Nonlinear Dynamics

, Volume 95, Issue 2, pp 1687–1692 | Cite as

Multiple-soliton solutions and lumps of a (3+1)-dimensional generalized KP equation

  • Jianping Yu
  • Fudong Wang
  • Wenxiu Ma
  • Yongli SunEmail author
  • Chaudry Masood Khalique
Original Paper


In this paper, we study a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation, which is physically meaningful. Applying the simplified Hirota’s method, we derive multiple-soliton solutions and lumps for this new model, where the coefficients of spatial variables are not constrained by any conditions. But the phase and the new model are dependent on all these coefficients. Moreover, this new model passes the Painlevé integrability test.


Simplified Hirota’s method Multiple-soliton solution Lumps Painlevé test 



All the authors deeply appreciate all the anonymous reviewers for their helpful and constructive suggestions, which can help improve this paper further. This work is supported by the National Natural Science Foundation of China (Nos. 11101029, 11271362 and 11375030), the Fundamental Research Funds for the Central Universities(No. 610806), Beijing City Board of Education Science and Technology Key Project (No. KZ201511232034), Beijing Nova program (No. Z131109000413029) and Beijing Finance Funds of Natural Science Program for Excellent Talents (No. 2014000026833ZK19).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Hirota, R.: The Direct Method in Soliton Theory. Cambridge University Press, Cambridge (2004)CrossRefzbMATHGoogle Scholar
  2. 2.
    Ma, W.X.: Generalized bilinear differential equations. Stud. Nonlinear Sci. 2, 140–144 (2011)Google Scholar
  3. 3.
    Ma, W.X.: Bilinear equations and resonant solutions characterized by Bell polynomials. Rep. Math. Phys. 72, 41–56 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ma, W.X., Abdeljabbar, A.: A bilinear Bäcklund transformation of a (3+1)-dimensional generalized KP equation. Appl. Math. Lett. 12, 1500–1504 (2012)CrossRefzbMATHGoogle Scholar
  5. 5.
    Wang, D.S., Zhang, H.Q.: Auto-Bäcklund transformation and new exact solutions of the (2+1)-dimensional Nizhniks-Novikovs-Veselov equation. Int. J. Mod. Phys. C 16, 393 (2005)CrossRefzbMATHGoogle Scholar
  6. 6.
    Lü, X., Ma, W.X., Khalique, C.M.: A direct bilinear Bäcklund transformation of a (2+1)-dimensional Korteweg-de Vries-like model. Appl. Math. Lett. 50, 37–42 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Matveev, V.B., Salle, M.A.: Darboux Transformations and Solitons. Springer, Berlin (1991)CrossRefzbMATHGoogle Scholar
  8. 8.
    Lü, X., Ma, W.X., Yu, J., Lin, F.H., Khalique, C.M.: Envelope bright- and dark-soliton solutions for the Gerdjikov-Ivanov model. Nonlinear Dyn. 82, 1211–1220 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Liu, H., Geng, X.G.: An integrable extension of TD hierarchy and generalized bi-Hamiltonian structures. Mod. Phys. Lett. B 29, 1550116 (2015)MathSciNetGoogle Scholar
  10. 10.
    Ablowitz, M.J., Kaup, D.J., Newell, A.C., Segur, H.: The inverse scattering transform-Fourier analysis for nonlinear problems. Stud. Appl. Math. 53, 249–315 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Baldwin, D., Hereman, W.: Symbolic software for the Painlevé test of nonlinear ordinary and partial differential equations. J. Nonlinear Math. Phys. 13(1), 90–110 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Wazwaz, A.M., Xu, G.Q.: Modified Kadomtsev-Petviashvili equation in (3+1) dimensions: multiple front-wave solutions. Commun. Theor. Phys. 63, 727–730 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Xu, G.Q.: The integrability for a generalized seventh-order KdV equation: Painlev’e property, soliton solutions. Lax pairs and conservation laws. Phys. Scr. 89, 125201 (2014)CrossRefGoogle Scholar
  14. 14.
    Biswas, A., Triki, H., Labidi, M.: Bright and dark solitons of the Rosenau–Kawahara equation with power law nonlinearity. Phys. Wave Phenom. 19(1), 24–29 (2011)CrossRefGoogle Scholar
  15. 15.
    Lü, X., Ma, W.X., Yu, J., Khalique, C.M.: Solitary waves with the Madelung fluid description: a generalized derivative nonlinear Schrdinger equation. Commun. Nonlinear Sci. Numer. Simul. 31, 40–46 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Wazwaz, A.M.: Multiple-soliton solutions for a (3+1)dimensional generalized KP equation. Commun. Nonlinear Sci. Numer. Simul. 17, 491–495 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ma, W.X.: Lump solutions to the Kadomtsev-Petviashvili equation. Phys. Lett. A 379, 1975–1978 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ma, W.X.: Lump-type solutions to the (3+1)-dimensional Jimbo–Miwa equation. JNSNS 17(7–8), 355–359 (2016)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Ma, W.X.: Abundant lump solutions and their interactions of (3+1)-dimensional linear PDEs. JGP 133, 10–16 (2018)Google Scholar
  20. 20.
    Ma, W.X., Zhou, Y.: Lump solutions to nonlinear partial differential equations via Hirota bilinear forms. JDE 264, 2633–2659 (2018). (in general dimensions)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Ma, W.X., Zhou, Y.: Lump-type solutions to nonlinear differential equations derived from generalized bilinear equations. Int. J. Modern Phys. B (2016)
  22. 22.
    Ma, W.X., Qin, Z.Y., Lü, X.: Lump solutions to dimensionally reduced p-gKP and p-gBKP equations. Nonlinear Dyn.
  23. 23.
    Kadomtsev, B.B., Petviashvili, V.I.: On the stability of solitary waves in weakly dispersive media. Sov. Phys. Dokl. 15, 539–541 (1970)zbMATHGoogle Scholar
  24. 24.
    Wazwaz, A.M.: Multi-front waves for extended form of modified Kadomtsev-Petviashvili equations. Appl. Math. Mech. 32(7), 875–880 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ma, W.X., Xia, T.: Pfaffianized systems for a generalized Kadomtsev-Petviashvili equation. Phys. Scr. 87, 055003 (2013)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Applied MathematicsUniversity of Science and Technology BeijingBeijingChina
  2. 2.Department of Mathematics and StatisticsUniversity of South FloridaTampaUSA
  3. 3.Department of MathematicsBeijing University of Chemical TechnologyBeijingChina
  4. 4.International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical SciencesNorth-West UniversityMmabathoSouth Africa

Personalised recommendations