignat, Fully discrete schemes for the Schrödinger equation. Dispersive properties, Math. Models Methods Appl. Sci. 17 (2007), no. 4, 567–591. MR 2316299, DOI 10.1142/S0218202507002029
- Liviu I. Ignat and Enrique Zuazua, Convergence rates for dispersive approximation schemes to nonlinear Schrödinger equations, J. Math. Pures Appl. (9) 98 (2012), no. 5, 479–517 (English, with English and French summaries). MR 2980459, DOI 10.1016/j.matpur.2012.01.001
- Liviu I. Ignat and Enrique Zuazua, Numerical dispersive schemes for the nonlinear Schrödinger equation, SIAM J. Numer. Anal. 47 (2009), no. 2, 1366–1390. MR 2485456, DOI 10.1137/070683787
- Palle E. T. Jorgensen, Essential self-adjointness of the graph-Laplacian, J. Math. Phys. 49 (2008), no. 7, 073510, 33. MR 2432048, DOI 10.1063/1.2953684
- Panayotis G. Kevrekidis, The discrete nonlinear Schrödinger equation, Springer Tracts in Modern Physics, vol. 232, Springer-Verlag, Berlin, 2009. Mathematical analysis, numerical computations and physical perspectives; Edited by Kevrekidis and with contributions by Ricardo Carretero-González, Alan R. Champneys, Jesús Cuevas, Sergey V. Dmitriev, Dimitri J. Frantzeskakis, Ying-Ji He, Q. Enam Hoq, Avinash Khare, Kody J. H. Law, Boris A. Malomed, Thomas R. O. Melvin, Faustino Palmero, Mason A. Porter, Vassilis M. Rothos, Atanas Stefanov and Hadi Susanto. MR 2742565, DOI 10.1007/978-3-540-89199-4
- Kay Kirkpatrick, Enno Lenzmann, and Gigliola Staffilani, On the continuum limit for discrete NLS with long-range lattice interactions, Comm. Math. Phys. 317 (2013), no. 3, 563–591. MR 3009717, DOI 10.1007/s00220-012-1621-x
- Y. Kuramoto, Chemical oscillations, waves, and turbulence, Springer Series in Synergetics, vol. 19, Springer-Verlag, Berlin, 1984. MR 762432, DOI 10.1007/978-3-642-69689-3
- Yoshiki Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, International Symposium on Mathematical Problems in Theoretical Physics (Kyoto Univ., Kyoto, 1975) Springer, Berlin, 1975, pp. 420–422. Lecture Notes in Phys., 39. MR 0676492
- M. A. Lohe, Higher-dimensional generalizations of the Watanabe-Strogatz transform for vector models of synchronization, J. Phys. A 51 (2018), no. 22, 225101, 24. MR 3803610, DOI 10.1088/1751-8121/aac030
- M. A. Lohe, Non-abelian Kuramoto models and synchronization, J. Phys. A 42 (2009), no. 39, 395101, 25. MR 2539317, DOI 10.1088/1751-8113/42/39/395101
- Renato E. Mirollo and Steven H. Strogatz, Synchronization of pulse-coupled biological oscillators, SIAM J. Appl. Math. 50 (1990), no. 6, 1645–1662. MR 1080514, DOI 10.1137/0150098
- Arkady Pikovsky, Michael Rosenblum, and Jürgen Kurths, Synchronization, Cambridge Nonlinear Science Series, vol. 12, Cambridge University Press, Cambridge, 2001. A universal concept in nonlinear sciences. MR 1869044, DOI 10.1017/CBO9780511755743
- A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol. 16 (1967), 15-42.
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Additional Information
Seung-Yeal Ha
Affiliation:
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea
MR Author ID:
684438
Email:
syha@snu.ac.kr
Gyuyoung Hwang
Affiliation:
Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea
MR Author ID:
1455927
Email:
hgy0407@snu.ac.kr
Dohyun Kim
Affiliation:
School of Mathematics, Statistics and Data Science, Sungshin Women’s University, Seoul 02844, Republic of Korea
MR Author ID:
858181
Email:
dohyunkim@sungshin.ac.kr
Keywords:
Continuum limit,
global existence,
quantum synchronization,
Schrödinger-Lohe model,
Strichartz estimates on lattice,
uniform stability
Received by editor(s):
March 15, 2022
Received by editor(s) in revised form:
April 8, 2022
Published electronically:
May 10, 2022
Additional Notes:
The work of the first author is supported by National Research Foundation of Korea (NRF-2020R1A2C3A01003881). The work of the last author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No.2021R1F1A1055929).
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