State transition of lump-type waves for the (2+1)-dimensional generalized KdV equation

  • Chuanjian WangEmail author
  • Hui Fang
  • Xiuxiu Tang
Original Paper


In this paper, we are mainly concerned with the (2+1)-dimensional generalized Korteweg–de Vries equation in fluid dynamics. Based on the translation transformation and Hirota bilinear method, we study the excitations of nonlinear lump-type waves on a constant background. A remarkable feature of these lump-type waves is that under some parameter conditions, these lump-type wave solutions can be converted into some amusing nonlinear wave structures, including the W-shaped solitary wave, double-peak solitary wave, parallel solitary wave, multi-peak solitary wave and periodic wave solutions. These results do not have an analog in the standard Kadomtsev–Petviashvili equation. The transition condition between the lump-type wave and other nonlinear wave solutions is presented. The dynamical behaviors of these nonlinear wave solutions are investigated analytically and illustrated graphically. Furthermore, the existence conditions for these nonlinear wave solutions are exhibited explicitly. Our results further enrich the nonlinear wave theories for the (2+1)-dimensional generalized Korteweg–de Vries equation.


(2+1)-Dimensional generalized Korteweg–de Vries equation Hirota bilinear method Lump-type wave State transition 



The authors would like to express their sincere thanks to the referees for their enthusiastic guidance and help. This work is supported by the National Natural Science Foundation of China (No. 11801240) and the Fund for Fostering Talents in Kunming University of Science and Technology (No. KKSY201707021).

Compliance with ethical standards

Conflicts of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.


  1. 1.
    Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge (1991)CrossRefzbMATHGoogle Scholar
  2. 2.
    Singh, N., Stepanyants, Y.: Obliquely propagating skew KP lumps. Wave Motion 64, 92–102 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Falcon, E., Laroche, C., Fauve, S.: Observation of depression solitary surface waves on a thin fluid layer. Phys. Rev. Lett. 89, 204501 (2002)CrossRefGoogle Scholar
  4. 4.
    Pelinovsky, D.E., Stepanyants, Y.A., Kivshar, Y.S.: Self-focusing of plane dark solitons in nonlinear defocusing media. Phys. Rev. E 51, 5016–5026 (1995)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Mironov, V.A., Smirnov, A.I., Smirnov, L.A.: Structure of vortex shedding past potential barriers moving in a Bose–Einstein condensate. J. Exp. Theor. Phys. 110, 877–889 (2010)CrossRefGoogle Scholar
  6. 6.
    Tauchert, T.R., Guzelsu, A.N.: An experimental study of dispersion of stress waves in a fiber-reinforced composite. Trans. ASME 39, 98–102 (1972)CrossRefGoogle Scholar
  7. 7.
    Zaharov, V.E.: Exact solutions in the problem of parametric interaction of three-dimensional wave packets. Dokl. Akad. Nauk SSSR 228, 1314–1316 (1976)MathSciNetGoogle Scholar
  8. 8.
    Ma, W.X.: Lump solutions to the Kadomtsev–Petviashvili equation. Phys. Lett. A 379, 1975–1978 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ma, W.X.: Lump-type solutions to the (3+1)-dimensional Jimbo–Miwa Equation. Int. J. Nonlinear Sci. Numer. 17, 355–359 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Yang, J.Y., Ma, W.X.: Abundant lump-type solutions of the Jimbo–Miwa equation in (3+1)-dimensions. Comput. Math. Appl. 73, 220–225 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ma, W.X., Qin, Z.Y., Lü, X.: Lump solutions to dimensionally reduced p-gKP and p-gBKP equations. Nonlinear Dyn. 84, 923–931 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Yang, J.Y., Ma, W.X.: Lump solutions to the BKP equation by symbolic computation. Int. J. Modern Phys. B 30, 1640028 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lü, X., Ma, W.X.: Study of lump dynamics based on a dimensionally reduced Hirota bilinear equation. Nonlinear Dyn. 85, 1217–1222 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Xu, Z.H., Chen, H.L., Dai, Z.D.: Rogue wave for the (2+1)-dimensional Kadomtsev–Petviashvili equation. Appl. Math. Lett. 37, 34–38 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Tan, W., Dai, Z.D.: Dynamics of kinky wave for (3+1)-dimensional potential Yu–Toda–Sasa–Fukuyama equation. Nonlinear Dyn. 85, 817–823 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Tan, W., Dai, Z.D.: Spatiotemporal dynamics of lump solution to the (1+1)-dimensional Benjamin–Ono equation. Nonlinear Dyn. 89, 2723–2728 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Lü, J.Q., Bilige, S., Chaolu, T.: The study of lump solution and interaction phenomenon to (2+1)-dimensional generalized fifth-order KdV equation. Nonlinear Dyn. 91, 1669–1676 (2018)CrossRefGoogle Scholar
  18. 18.
    Wang, C.J.: Spatiotemporal deformation of lump solution to (2+1)-dimensional KdV equation. Nonlinear Dyn. 84, 697–702 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Wang, C.J.: Lump solution and integrability for the associated Hirota bilinear equation. Nonlinear Dyn. 87, 2635–2642 (2017)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Osman, M.S., Machado, J.A.T.: New nonautonomous combined multi-wave solutions for (2+1)-dimensional variable coefficients KdV equation. Nonlinear Dyn. 93, 733–740 (2018)CrossRefzbMATHGoogle Scholar
  21. 21.
    Tang, Y.N., Tao, S.Q., Zhou, M.L., Guan, Q.: Interaction solutions between lump and other solitons of two classes of nonlinear evolution equations. Nonlinear Dyn. 89, 1–14 (2017)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Liu, J.G.: Interaction behaviors for the (2+1)-dimensional Sawada–Kotera equation. Nonlinear Dyn. 93, 741–747 (2018)CrossRefzbMATHGoogle Scholar
  23. 23.
    Guo, B.L., Ling, L.M., Liu, Q.P.: Nonlinear Schrödinger equation: generalized Darboux transformation and rogue wave solutions. Phys. Rev. E 85, 026607 (2012)CrossRefGoogle Scholar
  24. 24.
    Zhang, Y., Sun, Y.B., Xiang, W.: The rogue waves of the KP equation with self-consistent sources. Appl. Math. Comput. 263, 204–213 (2015)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Zhang, Y., Song, Y., Cheng, L., Ge, J.Y., Wei, W.W.: Exact solutions and Painlevé analysis of a new (2+1)-dimensional generalized KdV equation. Nonlinear Dyn. 68, 445–458 (2012)CrossRefzbMATHGoogle Scholar
  26. 26.
    Dai, Z.D., Liu, Z.J., Li, D.L.: Exact periodic solitary-wave solution for KdV equation. Chin. Phys. Lett. 25, 1531–1533 (2008)CrossRefGoogle Scholar
  27. 27.
    Ma, W.X., Zhou, R., Gao, L.: Exact one-periodic and two-periodic wave solutions to Hirota bilinear equations in (2+1) dimensions. Mod. Phys. Lett. A 24, 1677–1688 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Singh, M., Gupta, R.K.: Bäcklund transformations, Lax system, conservation laws and multi soliton solutions for Jimbo–Miwa equation with Bell polynomials. Commun. Nonlinear Sci. Numer. Simul. 37, 362–373 (2016)MathSciNetCrossRefGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsKunming University of Science and TechnologyKunmingChina

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