Please wait a minute...
Chin. Phys. B, 2014, Vol. 23(12): 120203    DOI: 10.1088/1674-1056/23/12/120203
GENERAL Prev   Next  

A conservative Fourier pseudospectral algorithm for the nonlinear Schrödinger equation

Lv Zhong-Quan (吕忠全)a b c, Zhang Lu-Ming (张鲁明)a, Wang Yu-Shun (王雨顺)c
a College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China;
b College of Science, Nanjing Forestry University, Nanjing 210037, China;
c School of Mathematical Sciences, Nanjing Normal University, Nanjing 210097, China
Abstract  In this paper, we derive a new method for a nonlinear Schrödinger system by using the square of the first-order Fourier spectral differentiation matrix D1 instead of the traditional second-order Fourier spectral differentiation matrix D2 to approximate the second derivative. We prove that the proposed method preserves the charge and energy conservation laws exactly. A deduction argument is used to prove that the numerical solution is second-order convergent to the exact solutions in ‖·‖2 norm. Some numerical results are reported to illustrate the efficiency of the new scheme in preserving the charge and energy conservation laws.
Keywords:  Fourier pseudospectral method      Schrödinger equation      conservation law      convergence  
Received:  21 April 2014      Revised:  18 August 2014      Accepted manuscript online: 
PACS:  02.60.Cb (Numerical simulation; solution of equations)  
  02.70.Bf (Finite-difference methods)  
  02.70.Jn (Collocation methods)  
  02.70.Hm (Spectral methods)  
Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 11271195, 41231173, and 11201169), the Postdoctoral Foundation of Jiangsu Province of China (Grant No. 1301030B), the Open Fund Project of Jiangsu Key Laboratory for NSLSCS (Grant No. 201301), and the Fund Project for Highly Educated Talents of Nanjing Forestry University (Grant No. GXL201320).
Corresponding Authors:  Lv Zhong-Quan     E-mail:  zhqlv@njfu.edu.cn

Cite this article: 

Lv Zhong-Quan (吕忠全), Zhang Lu-Ming (张鲁明), Wang Yu-Shun (王雨顺) A conservative Fourier pseudospectral algorithm for the nonlinear Schrödinger equation 2014 Chin. Phys. B 23 120203

[1] Menyuk C R 1998 J. Opt. Soc. Am. B: Opt. Phys. 5 392
[2] Wadati M, Izuka T and Hisakado M 1992 J. Phys. Soc. Jpn. 61 2241
[3] Feng K and Qin M Z 2010 Symplectic Geometric Algorithms for Hamiltonian Systems (Hangzhou: Science and Technology Press) p. 19 (in Chinese)
[4] Qin M Z and Wang Y S 2011 Structure-Preserving Algorithm for Partial Differential Equation (Hangzhou: Zhejiang Science and Technology Press) p. 420 (in Chinese)
[5] Cai J X and Wang Y S 2013 Chin. Phys. B 22 060207
[6] Akrivis G D 1993 IMA. J. Numer. Anal. 13 115
[7] Lv Z Q, Wang Y S and Song Y Z 2013 Chin. Phys. Lett. 30 030201
[8] Dehghan M and Taleei A 2010 Comput. Phys. Comm. 181 43
[9] Chang Q, Jia E and Sun W 1999 J. Comput. Phys. 148 397
[10] Wang T C and Guo B L 2011 Sci. China: Ser. A Math. 41 207
[11] Wang Y S, Li Q H and Song Y Z 2008 Chin. Phys. Lett. 25 1538
[12] Wang Y S and Li S T 2010 Int. J. Comput. Math. 87 775
[13] Cooley J W and Tukey J W 1965 Math. Comput. 19 297
[14] Canuto C, Hussani M and Quarteroni A 1988 Spectral Methods in Fluid Dynamics (New York: Springer-Verlag) p. 275
[15] Canuto C and Quarteroni A 1982 Math. Comput. 38 67
[16] Zhou Y L 1990 Application of Discrete Functional Analysis to the Diserence Methods (Beijing: International Academic Publishers)
[17] Chen J B and Qin M Z 2001 Electr. Trans. Numer. Anal. 12 193
[1] Meshfree-based physics-informed neural networks for the unsteady Oseen equations
Keyi Peng(彭珂依), Jing Yue(岳靖), Wen Zhang(张文), and Jian Li(李剑). Chin. Phys. B, 2023, 32(4): 040208.
[2] Phase-coherence dynamics of frequency-comb emission via high-order harmonic generation in few-cycle pulse trains
Chang-Tong Liang(梁畅通), Jing-Jing Zhang(张晶晶), and Peng-Cheng Li(李鹏程). Chin. Phys. B, 2023, 32(3): 033201.
[3] All-optical switches based on three-soliton inelastic interaction and its application in optical communication systems
Shubin Wang(王树斌), Xin Zhang(张鑫), Guoli Ma(马国利), and Daiyin Zhu(朱岱寅). Chin. Phys. B, 2023, 32(3): 030506.
[4] Quantitative analysis of soliton interactions based on the exact solutions of the nonlinear Schrödinger equation
Xuefeng Zhang(张雪峰), Tao Xu(许韬), Min Li(李敏), and Yue Meng(孟悦). Chin. Phys. B, 2023, 32(1): 010505.
[5] Data-driven parity-time-symmetric vector rogue wave solutions of multi-component nonlinear Schrödinger equation
Li-Jun Chang(常莉君), Yi-Fan Mo(莫一凡), Li-Ming Ling(凌黎明), and De-Lu Zeng(曾德炉). Chin. Phys. B, 2022, 31(6): 060201.
[6] Exact solutions of the Schrödinger equation for a class of hyperbolic potential well
Xiao-Hua Wang(王晓华), Chang-Yuan Chen(陈昌远), Yuan You(尤源), Fa-Lin Lu(陆法林), Dong-Sheng Sun(孙东升), and Shi-Hai Dong(董世海). Chin. Phys. B, 2022, 31(4): 040301.
[7] Closed form soliton solutions of three nonlinear fractional models through proposed improved Kudryashov method
Zillur Rahman, M Zulfikar Ali, and Harun-Or Roshid. Chin. Phys. B, 2021, 30(5): 050202.
[8] Collapse arrest in the space-fractional Schrödinger equation with an optical lattice
Manna Chen(陈曼娜), Hongcheng Wang(王红成), Hai Ye(叶海), Xiaoyuan Huang(黄晓园), Ye Liu(刘晔), Sumei Hu(胡素梅), and Wei Hu(胡巍). Chin. Phys. B, 2021, 30(10): 104206.
[9] Variation of electron density in spectral broadening process in solid thin plates at 400 nm
Si-Yuan Xu(许思源), Yi-Tan Gao(高亦谈), Xiao-Xian Zhu(朱孝先), Kun Zhao(赵昆), Jiang-Feng Zhu(朱江峰), and Zhi-Yi Wei(魏志义). Chin. Phys. B, 2021, 30(10): 104205.
[10] A meshless algorithm with the improved moving least square approximation for nonlinear improved Boussinesq equation
Yu Tan(谭渝) and Xiao-Lin Li(李小林). Chin. Phys. B, 2021, 30(1): 010201.
[11] Two integrable generalizations of WKI and FL equations: Positive and negative flows, and conservation laws
Xian-Guo Geng(耿献国), Fei-Ying Guo(郭飞英), Yun-Yun Zhai(翟云云). Chin. Phys. B, 2020, 29(5): 050201.
[12] An extension of integrable equations related to AKNS and WKI spectral problems and their reductions
Xian-Guo Geng(耿献国), Yun-Yun Zhai(翟云云). Chin. Phys. B, 2018, 27(4): 040201.
[13] A local energy-preserving scheme for Zakharov system
Qi Hong(洪旗), Jia-ling Wang(汪佳玲), Yu-Shun Wang(王雨顺). Chin. Phys. B, 2018, 27(2): 020202.
[14] Truncated series solutions to the (2+1)-dimensional perturbed Boussinesq equation by using the approximate symmetry method
Xiao-Yu Jiao(焦小玉). Chin. Phys. B, 2018, 27(10): 100202.
[15] A novel stable value iteration-based approximate dynamic programming algorithm for discrete-time nonlinear systems
Yan-Hua Qu(曲延华), An-Na Wang(王安娜), Sheng Lin(林盛). Chin. Phys. B, 2018, 27(1): 010203.
No Suggested Reading articles found!