Nonlinear Dynamics

, Volume 92, Issue 2, pp 487–497 | Cite as

Interaction solutions for a reduced extended \(\mathbf{(3}\varvec{+}{} \mathbf{1)}\)-dimensional Jimbo–Miwa equation

  • Yun-Hu Wang
  • Hui Wang
  • Huan-He Dong
  • Hong-Sheng Zhang
  • Chaolu Temuer
Original Paper


In this paper, the exact solutions of a reduced extended \((3+1)\)-dimensional Jimbo–Miwa equation are investigated with the help of its bilinear representation and symbolic computation. Firstly, a kind of bright–dark lump wave solutions is directly obtained by taking the solution F in bilinear equation as a quadratic function. Furthermore, the interaction solutions between one lump wave and one stripe wave are also presented by taking F as a combination of quadratic function and exponential function. Finally, by taking F as a combination of quadratic function and hyperbolic cosine function, the rogue wave which aroused by the interaction between lump soliton and a pair of stripe solitons are obtained. The dynamic properties of the above three kinds of exact solutions are displayed vividly by figures.


Interaction solutions Lump wave solution Rogue wave solution Jimbo–Miwa equation 



The authors would like to express their sincere thanks to the referees for their valuable comments. This work is supported by the National Natural Science Foundation of China (Nos. 11405103, 11571008, 51679132, 11601321 and 11526137), and the Shanghai Science and Technology Committee (No. 17040501600).

Compliance with ethical standards

Conflict of interest

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


  1. 1.
    Jimbo, M., Miwa, T.: Solitons and infinite dimensional Lie algebras. Publ. RIMS Kyoto Univ. 19, 943 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Wazwaz, A.M.: Multiple-soliton solutions for extended \((3+1)\)-dimensional Jimbo–Miwa equations. Appl. Math. Lett. 64, 21 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Sun, H.Q., Chen, A.H.: Lump and lump–kink solutions of the \((3+1)\)-dimensional Jimbo–Miwa and two extended Jimbo–Miwa equations. Appl. Math. Lett. 68, 55 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Hirota, R.: The Direct Method in Soliton Theory. Cambridge University Press, New York (2004)CrossRefzbMATHGoogle Scholar
  5. 5.
    Ma, W.X.: Lump solutions to the Kadomtsev–Petviashvili equation. Phys. Lett. A 379, 1975 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Satsuma, J., Ablowitz, M.J.: Two-dimensional lumps in nonlinear dispersive systems. J. Math. Phys. 20, 1496 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kaup, D.J.: The lump solutions and the Bäklund transformation for the three-dimensional three-wave resonant interaction. J. Math. Phys. 22, 1176 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gilson, C.R., Nimmo, J.J.C.: Lump solutions of the BKP equation. Phys. Lett. A 147, 472 (1990)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Imai, K.: Dromion and lump solutions of the Ishimori-I equation. Prog. Theor. Phys. 98, 1013 (1997)CrossRefGoogle Scholar
  10. 10.
    Lü, X., Ma, W.X.: Study of lump dynamics based on a dimensionally reduced Hirota bilinear equation. Nonlinear Dyn. 85, 1217 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Zhang, X.E., Chen, Y., Tang, X.Y.: Rogue wave and a pair of resonance stripe solitons to a reduced generalized \((3+1)\)-dimensional KP equation. arXiv:1610.09507 (2016)
  12. 12.
    Zhang, X.E., Chen, Y.: Rogue wave and a pair of resonance stripe solitons to a reduced \((3+1)\)-dimensional Jimbo–Miwa equation. Commun. Nonlinear Sci. Numer. Simul. 52, 24 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Zhao, H.Q., Ma, W.X.: Mixed lump–kink solutions to the KP equation. Comput. Math. Appl. 74, 1399 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Zhao, Z.L., Chen, Y., Han, B.: Lump soliton, mixed lump stripe and periodic lump solutions of a \((2+1)\)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation. Mod. Phys. Lett. B 31, 1750157 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Wang, C.J.: Spatiotemporal deformation of lump solution to \((2+1)\)-dimensional KdV equation. Nonlinear Dyn. 84, 697 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Zhang, Y., Dong, H.H., Zhang, X.E., Yang, H.W.: Rational solutions and lump solutions to the generalized \((3 + 1)\)-dimensional shallow water-like equation. Comput. Math. Appl. 73, 246 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Zhang, H.Q., Ma, W.X.: Lump solutions to the \((2+1)\)-dimensional Sawada–Kotera equation. Nonlinear Dyn. 87, 2305 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Cheng, L., Zhang, Y.: Lump-type solutions for the \((4+1)\)-dimensional Fokas equation via symbolic computations. Mod. Phys. Lett. B 31, 1750224 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Gilson, C., Lambert, F., Nimmo, J., Willox, R.: On the combinatorics of the Hirota \(D\)-operators. Proc. R. Soc. Lond. A 452, 223 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    EG, Fan: The integrability of nonisospectral and variable-coefficient KdV equation with binary Bell polynomials. Phys. Lett. A 375, 493 (2011)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Ma, W.X.: Bilinear equations, Bell polynomials and linear superposition principle. J. Phys. Conf. Ser. 411, 012021 (2013)CrossRefGoogle Scholar
  22. 22.
    Wang, Y.H., Chen, Y.: Integrability of the modified generalised Vakhnenko equation. J. Math. Phys. 53, 123504 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Miao, Q., Wang, Y.H., Chen, Y., Yang, Y.Q.: PDEBellII: a Maple package for finding bilinear forms, bilinear Bäcklund transformations, Lax pairs and conservation laws of the KdV-type equations. Comput. Phys. Commun. 185, 357 (2013)CrossRefzbMATHGoogle Scholar
  24. 24.
    Wang, Y.H., Wang, H., Chaolu, T.: Lax pair, conservation laws, and multi-shock wave solutions of the DJKM equation with Bell polynomials and symbolic computation. Nonlinear Dyn. 78, 1101 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Wang, S., Tang, X.Y., Lou, S.Y.: Soliton fission and fusion: Burgers equation and Sharma–Tasso–Olver equation. Chaos Solitons Fractals 21, 231 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Or-Roshid, H., Rashidi, M.M.: Multi-soliton fusion phenomenon of Burgers equation and fission, fusion phenomenon of Sharma–Tasso–Olver equation. J. Ocean Eng. Sci. 2, 120 (2017)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Art and SciencesShanghai Maritime UniversityShanghaiPeople’s Republic of China
  2. 2.Department of MathematicsShanghai UniversityShanghaiPeople’s Republic of China
  3. 3.College of Mathematics and Systems ScienceShandong University of Science and TechnologyQingdaoPeople’s Republic of China
  4. 4.College of Ocean Science and EngineeringShanghai Maritime UniversityShanghaiPeople’s Republic of China

Personalised recommendations