Nonlinear Dynamics

, Volume 87, Issue 4, pp 2635–2642 | Cite as

Lump solution and integrability for the associated Hirota bilinear equation

  • Chuanjian Wang
Original Paper


This paper studies lump solution and integrability for the associated Hirota bilinear equation. The integrability in the sense of Lax pair and the bilinear Bäcklund transformations is presented by the binary Bell polynomial method. The lump solution is derived when the period of complexiton solution goes to infinite. Conversely, complexiton solution can also be derived from the lump solution. Complexiton solution is a superposition structure of lump solutions. The dynamics of the lump solution are investigated and exhibited mathematically and graphically. These results further supplement and enrich the theories for the associated Hirota bilinear equation. It is hoped that these results might provide us with useful information on the dynamics of the relevant fields in nonlinear science.


Binary Bell polynomial Hirota’s bilinear form Bäcklund transformation Lax pair Complexiton solution Lump solution 



The author would like to express his sincere thanks to referees for their enthusiastic guidance and help. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11501266, 11661047) and the Fund for Fostering Talents in Kunming University of Science and Technology (No: KKSY201403049).


  1. 1.
    Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering, vol. 149. Cambridge University Press, Cambridge (1999)MATHGoogle Scholar
  2. 2.
    Jin, J.H.: Multiple solutions of the Kirchhoff-type problem in \(R^{N}\). Appl. Math. Nonlinear Sci. 1, 229–238 (2016)CrossRefGoogle Scholar
  3. 3.
    Dai, Z.D., Liu, J., Liu, Z.J.: Exact periodic kink-wave and degenerative soliton solutions for potential Kadomtsev–Petviashvili equation. Commun. Nonlinear Sci. Numer. Simul. 15, 2331–2336 (2010)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Wang, C.J.: Dynamic behavior of traveling waves for the Sharma–Tasso–Olver equation. Nonlinear Dyn. 85, 1119–1126 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Xu, Z., Chen, H., Jiang, M., Dai, Z., Chen, W.: Resonance and deflection of multi-soliton to the (2+1)-dimensional Kadomtsev–Petviashvili equation. Nonlinear Dyn. 78, 461–466 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ma, W.X., Zhu, Z.N.: Solving the (3+1)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm. Appl. Math. Comput. 218, 11871–11879 (2012)MathSciNetMATHGoogle Scholar
  7. 7.
    Singh, J., Kumar, D., Kıçıman, A.: Numerical solutions of nonlinear fractional partial differential equations arising in spatial diffusion of biological populations. Abstr. Appl. Anal. 2014, 535793 (2014). doi: 10.1155/2014/535793 MathSciNetGoogle Scholar
  8. 8.
    Kumar, D., Singh, J., Baleanu, D.: Numerical computation of a fractional model of differential-difference equation. J. Comput. Nonlinear Dyn. 11, 061004 (2016)CrossRefGoogle Scholar
  9. 9.
    Kumar, D., Singh, J., Kumar, S.: A fractional model of Navier–Stokes equation arising in unsteady flow of a viscous fluid. J. Assoc. Arab Univ. Basic Appl. Sci. 17, 14–19 (2015)Google Scholar
  10. 10.
    Singh, J., Kumar, D., Nieto, J.J.: A reliable algorithm for local fractional Tricomi equation arising in fractal transonic flow. Entropy 18, 206 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Vishwanath, B.A., Shankar Naik, B., Mahesh Kumar, N.: Multigrid method for the solution of EHL line contact with bio-based oils as lubricants. Appl. Math. Nonlinear Sci. 1, 359–368 (2016)Google Scholar
  12. 12.
    Imai, K.: Dromion and lump solutions of the Ishimori-I equation. Prog. Theor. Phys. 98, 1013–1023 (1997)CrossRefGoogle Scholar
  13. 13.
    Ma, W.X.: Lump solutions to the Kadomtsev–Petviashvili equation. Phys. Lett. A 379, 1975–1978 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Ma, W.X., Qin, Z.Y., L, X.: Lump solutions to dimensionally reduced p-gKP and p-gBKP equations. Nonlinear Dyn. 84, 923–931 (2016)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Lü, X., Ma, W.X.: Study of lump dynamics based on a dimensionally reduced Hirota bilinear equation. Nonlinear Dyn. 85, 1217–1222 (2016)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Bell, E.T.: Exponential polynomials. Ann. Math. 35, 258–277 (1934)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Ma, W.X.: Bilinear equations and resonant solutions characterized by Bell polynomials. Rep. Math. Phys. 72, 41–56 (2013)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Ma, W.X.: Trilinear equations, Bell polynomials, and resonant solutions. Front. Math. China 8, 1139–1156 (2013)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Lü, X., Tian, B., Qi, F.H.: Bell-polynomial construction of Bäcklund transformations with auxiliary independent variable for some soliton equations with one Tau-function. Nonlinear Anal. Real World Appl. 13, 1130–1138 (2012)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Singh, M., Gupta, R.K.: Bäcklund transformations, Lax system, conservation laws and multisoliton solutions for Jimbo–Miwa equation with Bell polynomials. Commun. Nonlinear Sci. Numer. Simul. 37, 362–373 (2016)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Ma, W.X.: Complexiton solutions to the Korteweg–de Vries equation. Phys. Lett. A 301, 35–44 (2002)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Ma, W.X.: Complexiton solutions to integrable equations. Nonlinear Anal. 63, e2461–e2471 (2005)CrossRefMATHGoogle Scholar
  23. 23.
    Chow, K.W., Wu, C.F.: The superposition of algebraic solitons for the modified Korteweg–de Vries equation. Commun. Nonlinear Sci. Numer. Simul. 19, 49–52 (2014)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Jeffrey, A., Zwillenger, D.: Table of Integrals, Series and Products. Academic Press, New York (2014)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.School of ScienceKunming University of Science and TechnologyKunmingChina

Personalised recommendations