, 92:10 | Cite as

Dynamics of dark multisoliton and rational solutions for three nonlinear differential-difference equations

  • Haotian Wang
  • Xiao-Yong WenEmail author


In this paper, three nonlinear differential-difference equations (NDDEs) from the same hierarchy are investigated using the generalised perturbation \((n,N-n)\)-fold Darboux transformation (DT) technique. The dark multisoliton solutions in terms of determinants for three equations are obtained by means of the discrete N-fold DT. Propagation and elastic interaction structures of such soliton solutions are shown graphically. The details of their evolutions are studied through numerical simulations. Numerical results show the accuracy of our numerical scheme and the stable evolutions of such dark multisolitons without a noise. We find that the solutions of lower-order NDDEs in the same hierarchy are more robust against a small noise than their corresponding higher-order NDDEs. The discrete generalised perturbation \((1,N-1)\)-fold DT is used to derive some discrete rational and semirational solutions of the first equation, and a few mathematical features are also discussed. Results in this paper might be helpful for understanding some physical phenomena.


Nonlinear differential-difference equations generalised perturbation (n, Nn)-fold Darboux transformation dark multisoliton solutions rational and semirational solutions numerical simulations 


02.30.Ik 05.45.Yv 02.60.Cb 04.20.Jb 



This work was partially supported by Qin Xin Talents Cultivation Program of Beijing Information Science and Technology University (QXTCP-B201704), the NSFC under Grant Nos 11375030 and 61178091, the Beijing Natural Science Foundation under Grant No. 1153004.


  1. 1.
    M J Ablowitz and P A Clarkson, Solitons, nonlinear evolution equations and inverse scattering (Cambridge University Press, Cambridge, 1991)CrossRefGoogle Scholar
  2. 2.
    A M Wazwaz, Pramana – J. Phys. 77, 233 (2011)ADSCrossRefGoogle Scholar
  3. 3.
    A M Wazwaz, Pramana – J. Phys. 87: 68 (2016)ADSCrossRefGoogle Scholar
  4. 4.
    Y Z Li and J G Liu, Pramana – J. Phys. 90: 71 (2018)Google Scholar
  5. 5.
    Y K Liu and B Li, Pramana – J. Phys. 88: 57 (2017)Google Scholar
  6. 6.
    Z Du, B Tian, X Y Xie, J Chai and X Y Wu, Pramana – J. Phys. 90: 45 (2018)Google Scholar
  7. 7.
    D W Zuo, H X Jia and D M Shan, Superlattices Microst. 101, 522 (2017)ADSCrossRefGoogle Scholar
  8. 8.
    D W Zuo and H X Jia, Optik 127, 11282 (2016)ADSCrossRefGoogle Scholar
  9. 9.
    D W Zuo, H X Mo and H P Zhou, Z. Naturforsch A 71, 305 (2016)ADSCrossRefGoogle Scholar
  10. 10.
    D W Zuo, Appl. Math. Lett. 79, 182 (2018)MathSciNetCrossRefGoogle Scholar
  11. 11.
    M Wadati, Prog. Theor. Phys. Suppl. 59, 36 (1977)ADSCrossRefGoogle Scholar
  12. 12.
    V Zakhov, S Musher and A Rubenshik, Sov. Phys. Lett. 19, 151 (1974)ADSGoogle Scholar
  13. 13.
    M Toda, Theory of nonlinear lattices (Springer, Berlin, 1989)CrossRefGoogle Scholar
  14. 14.
    D J Kaup, Math. Comput. Simul. 69, 322 (2005)CrossRefGoogle Scholar
  15. 15.
    R Hirota, J. Phys. Soc. Jpn. 35, 289 (1973)ADSCrossRefGoogle Scholar
  16. 16.
    N Liu and X Y Wen, Mod. Phys. Lett. B 32, 1850085 (2018)ADSCrossRefGoogle Scholar
  17. 17.
    F J Yu and S Feng, Math. Methods Appl. Sci. 40, 5515 (2017)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    F J Yu, Chaos 27, 023108 (2017)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    W X Ma and X X Xu, J. Phys. A 37, 1323 (2004)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    W X Ma, J. Phys. A  40, 15055 (2007)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    X Y Wen, Z Y Yan and B A Malomed, Chaos 26, 013105 (2016)MathSciNetCrossRefGoogle Scholar
  22. 22.
    X Y Wen and D S Wang, Wave Motion 79, 84 (2018)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Y F Zhang and W J Rui, Rep. Math. Phys. 78, 19 (2016)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Y F Zhang, X Z Zhang, Y Wang and J G Liu, Z. Naturforsch. A 72, 77 (2017)Google Scholar
  25. 25.
    Y F Zhang, X Z Zhang and H H Dong, Commun. Theor. Phys. 68, 755 (2017)ADSCrossRefGoogle Scholar
  26. 26.
    M Wadati, Prog. Theor. Phys. Suppl. 59, 36 (1976)ADSCrossRefGoogle Scholar
  27. 27.
    Y T Wu and X G Geng, J. Phys. A 31, 677 (1998)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    L N Trefethen, Spectral methods in MATLAB (SIAM, Philadelphia, 2000).CrossRefGoogle Scholar

Copyright information

© Indian Academy of Sciences 2018

Authors and Affiliations

  1. 1.School of Applied ScienceBeijing Information Science and Technology UniversityBeijingChina

Personalised recommendations