Nonlinear Dynamics

, Volume 96, Issue 4, pp 2463–2473 | Cite as

Fission and fusion collision of high-order lumps and solitons in a \((3+1)\)-dimensional nonlinear evolution equation

  • Wei LiuEmail author
  • Xiaoxiao Zheng
  • Chu Wang
  • Shengqi Li
Original Paper


Multiple dark soliton solutions and semi-rational solutions to a \((3+1)\)-dimensional nonlinear evolution equation are obtained by a combination of the Kadomtsev–Petviashvili hierarchy reduction method and the Hirota’s bilinear method. The collision phenomena of fission and fusion of high-order lumps and solitons in the \((3+1)\)-dimensional nonlinear evolution equation are described by these semi-rational solutions. After the collision of higher-order lumps and solitons, the lumps would fuse into or fissure from the line solitons. As lumps are created or annihilated, the exchange of energy occurs between the lumps and the line solitons.


(3+1)-Dimensional nonlinear evolution Hirota bilinear method Lump–soliton solution KP hierarchy reduction method 



This work is supported by the National Natural Science Foundation of China under Grant No. 11801321.

Compliance with ethical standards

Conflict of interest

All authors declare that they have no conflict of interest.


  1. 1.
    Khan, N.S., Islam, S., Gul, T., Khan, W., Khan, I., Ali, L.: Thin film flow of a second grade fluid in a porous medium past a stretching sheet with heat transfer. Alex. Eng. J. 57(2), 1019–1031 (2018)CrossRefGoogle Scholar
  2. 2.
    Khan, N.S., Islam, S., et al.: Thermophoresis and thermal radiation with heat and mass transfer in a magnetohydro dynamic thin film second-grade fluid of variable properties past a stretching sheet. Eur. Phys. J. Plus 132(11), 11 (2017)CrossRefGoogle Scholar
  3. 3.
    Zuhra, S., Khan, N.S., Khan, M.A., et al.: Flow and heat transfer in water based liquid film fluids dispensed with graphene nanoparticles. Results Phys. 8, 1143–1157 (2018)CrossRefGoogle Scholar
  4. 4.
    Xu, F.Y., Zhang, X.G., Wu, Y.H., Liu, L.S.: Global existence and the optimal decay rates for the three dimensional compressible nematic liquid crystal flow. Acta Appl. Math. 150(1), 1–14 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Khan, N.S., Gul, T., Islam, S., Khan, W., et al.: Magnetohydro dynamic nanoliquid thin film sprayed on a stretching cylinder with heat transfer. Appl. Sci. 7(3), 271 (2017)CrossRefGoogle Scholar
  6. 6.
    Khan, N.S., Gul, T., Khan, W., Bonyah, E., Islam, S.: Mixed convection in gravity-driven thin film non-Newtonian nanofluids flow with gyrotactic microorganisms. Results Phys. 7, 4033–4049 (2017)CrossRefGoogle Scholar
  7. 7.
    Khan, N.S., Gul, T., Islam, S., et al.: Brownian motion and thermophoresis effects on MHD mixed convective thin film second-grade nanofluid flow with Hall effect and heat transfer past a stretching sheet. J. Nanofluids 6(5), 812–829 (2017)CrossRefGoogle Scholar
  8. 8.
    Palwasha, Z., Islam, S., Khan, N.S., Ayaz, H.: Non-Newtonian nanoliquids thin-film flow through a porous medium with magnetotactic microorganisms. Appl. Nanosci. 8(6), 1523–1544 (2018)CrossRefGoogle Scholar
  9. 9.
    Zuhra, S., Saeed, K.N., Saeed, I.: Magnetohydro dynamic second grade nanofluid flow containing nanoparticles and gyrotactic microorganisms. Comput. Appl. Math. 37, 6332–6358 (2018)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Saeed, K.N.: Bioconvection in second grade nanofluid flow containing nanoparticles and gyrotactic micro organisms. Braz. J. Phys. 43(4), 227–241 (2018)Google Scholar
  11. 11.
    Khan, N.S., Zuhra, S., Shah, Z., Bonyah, E., Khan, W., Islam, S.: Slip flow of Eyring–Powell nanoliquid film containing graphene nanoparticles due to an unsteady stretching sheet with heat transfer. AIP Adv. 8(11), 115302 (2018)CrossRefGoogle Scholar
  12. 12.
    Wang, Y., Liu, L.S., Zhang, X.G., Wu, Y.H.: Positive solutions of a fractional semipositone differential system arising from the study of HIV infection models. Appl. Math. Comput. 258, 312–324 (2015)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Fokas, A.S., Pelinovsky, D.E., Sulaem, C.: Interaction of lumps with a line soliton for the DSII equation. Physica D 152, 189–198 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gorshkov, K.A., Pelinovsky, D.E., Stepanyants, Y.A.: Normal and anomalous scattering, formation and decay of bound states of two-dimensional solitons described by the Kadomtsev–Petviashvili equation. JETP 77(2), 237–245 (1993)Google Scholar
  15. 15.
    Lu, Z., Tian, E., Grimshaw, R.: Interaction of two lump solitons described by the Kadomtsev–Petviashvili I equation. Wave Motion 40, 123–135 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Fokas, A.S., Pogrebkov, A.K.: Inverse scattering transform for the KPI equation on the background of a one-line soliton. Nonlinearity 16, 771–783 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Rao, J., Cheng, Y., He, J.: Rational and semirational solutions of the nonlocal Dave–Stewartson equation. Stud. Appl. Math. 139, 568–598 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Rao, J., Porsezian, K., He, J., Kanna, T.: Dynamics of lumps and dark–dark solitons in the multi-component long-wave–short-wave resonance interaction system. Proc. R. Soc. A 474, 20170627 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Rao, J., Porsezian, K., He, J.: Semi-rational solutions of the third-type Davey–Stewartson equation. Chaos 27, 083115 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ma, W.X., Qin, Z., Lü, X.: Lump solutions to dimensionally reduced p-gKP and p-gBKP equations. Nonlinear Dyn. 84, 923 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Zhang, X., 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, 24–31 (2017)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Jia, M., Lou, Y.: A novel type of rogue waves with predictability in nonlinear physics. Preprint. arXiv:1710.06604 [nlin.SI] (2017)
  23. 23.
    Zheng, X.X., Shang, Y.D., Peng, X.M.: Orbital stability of periodic travelling wave solutions of the generalized Zakharov equations. Acta Math. Sci. 37, 998–1018 (2017)CrossRefzbMATHGoogle Scholar
  24. 24.
    Zheng, X.X., Shang, Y.D., Peng, X.M.: Orbital stability of solitary waves of the coupled Klein–Gordon–Zakharov equations. Math. Methods Appl. Sci. 40, 2623–2633 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Zheng, X.X., Shang, Y.D., Di, H.F.: The time-periodic solutions to the modified Zakharov equations with a quantum correction. Mediterr. J. Math. 14, 152 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Liu, W., Zheng, X.X., Li, X.L.: Bright and dark soliton solutions to the partial reverse space-time nonlocal Mel’nikov equation. Nonlinear Dyn. 94, 2177–2189 (2018)CrossRefGoogle Scholar
  27. 27.
    Liu, W., Wazwaz, A.M., Zheng, X.X.: Families of semi-rational solutions to the Kadomtsev–Petviashvili I equation. Commun. Nonlinear Sci. Numer. Simul. 67, 480–491 (2019)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Geng, X.: Algebraic-geometrical solutions of some multidimensional nonlinear evolution equations. J. Phys. A Math. Gen. 36, 2289 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Geng, X., Ma, Y.: \(N\)-soliton solution and its Wronskian form of a \((3 + 1)\)-dimensional nonlinear evolution equation. Phys. Lett. A 369, 285–289 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Wazwaz, A.M.: A \((3 + 1)\)-dimensional nonlinear evolution equation with multiple soliton solutions and multiple singular soliton solutions. Appl. Math. Comput. 215(25), 1548–1552 (2009)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Wazwaz, A.M.: A variety of distinct kinds of multiple soliton solutions for a \((3+1)\)-dimensional nonlinear evolution equation. Math. Methods Appl. Sci. 36(26), 349–357 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Wazwaz, A.M.: New \((3+1)\)-dimensional nonlinear evolution equation: multiple soliton solutions. Cent. Eur. J. Eng. 4, 352–356 (2014)Google Scholar
  33. 33.
    Chen, M., Li, X., Wang, Y., Li, B.: A pair of resonance stripe solitons and lump solutions to a reduced \((3+1)\)-dimensional nonlinear evolution equation. Commun. Theor. Phys. 54, 947 (2010)CrossRefGoogle Scholar
  34. 34.
    Zha, Q.: Rogue waves and rational solutions of a \((3+ 1)\)-dimensional nonlinear evolution equation. Phys. Lett. A 377, 3021 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Shi, Y., Zhang, Y.: Rogue waves of a \((3+ 1)\)-dimensional nonlinear evolution equation. Commun. Nonlinear Sci. Numer. Simul. 44, 120–129 (2017)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Tang, Y., Tao, S., Zhou, M., Guan, Q.: Interaction solutions between lump and other solitons of two classes of nonlinear evolution equations. Nonlinear Dyn. 89, 429–442 (2017)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Hirota, R.: The Direct Method in Soliton Theory. Cambridge University Press, Cambridge (2004)CrossRefzbMATHGoogle Scholar
  38. 38.
    Matsuno, Y.: Bilinear Transformation Method. Academic, New York (1984)zbMATHGoogle Scholar
  39. 39.
    Ohta, Y., Wang, D., Yang, J.: General N-dark–dark solitons in the coupled nonlinear Schrödinger equations. Stud. Appl. Math. 127, 345 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Ohta, Y., Yang, J.: General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation. Proc. R. Soc. A 468, 1716 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Ohta, Y., Yang, J.: Rogue waves in the Davey–Stewartson I equation. Phys. Rev. E 86, 036604 (2012)CrossRefGoogle Scholar
  42. 42.
    Ohta, Y., Yang, J.: Dynamics of rogue waves in the Davey–Stewartson II equation. J. Phys. A Math. Theor. 46, 105202 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.College of Mathematic and Information ScienceShandong Technology and Business UniversityYantaiPeople’s Republic of China
  2. 2.School of Mathematical SciencesQufu Normal UniversityQufuPeople’s Republic of China

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