Nonlinear Dynamics

, Volume 93, Issue 4, pp 2533–2541 | Cite as

M-lump and interactive solutions to a (3 \({+}\) 1)-dimensional nonlinear system

  • Yan Zhang
  • Yinping LiuEmail author
  • Xiaoyan Tang
Original Paper


This paper aims at computing M-lump solutions for the \((3+1)\)-dimensional nonlinear evolution equation. These solutions in all directions decline to an identical state obtained by employing the “long wave” limit with respect to the N-soliton solutions which are got by using the direct methods. Subsequently, we discuss the dynamic properties of the M-lump solutions which describe the multiple collisions of lumps. Based on the obtained lump solutions, the lump–kink solutions are also obtained. In addition, the periodic interactive solutions are given.


(3 \({+}\) 1)-dimensional nonlinear evolution equation Lump solution Lump–kink solution Interaction 



The work is supported by the National Natural Science Foundation of China (Nos. 11675055 and 11435005) and Shanghai Knowledge Service Platform for Trustworthy Internet of Things (No. ZF1213).

Compliance with ethical standards

Conflict of interest

The authors declare that there are no conflicts of interest between this manuscript and published articles mostly for technical terms, mathematical expressions and explanations on mathematical terms.


  1. 1.
    Manakov, S.V., Zakharov, V.E., Bordag, L.A., Its, A.R., Matveev, V.B.: Two-dimensional solitons of the Kadomtsev–Petviashvili equation and their interaction. Phys. Lett. A 63(3), 205–206 (1977)CrossRefGoogle Scholar
  2. 2.
    Krichever, I.M.: Rational solutions of the Kadomtsev–Petviashvili equation and integrable systems of n particles on a line. Funct. Anal. Appl. 12(1), 59–61 (1978)CrossRefzbMATHGoogle Scholar
  3. 3.
    Satsuma, J., Ablowitz, M.J.: Two-dimensional lumps in nonlinear dispersive systems. J. Math. Phys. 20(7), 1496–1503 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Villarroel, J., Ablowitz, M.J.: On the discrete spectrum of the nonstationary schrödinger equation and multipole lumps of the Kadomtsev–Petviashvili i equation. Commun. Math. Phys. 207(1), 1–42 (1999)CrossRefzbMATHGoogle Scholar
  5. 5.
    Imai, K.: Dromion and lump solutions of the Ishimori-i equation. Prog. Theor. Phys. 98(5), 1013–1023 (1997)CrossRefGoogle Scholar
  6. 6.
    Zhang, H.Q., Ma, W.X.: Lump solutions to the (\(2+1\))-dimensional Sawada–Kotera equation. Nonlinear Dyn. 87(4), 2305–2310 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Lv, J.Q., Bilige, S.D.: Lump solutions of a (\(2+1\))-dimensional bsk equation. Nonlinear Dyn. 90(3), 2119–2124 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ma, W.X.: Lump solutions to the Kadomtsev–Petviashvili equation. Phys. Lett. A 379(36), 1975–1978 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ma, W.X., Qin, Z.Y., Xing, L.: Lump solutions to dimensionally reduced p-gkp and p-gbkp equations. Nonlinear Dyn. 84(2), 923–931 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ma, W.X., Zhou, Y.: Lump solutions to nonlinear partial differential equations via Hirota bilinear forms. J. Differ. Equ. 264(4), 2633–2659 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ma, W.X., Zhou, Y., Dougherty, R.: Lump-type solutions to nonlinear differential equations derived from generalized bilinear equations. Int. J. Mod. Phys. B 30(28n29), 1640018 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    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, 55–61 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Zhang, X.E., Chen, Y.: Deformation rogue wave to the (\(2+1\))-dimensional KdV equation. Nonlinear Dyn. 1, 755–763 (2017)MathSciNetGoogle Scholar
  14. 14.
    Yan, Z.Y.: New families of nontravelling wave solutions to a new (\(3+1\))-dimensional potential-YTSF equation. Phys. Lett. A 318(12), 78–83 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Wazwaz, A.M.: Multiple-soliton solutions for the Calogero–Bogoyavlenskii–Schiff, Jimbo–Miwa and YTSF equations. Appl. Math. Comput. 203(2), 592–597 (2008)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Zhang, T.X., Xuan, H.N., Zhang, D.F., Wang, C.J.: Non-travelling wave solutions to a (\(3+1\))-dimensional potential-YTSF equation and a simplified model for reacting mixtures. Chaos Solitons Fractals 34(3), 1006–1013 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Yin, H.M., Tian, B., Chai, J., Wu, X.Y., Sun, W.R.: Solitons and bilinear backlund transformations for a (\(3+1\))-dimensional Yu–Toda–Sasa–Fukuyama equation in a liquid or lattice. Appl. Math. Lett. 58, 178–183 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hu, Y.J., Chen, H.L., Dai, Z.D.: New kink multi-soliton solutions for the (\(3+1\))-dimensional potential-Yu–Toda–Sasa–Fukuyama equation. Appl. Math. Comput. 234, 548–556 (2014)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Tu, J.M., Tian, S.F., Xu, M.J., Song, X.Q., Zhang, T.T.: Backlund transformation, infinite conservation laws and periodic wave solutions of a generalized (\(3+1\))-dimensional nonlinear wave in liquid with gas bubbles. Nonlinear Dyn. 83(3), 1199–1215 (2016)CrossRefzbMATHGoogle Scholar
  20. 20.
    Xu, M.J., Tian, S.F., Tu, J.M., Ma, P.L., Zhang, T.T.: On quasiperiodic wave solutions and integrability to a generalized (\(2+1\))-dimensional Korteweg-de Vries equation. Nonlinear Dyn. 82(4), 1–19 (2015)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Wang, X.B., Tian, S.F., Feng, L.L., Yan, H., Zhang, T.T.: Quasiperiodic waves, solitary waves and asymptotic properties for a generalized (\(3 + 1\))-dimensional variable-coefficient b-type Kadomtsev–Petviashvili equation. Nonlinear Dyn. 88(3), 2265–2279 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

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

Authors and Affiliations

  1. 1.School of Computer Science and Software EngineeringEast China Normal UniversityShanghaiChina

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