, 90:45 | Cite as

Bäcklund transformation and soliton solutions in terms of the Wronskian for the Kadomtsev–Petviashvili-based system in fluid dynamics

  • Zhong Du
  • Bo Tian
  • Xi-Yang Xie
  • Jun Chai
  • Xiao-Yu Wu


In this paper, investigation is made on a Kadomtsev–Petviashvili-based system, which can be seen in fluid dynamics, biology and plasma physics. Based on the Hirota method, bilinear form and Bäcklund transformation (BT) are derived. N-soliton solutions in terms of the Wronskian are constructed, and it can be verified that the N-soliton solutions in terms of the Wronskian satisfy the bilinear form and Bäcklund transformation. Through the N-soliton solutions in terms of the Wronskian, we graphically obtain the kink-dark-like solitons and parallel solitons, which keep their shapes and velocities unchanged during the propagation.


Fluid dynamics Kadomtsev–Petviashvili-based system Wronskian Hirota method soliton solutions Bäcklund transformation 


05.45.Yv 47.35.Fg 02.30.Jr 



This work has been supported by the National Natural Science Foundation of China under Grant Nos 11772017, 11272023 and 11471050, by the Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), China (IPOC: 2017ZZ05) and by the Fundamental Research Funds for the Central Universities of China under Grant No. 2011BUPTYB02.


  1. 1.
    R K Bullough and P J Caudrey, Solitons (Springer, Berlin, 1980)CrossRefzbMATHGoogle Scholar
  2. 2.
    M P Barnett, J F Capitani, J Von Zur Gathen and J Gerhard, Int. J. Quant. Chem. 100, 80 (2004)CrossRefGoogle Scholar
  3. 3.
    G Das and J Sarma, Phys. Plasmas 6, 4394 (1999)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    M J Ablowitz and P A Clarkson, Solitons, nonlinear evolution equations and inverse scattering (Cambridge University Press, Cambridge, 1992)zbMATHGoogle Scholar
  5. 5.
    M Iwao and R Hirota, J. Phys. Soc. Jpn 66, 577 (1997)ADSCrossRefGoogle Scholar
  6. 6.
    M Wadati, J. Phys. Soc. Jpn 32, 1681 (1972)ADSCrossRefGoogle Scholar
  7. 7.
    T Xu, J Li, H Q Zhang, Y X Zhang, Z Z Yao and B Tian, Phys. Lett. A 369, 458 (2007) M Li, T Xu and D X Meng, J. Phys. Soc. Jpn 85(12), 124001 (2016)Google Scholar
  8. 8.
    F Caruello and M Tabor, Physica D 39, 77 (1989)Google Scholar
  9. 9.
    X Y Tang and S Y Lou, J. Math. Phys. 44, 4000 (2003)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    M Wadati, J. Phys. Soc. Jpn 38, 673 (1975)ADSCrossRefGoogle Scholar
  11. 11.
    R Hirota, Direct method in soliton theory (Springer, Berlin, 1980)CrossRefGoogle Scholar
  12. 12.
    S Zhang, C Tian and W Y Qian, Pramana – J. Phys. 86, 1259 (2016)ADSCrossRefGoogle Scholar
  13. 13.
    V B Matveev and M A Salle, Darboux transformation and soliton (Springer, Berlin,1991)CrossRefzbMATHGoogle Scholar
  14. 14.
    M Li and T Xu, Phys. Rev. E 91, 033202 (2015)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    T Xu, M Li and L Li, EPL 109, 30006 (2015)ADSCrossRefGoogle Scholar
  16. 16.
    T Xu and Y Zhang, Nonlinear Dyn. 73, 485 (2013)CrossRefGoogle Scholar
  17. 17.
    M Li, T Xu, L Wang and F H Qi, Appl. Math. Lett. 60, 8 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    T Xu, H J Li, H J Zhang, M Li and S Lan, Appl. Math. Lett. 63, 88 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    T Xu, C J Liu, F H Qi, C X Li and D X Meng, J. Nonlin. Math. Phys. 24, 116 (2017)CrossRefGoogle Scholar
  20. 20.
    J J Nimmo and N C Freeman, Phys. Lett. A 95, 4 (1983)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    N C Freeman and J J Nimmo, Phys. Lett. A 95, 1 (1983)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    B Qin, B Tian, L C Liu, X H Meng and W J Liu, Commun. Theor. Phys. 54, 1059 (2010)ADSCrossRefGoogle Scholar
  23. 23.
    B Kadomtsev and V Petviashvili, Sov. Phys. Dokl. 15, 539 (1970)ADSGoogle Scholar
  24. 24.
    Y Nakamura, H Bailung, K E Lonngren, Phys. Plasmas 6, 3466 (1999)ADSCrossRefGoogle Scholar
  25. 25.
    S Y Lou and X B Hu, J. Math. Phys. 38, 6401 (1997)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    C L Bai and H Zhao, Phys. Scr. 73, 429 (2006)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    C Q Dai and C Y Liu, Nonlinear Anal. 17, 271 (2012)Google Scholar
  28. 28.
    J Q Mei, D S Li and H Q Zhang, Chaos Solitons Fractals 22, 669 (2004)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    F Galas, J. Phys. A 25, 981 (1995)MathSciNetCrossRefGoogle Scholar
  30. 30.
    G Biondini and S Chakravarty, J. Math. Phys. 47, 033514 (2006)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    S Chakravarty and Y Kodama, J. Phys. A 41, 275209 (2008)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    M S Khatun, M F Hoque and M A Rahman, Pramana – J. Phys. 88, 86 (2017)ADSCrossRefGoogle Scholar

Copyright information

© Indian Academy of Sciences 2018

Authors and Affiliations

  • Zhong Du
    • 1
  • Bo Tian
    • 1
  • Xi-Yang Xie
    • 1
  • Jun Chai
    • 1
  • Xiao-Yu Wu
    • 1
  1. 1.State Key Laboratory of Information Photonics and Optical Communications, and School of ScienceBeijing University of Posts and TelecommunicationsBeijingChina

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