Motion and Interaction of Envelope Solitons in Schrödinger Equation Simulated by Symplectic Algorithm

Lai Lianyou; Xu Weijian

doi:doi:10.11648/j.ajpa.20190701.11

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Motion and Interaction of Envelope Solitons in Schrödinger Equation Simulated by Symplectic Algorithm

Lai Lianyou, Xu Weijian

Published in American Journal of Physics and Applications (Volume 7, Issue 1)

Received: 18 November 2018 Accepted: 14 December 2018 Published: 21 January 2019

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Abstract

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.

Published in	American Journal of Physics and Applications (Volume 7, Issue 1)
DOI	10.11648/j.ajpa.20190701.11
Page(s)	1-7
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2019. Published by Science Publishing Group

Keywords

Schrödinger Equation, Soliton, Symplectic Algorithm, Optical Communication

References

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[4]	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.
[5]	Mattsson K , Werpers J . “High-fidelity numerical simulation of solitons in the nerve axon,” Journal of Computational Physics, 2016, 305: 793-816.
[6]	Yu, Fajun. “Nonautonomous soliton, controllable interaction and numerical simulation for generalized coupled cubic–quintic nonlinear Schrödinger equations,” Nonlinear Dynamics, 2016, 85(2): 1203-1216.
[7]	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.
[8]	FENG Kang, QINMengzhao. “Hamiltonian algorithms for hamiltonian systems and a comparative numerical study,” Comput. Phys. Comm., 1991, 65:173~187.
[9]	FENG Kang, et al. “Symplectic Algorithm in Hamilton System,” Hangzhou: Zhejiang Science and Technology Publishing House, 2003: 358-359.
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APA Style

Lai Lianyou, Xu Weijian. (2019). Motion and Interaction of Envelope Solitons in Schrödinger Equation Simulated by Symplectic Algorithm. American Journal of Physics and Applications, 7(1), 1-7. https://doi.org/10.11648/j.ajpa.20190701.11

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ACS Style

Lai Lianyou; Xu Weijian. Motion and Interaction of Envelope Solitons in Schrödinger Equation Simulated by Symplectic Algorithm. Am. J. Phys. Appl. 2019, 7(1), 1-7. doi: 10.11648/j.ajpa.20190701.11

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AMA Style

Lai Lianyou, Xu Weijian. Motion and Interaction of Envelope Solitons in Schrödinger Equation Simulated by Symplectic Algorithm. Am J Phys Appl. 2019;7(1):1-7. doi: 10.11648/j.ajpa.20190701.11

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@article{10.11648/j.ajpa.20190701.11,
  author = {Lai Lianyou and Xu Weijian},
  title = {Motion and Interaction of Envelope Solitons in Schrödinger Equation Simulated by Symplectic Algorithm},
  journal = {American Journal of Physics and Applications},
  volume = {7},
  number = {1},
  pages = {1-7},
  doi = {10.11648/j.ajpa.20190701.11},
  url = {https://doi.org/10.11648/j.ajpa.20190701.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajpa.20190701.11},
  abstract = {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.},
 year = {2019}
}

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TY  - JOUR
T1  - Motion and Interaction of Envelope Solitons in Schrödinger Equation Simulated by Symplectic Algorithm
AU  - Lai Lianyou
AU  - Xu Weijian
Y1  - 2019/01/21
PY  - 2019
N1  - https://doi.org/10.11648/j.ajpa.20190701.11
DO  - 10.11648/j.ajpa.20190701.11
T2  - American Journal of Physics and Applications
JF  - American Journal of Physics and Applications
JO  - American Journal of Physics and Applications
SP  - 1
EP  - 7
PB  - Science Publishing Group
SN  - 2330-4308
UR  - https://doi.org/10.11648/j.ajpa.20190701.11
AB  - 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.
VL  - 7
IS  - 1
ER  -

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Author Information

Lai Lianyou

College of Information Engineering, Jimei University, Xiamen, China
Xu Weijian

College of Mechanical Engineering and Automation, Huaqiao University, Xiamen, China

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APA Style

Lai Lianyou, Xu Weijian. (2019). Motion and Interaction of Envelope Solitons in Schrödinger Equation Simulated by Symplectic Algorithm. American Journal of Physics and Applications, 7(1), 1-7. https://doi.org/10.11648/j.ajpa.20190701.11

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ACS Style

Lai Lianyou; Xu Weijian. Motion and Interaction of Envelope Solitons in Schrödinger Equation Simulated by Symplectic Algorithm. Am. J. Phys. Appl. 2019, 7(1), 1-7. doi: 10.11648/j.ajpa.20190701.11

Copy | Download

AMA Style

Lai Lianyou, Xu Weijian. Motion and Interaction of Envelope Solitons in Schrödinger Equation Simulated by Symplectic Algorithm. Am J Phys Appl. 2019;7(1):1-7. doi: 10.11648/j.ajpa.20190701.11

Copy | Download

@article{10.11648/j.ajpa.20190701.11,
  author = {Lai Lianyou and Xu Weijian},
  title = {Motion and Interaction of Envelope Solitons in Schrödinger Equation Simulated by Symplectic Algorithm},
  journal = {American Journal of Physics and Applications},
  volume = {7},
  number = {1},
  pages = {1-7},
  doi = {10.11648/j.ajpa.20190701.11},
  url = {https://doi.org/10.11648/j.ajpa.20190701.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajpa.20190701.11},
  abstract = {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.},
 year = {2019}
}

Copy | Download

TY  - JOUR
T1  - Motion and Interaction of Envelope Solitons in Schrödinger Equation Simulated by Symplectic Algorithm
AU  - Lai Lianyou
AU  - Xu Weijian
Y1  - 2019/01/21
PY  - 2019
N1  - https://doi.org/10.11648/j.ajpa.20190701.11
DO  - 10.11648/j.ajpa.20190701.11
T2  - American Journal of Physics and Applications
JF  - American Journal of Physics and Applications
JO  - American Journal of Physics and Applications
SP  - 1
EP  - 7
PB  - Science Publishing Group
SN  - 2330-4308
UR  - https://doi.org/10.11648/j.ajpa.20190701.11
AB  - 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.
VL  - 7
IS  - 1
ER  -

Copy | Download