Motion and Interaction of Envelope Solitons in Schrödinger Equation Simulated by Symplectic Algorithm
American Journal of Physics and Applications
Volume 7, Issue 1, January 2019, Pages: 1-7
Received: Nov. 18, 2018;
Accepted: Dec. 14, 2018;
Published: Jan. 21, 2019
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Lai Lianyou, College of Information Engineering, Jimei University, Xiamen, China; College of Mechanical Engineering and Automation, Huaqiao University, Xiamen, China
Xu Weijian, College of Mechanical Engineering and Automation, Huaqiao University, Xiamen, China
The expression of Gaussian envelope soliton in Schrödinger equations are given and proved in this paper. According to the characteristics of the Gauss envelope soliton, further proposed that the interaction between Gaussian envelope solitons exists in Schrödinger equation. The symplectic algorithm for solving Schrödinger equation is proposed after analysis characteristics of Schrödinger equation. First, the Schrödinger equation is transformed into the standard Hamiltonian canonical equation by separating the real and imaginary parts of wave function. Secondly, the symplectic algorithm is implemented by using the Euler center difference method for the canonical equation. The conserved quantity of symplectic algorithm is given, and the stability of symplectic algorithm is proved. The numerical simulation experiment was carried out on Schrödinger equation in Gauss envelope soliton motion and multi solitons interaction. The experimental results show that the proposed method is correct and the symplectic algorithm is effective.
Motion and Interaction of Envelope Solitons in Schrödinger Equation Simulated by Symplectic Algorithm, American Journal of Physics and Applications.
Vol. 7, No. 1,
2019, pp. 1-7.
Copyright © 2019 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/
) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Trocha P , Karpov M , Ganin D , et al. “Ultrafast optical ranging using microresonator soliton frequency combs,” Science, 2018, 359(6378): 887-891.
Biswas A , Hubert M B , Justin M , et al. “Chirped dispersive bright and singular optical solitons with Schrödinger–Hirota equation,” Optik, 2018, 168: 192-195.
Biswas A , Ekici M , Sonmezoglu A , et al. “Optical solitons in parabolic law medium with weak non-local nonlinearity by extended trial function method,” Optik, 2018, 163: 56-61.
Anjan B , Mehmet E , Abdullah S , et al. “Solitons in optical metamaterials with anti-cubic nonlinearity,” The European Physical Journal Plus, 2018, 133(5): 204.
Mattsson K , Werpers J . “High-fidelity numerical simulation of solitons in the nerve axon,” Journal of Computational Physics, 2016, 305: 793-816.
Yu, Fajun. “Nonautonomous soliton, controllable interaction and numerical simulation for generalized coupled cubic–quintic nonlinear Schrödinger equations,” Nonlinear Dynamics, 2016, 85(2): 1203-1216.
FENG Kang. Proceedings of the 1984 Beijing symposium. “on differential geometry and differential equation computation of partial differential equations,” Beijing: Science Press, 1985. 42-58.
FENG Kang, QINMengzhao. “Hamiltonian algorithms for hamiltonian systems and a comparative numerical study,” Comput. Phys. Comm., 1991, 65:173~187.
FENG Kang, et al. “Symplectic Algorithm in Hamilton System,” Hangzhou: Zhejiang Science and Technology Publishing House, 2003: 358-359.
Blanes S , Casas F , Murua A . “Symplectic time-average propagators for the Schrödinger equation with a time-dependent Hamiltonian,” The Journal of Chemical Physics, 2017, 146(11): 114109.
Chen Q , Qin H , Liu J , et al. “Canonical symplectic structure and structure-preserving geometric algorithms for Schrödinger-Maxwell systems,” Journal of Computational Physics, 2017, 349: 441-452.
He Y , Zhou Z , Sun Y , et al. “Explicit, K-symplectic algorithms for charged particle dynamics,” Physics Letters A, 2017, 381(6): 568-573.
Gaset J , RománRoy, Narciso. “Multisymplectic unified formalism for Einstein-Hilbert Gravity,” Journal of Mathematical Physics, 2018, 59(3).
Shifler R M . “Equivariant Quantum Cohomology of the Odd Symplectic Grassmannian,” Mathematische Zeitschrift, 2017: 1-35.
He Z F , Liu S , Chen S , et al. “Application of symplectic finite-difference time-domain scheme for anisotropic magnetised plasma,” IET Microwaves, Antennas & Propagation, 2017, 11(5): 600-606.