Jump to ContentJump to Main Navigation
Show Summary Details
More options …

International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

Editorial Board: Armbruster, Dieter / Chen, Xi / Bessaih, Hakima / Chou, Tom / Grauer, Rainer / Marzocchella, Antonio / Rangarajan, Govindan / Trivisa, Konstantina / Weikard, Rudi

IMPACT FACTOR 2017: 1.162

CiteScore 2017: 1.41

SCImago Journal Rank (SJR) 2017: 0.382
Source Normalized Impact per Paper (SNIP) 2017: 0.636

Mathematical Citation Quotient (MCQ) 2017: 0.12

See all formats and pricing
More options …
Volume 19, Issue 3-4


Numerical Methods for the Derivative Nonlinear Schrödinger Equation

Shu-Cun Li / Xiang-Gui Li
  • Corresponding author
  • School of Applied Science,Beijing Information Science and Technology University,Beijing,100192 P.R. China
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Fang-Yuan Shi
Published Online: 2018-03-31 | DOI: https://doi.org/10.1515/ijnsns-2016-0184


In this work, a second-order accuracy in both space and time Crank–Nicolson (C-N)-type scheme, a fourth-order accuracy in space and second-order accuracy in time compact scheme and a sixth-order accuracy in space and second-order accuracy in time compact scheme are proposed for the derivative nonlinear Schrödinger equation. The C-N-type scheme is tested to satisfy the conservation of discrete mass. For the two compact schemes, the iterative algorithm and the Thomas algorithm in block matrix form are adopted to enhance the computational efficiency. Numerical experiment is given to test the mass conservation for the C-N-type scheme as well as the accuracy order of the three schemes. In addition, the numerical simulation of binary collision and the influence on the solitary solution by adding a small random perturbation to the initial condition are also discussed.

Keywords: derivative nonlinear Schrödinger equation; compact methods; binary collision; random perturbation

MSC 2010: 65M06; 35L05; 81Q05; 81-08


  • [1]

    Chen H. H., Y. Lee C. and Liu C. S., Integrability of nonlinear Hamiltonian systems by inverse scattering method, Phys. Scripta 20 (1979), 490-492.CrossrefGoogle Scholar

  • [2]

    E. Mjø husa, A note on the modulational instability of long Alfvé waves parallel to the magnetic field, J. Plasma Phys. 19 (1978), 437-447.Crossref

  • [3]

    Guo B. L., Ling L. M. and Liu Q. P., High-order solutions and generalized Darboux transformations of derivative nonlinear Schrödinger equations, Stud. Appl. Math. 130 (2013), 317-344.Crossref

  • [4]

    Kaup D. J. and Newell A. C., An exact solution for a derivative nonlinear Schrödinger equation, J. Math. Phys. 19 (1978), 708-801.

  • [5]

    Liu,B Y. X.. F. Yang and Cai H., Soliton solutions of DNLS equation found by IST anew and its verification in Marchenko formalism, Theor Int. J.. Phys. 45 (2006), 1836-1845.Crossref

  • [6]

    Nakamura A. and Chen H. H., Multi-soliton solution of a derivative non-linear Schrödinger equation, J. Phys. Soc. Jpn. 49 (1980), 813-816.Crossref

  • [7]

    Hirota R., The direct method in soliton theory, Press Cambridge University, York New, 2004.Google Scholar

  • [8]

    Zhang Y. S., Guo L. J., He J. S. and Zhou Z. X., Darboux transformation of the second-type derivative nonlinear Schrödinger equation, Lett. Math. Phys. 105 (2015), 853-891.Crossref

  • [9]

    Tsuchida T. and Wadati M., New integrable systems of derivative nonlinear Schrödinger equations with multiple components, Phys. Lett. A 257 (1999), 53-64.CrossrefGoogle Scholar

  • [10]

    Lai W. C. and Chow K. W., Special derivative nonlinear Schrödinger (DNLS) systems exhibiting 2-soliton solutions, Chaos Soliton. Fract. 11 (2000), 2055-2066.Crossref

  • [11]

    Basu-Mallick B. and Bhattacharyya T., Jost solutions and quantum conserved quantities of an integrable derivative nonlinear Schrödinger model, Nucl. Phys. B 668 (2003), 415-446.Crossref

  • [12]

    Li S. C., X. G. Li J. J. Cao and W. B. Li, High-order numerical method for the derivative nonlinear Schrödinger equation, Int. J. Model. Simul. Sci. Comput. 8 (2017), article 1750017.

  • [13]

    Chu P. C. and Fan C., A three-point combined compact difference scheme, J. Comput. Phys. 140 (1998), 370-399.CrossrefGoogle Scholar

  • [14]

    Lele S. K., Compact finite difference schemes with spectral-like resolution, J. Comput. Phys. 103 (1992), 16-42.CrossrefGoogle Scholar

  • [15]

    Bao W. Z. and Cai Y. Y., Ground states and dynamics of spin-orbit-coupled Bose-Einstein condensates, SIAM J. Appl. Math. 75 (2015), 492-517.Web of ScienceCrossrefGoogle Scholar

  • [16]

    Li X. G., J. Zhu, R. Zhang P. and Cao S. S., A combined discontinuous Galerkin method for the dipolar Bose-Einstein condensation, J. Comput. Phys. 275 (2014), 363-376.CrossrefWeb of ScienceGoogle Scholar

  • [17]

    Hua D. Y., Li X. G. and Zhu J., A mass conserved splitting method for the nonlinear Schrödinger equation, Adv. Differ. Equ. 2012 (2012), 85.Crossref

  • [18]

    Hua D. Y. and Li X. G., The finite element method for computing the ground states of the dipolar Bose-Einstein condensates, Appl. Math. Comput. 234 (2014), 214-222.Web of Science

  • [19]

    Wang T. C. and Guo B. L., Xu Q. B., Fourth-order compact and energy conservative difference schemes for the nonlinear Schrödinger equation in two dimensions, Comput J.. Phys. 243 (2013), 382-399.Crossref

  • [20]

    Wen X. Y., Yang Y. Q. and Yan Z. Y., Generalized perturbation (n,M)-fold Darboux transformations and multi-rogue-wave structures for the modified self-steepening nonlinear Schrödinger equation, Phys. Rev. E 92 (2015), article 012917.

About the article

Published Online: 2018-03-31

Published in Print: 2018-06-26

This work is supported by National Natural Science Foundation of China (No. 11671044), the Science Challenge Project (No. TZ2016001) and Beijing Municipal Commission of Education (No. PXM2017\_014224\_000020).

Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, Volume 19, Issue 3-4, Pages 239–249, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339, DOI: https://doi.org/10.1515/ijnsns-2016-0184.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Shu-Cun Li and Xiang-Gui Li
Computational and Applied Mathematics, 2018

Comments (0)

Please log in or register to comment.
Log in