薛定谔方程中包络孤子运动及相互作用的辛算法模拟

J4 ›› 2017, Vol. 34 ›› Issue (2): 231-240.

薛定谔方程中包络孤子运动及相互作用的辛算法模拟

赖联有

集美大学

收稿日期:2016-02-29 修回日期:2016-08-27 出版日期:2017-03-28 发布日期:2017-04-06
通讯作者: 赖联有

Simulating the motion and interaction of envelope solitons in Schr?dinger equation by symplectic algorithm

Received:2016-02-29 Revised:2016-08-27 Published:2017-03-28 Online:2017-04-06

摘要/Abstract

摘要： 给出了Schr?dinger方程中高斯包络孤子的表达式，并对该表达式进行了证明。针对该高斯包络孤子的特点，进一步提出Schr?dinger方程中存在高斯包络孤子相互作用的情况。针对Schr?dinger方程的特点，提出Schr?dinger方程的辛算法。首先，通过波函数实部和虚部分离的方法把Schr?dinger方程变换成标准的哈密顿正则方程组。然后，通过对正则方程进行欧拉中心差分离散实现辛算法。给出了辛算法的守恒量，并证明了辛算法的稳定性。对Schr?dinger方程中的高斯包络孤子运动及多孤子相互作用过程进行了数值仿真实验，实验结果证明了所提观点的正确性及辛算法的有效性。

关键词: 光通信, 孤子, 辛算法, 薛定谔方程, optical communication, soliton, symplectic algorithm, Schr?dinger equation

Abstract: The expression of Gaussian envelope soliton in Schr?dinger 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 Schr?dinger equation. The symplectic algorithm for solving Schr?dinger equation is proposed after analysis characteristics of Schr?dinger equation. First, the Schr?dinger 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 Schr?dinger 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.

赖联有. 薛定谔方程中包络孤子运动及相互作用的辛算法模拟[J]. J4, 2017, 34(2): 231-240.

参考文献

[1] MA RCUSE D. Simulation of single-channel optical sustems at 100CB/s[J]. Journal of Lightwave Technology, 1999, 17(4): 564-569.
[2] HASEGAWA A. Optical solitons in fibers for communication systems[J]. Optics and Photonics News, 2002, 13(2): 33-37.
[3] DIMITREG O. Generation of megawatt optical solitons in Hollow-Core photonic band-gap fibers [J]. Science, 2003, 301(5640): 1702-1704.
[4] Kivshar Y S, Agrawal G P. Optical Solitons-From Fibers to Photonic Crystals[M]. San Diego, Academic Press, 2003.
[5] MIAO Run-cai, WANG Fei, ZENG Xiang-mei. Soliton decay induced by interaction between two neighboring second-order solitons[J]. Acta Photonica Sinica, 2004, 33(8):927-930. 苗润才, 王飞, 曾祥梅. 二阶孤子间相互作用引起孤子衰变[J]. 光子学报, 2004, 33(8):927-930.
[6] CLAUDIO C, MARCO P. Observation of optical spatial solitons in a highly nonlocal medium [J]. Phys Rev Lett, 2004, 92:113902-113905.
[7] Xu Siliu , Liu Huiping, Yi Lin. Two-dimensional Kummer-Gaussian soliton clusters in strongly nonlocal nonlineaxmedia[J]. Acta Physica sinica(物理学报), 2010,59(2):138-143(in Chinese).
[8] Hao Yueyu. Nonlinear management for optical fiber soliton with the structure double pitfalls [J]. Chinese Journal of Quantum Electronics(量子电子学报), 2012, 29(6): 754-758 (in Chinese).
[9] FENG Kang. Proceedings of the 1984 Beijing symposium. on differential geometry and differential equation computation of partial differential equations [C]. Beijing: Science Press, 1985. 42-58.
[10] Feng Kang, in Proc. 10th Inter. Conf. Numerical Methods in Fluid Dynamics, Ed. F. G. Zhuang, Lect Notes in Phys, 1986, 264: 1~7
[11] Feng Kang, Qin Mengzhao, in Proc. Numerical Methods for PDE's, Ed. Zhou Youlan, Lect Notes in Phys, Springer 1987, 1297: 1~37
[12] Feng Kang, Qin Mengzhao. Hamiltonian algorithms for hamiltonian systems and a comparative numerical study[J]. Comput. Phys. Comm., 1991, 65:173~187
[13] Feng Kang. Symplectic, Contact and Volume-Preserving Algorithms, Proc. 1st China-Japan Conf. on Numer. Math., Beijin, 1992, World Scientific , Singapor, 1993
[14] Ding Peizhu, etc. Symplectic Algorithm of Quantum Systems[J]. Journal of Jilin University, 1993(4): 75～78. 丁培柱, 等. 量子系统的辛算法[J]. 吉林大学自然科学学报, 1993(4): 75～78 .

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