Skip to main content
SpringerLink
Log in

Variety interaction solutions comprising lump solitons for a generalized BK equation by trilinear analysis

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract

In this paper, we study a new generalized (2+1)-dimensional Bogoyavlensky–Konopelchenko equation which is considered in soliton theory and generated by considering the Hirota trilinear operators. We retrieve some novel exact analytical solutions, including lump-type wave, breather wave as well as breather-kink-type wave, and lump-kink wave solutions for a (2+1)-dimensional \({\overline{q}}\)-gBK equation by Maple symbolic computations. The required conditions of the analyticity and positivity of the solutions can be easily achieved by taking special choices of the involved parameters. The main ingredients for this scheme are to recover the Hirota trilinear forms and their generalized equivalences. Through the way of resetting different space constants, we adjust the coordinates of kink-type waves in order for them colliding with the breather wave; after that, the transformed kink-type waves gradually swallow the breather wave. Also, the N-soliton wave solutions for the (2+1)-dimensional gBK equation are successfully constructed through using a perturbation method. Lastly, the graphical simulations of the exact solutions are depicted.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19

Similar content being viewed by others

References

  1. W.X. Ma, Z. Zhu, Solving the \((3+1)\)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm. Appl. Math. Comput. 218, 11871–11879 (2012)

    MathSciNet  MATH  Google Scholar 

  2. M.R. Foroutan, J. Manafian, A. Ranjbaran, Lump solution and its interaction to (3+1)-D potential-YTSF equation. Nonlinear Dyn. 92(4), 2077–2092 (2018)

    Article  Google Scholar 

  3. J. Manafian, M. Lakestani, Abundant soliton solutions for the Kundu–Eckhaus equation via \(tan(\phi /2)\)-expansion method. Optik 127, 5543–5551 (2016)

    Article  ADS  Google Scholar 

  4. H.M. Baskonus, H. Bulut, Exponential prototype structures for (2+1)-dimensional Boiti–Leon–Pempinelli systems in mathematical physics. Waves Rand. Comp. Media 26, 201–208 (2016)

    MathSciNet  MATH  Google Scholar 

  5. M. Dehghan, J. Manafian, The solution of the variable coefficients fourth-order parabolic partial differential equations by homotopy perturbation method. Zeitschrift für Naturforschung A 64a, 420–30 (2009)

    Article  ADS  Google Scholar 

  6. S. Saha Ray, On conservation laws by Lie symmetry analysis for (2+1)-dimensional Bogoyavlensky–Konopelchenko equation in wave propagation. Comput. Math. Appl. 74, 1158–1165 (2017)

    Article  MathSciNet  Google Scholar 

  7. X.H. Zhao, B. Tian, X.Y. Xie, X.Y. Wu, Y. Sun, Y.J. Guo, Solitons, Bäcklund transformation and Lax pair for a (2+1)-dimensional Davey–Stewartson system on surface waves of finite depth. Wave Rand. Comp. 28, 356–366 (2018)

    Article  Google Scholar 

  8. J. Manafian, M. Lakestani, \(N\)-lump and interaction solutions of localized waves to the (2+1)-dimensional variable-coefficient Caudrey–Dodd–Gibbon–Kotera–Sawada equation. J. Geo. Phys. 150, 103598 (2020)

    Article  MathSciNet  Google Scholar 

  9. J. Manafian, B. Mohammadi-Ivatlo, M. Abapour, Lump-type solutions and interaction phenomenon to the (2+1)-dimensional Breaking Soliton equation. Appl. Math. Comput. 13, 13–41 (2019)

    MathSciNet  MATH  Google Scholar 

  10. O.A. Ilhan, J. Manafian, M. Shahriari, Lump wave solutions and the interaction phenomenon for a variable-coefficient Kadomtsev–Petviashvili equation. Comput. Math. Appl. 78(8), 2429–2448 (2019)

    Article  MathSciNet  Google Scholar 

  11. O.A. Ilhan, J. Manafian, Periodic type and periodic cross-kink wave solutions to the (2+1)-dimensional breaking soliton equation arising in fluid dynamics. Mod. Phys. Lett. B 33, 1950277 (2019). https://doi.org/10.1142/S0217984919502774

    Article  ADS  MathSciNet  Google Scholar 

  12. X. Geng, G. He, L. Wu, Riemann theta function solutions of the Caudrey–Dodd–Gibbon–Sawada–Kotera hierarchy. J. Geo. Phys. 140, 85–103 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  13. W.X. Ma, Lump solutions to the Kadomtsev–Petviashvili equation. Phys. Let. A 379, 1975–1978 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  14. R. Hirota, The Direct Method in Soliton Theory (Cambridge Univ. Press, Cambridge, 2004)

    Book  Google Scholar 

  15. J.Y. Yang, W.X. Ma, Lump solutions to the bKP equation by symbolic computation. Int. J. Mod. Phys. B 30, 1640028 (2016). https://doi.org/10.1142/S0217979216400282

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. W.X. Ma, Z.Y. Qin, X. Lv, Lump solutions to dimensionally reduced p-gKP and p-gbKP equations. Nonlinear Dyn. 84, 923–931 (2016). https://doi.org/10.1007/s11071-015-2539-6

    Article  MathSciNet  MATH  Google Scholar 

  17. J. Lü, S. Bilige, T. Chaolu, The study of lump solution and interaction phenomenon to \((2+1)\)-dimensional generalized fifth-order KdV equation. Nonlinear Dyn. 91, 1669–1676 (2018)

    Article  Google Scholar 

  18. J. Manafian, O.A. Ilhan, L. Avazpour, A. Alizadeh, \(M\) lump and interaction between \(M\) lump and N stripe for the third-order evolution equation arising in the shallow water. Adv. Diff. Eq. 2020, 207 (2020). https://doi.org/10.1186/s13662-020-02669-y

    Article  MathSciNet  Google Scholar 

  19. J. Manafian, M. Lakestani, Lump-type solutions and interaction phenomenon to the bidirectional Sawada–Kotera equation. Pramana 92, 41 (2019)

    Article  ADS  Google Scholar 

  20. J. Manafian, Novel solitary wave solutions for the (3+1)-dimensional extended Jimbo–Miwa equations. Comput. Math. Appl. 76(5), 1246–1260 (2018)

    Article  MathSciNet  Google Scholar 

  21. Y.J. Feng, Y.T. Gaoa, L.Q. Li, T.T. Jia, Bilinear form, solitons, breathers and lumps of a (3+1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup– Kupershmidt equation in ocean dynamics, fluid mechanics and plasma physics. Eur. Phys. J. Plus 135, 272 (2020)

    Article  Google Scholar 

  22. J. Manafian, O.A. Ilhan, L. Avazpour, A. Alizadeh, \(N\)-lump and interaction solutions of localized waves to the (2+1)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation arise from a model for an incompressible fluid. Math. Math. Appl. Sci. 43, 9904–9927 (2020)

    Article  MathSciNet  Google Scholar 

  23. O.A. Ilhan, J. Manafian, M. Shahriari, Lump wave solutions and the interaction phenomenon for a variable-coefficient Kadomtsev–Petviashvili equation. Comput. Math. Appl. 78(8), 2429–2448 (2019)

    Article  MathSciNet  Google Scholar 

  24. W.X. Ma, Y. Zhou, Lump solutions to nonlinear partial differential equations via Hirota bilinear forms. J. Diff. Eqn. 264, 2633–2659 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  25. W.X. Ma, A search for lump solutions to a combined fourthorder nonlinear PDE in (2+1)-dimensions. J. Appl. Anal. Comput. 9, 1319–1332 (2019)

    MathSciNet  MATH  Google Scholar 

  26. W.X. Ma, Interaction solutions to Hirota–Satsuma–Ito equation in (2+1)-dimensions. Front. Math. China 14, 619–629 (2019)

    Article  MathSciNet  Google Scholar 

  27. J.Y. Yang, W.X. Ma, C.M. Khalique, Determining lump solutions for a combined soliton equation in (2+1)-dimensions. Eur. Phys. J. Plus 135, 494 (2020)

    Article  Google Scholar 

  28. W.X. Ma, Y. Zhang, Y. Tang, Symbolic computation of lump solutions to a combined equation involving three types of nonlinear terms, East Asian. J. Appl. Math. 10, 732–745 (2020)

    MathSciNet  MATH  Google Scholar 

  29. W.X. Ma, \(N\)-soliton solutions and the Hirota conditions in (2+1)-dimensions. Opt. Quant. Elec. 52, 511 (2020)

    Article  Google Scholar 

  30. S.T. Chen, W.X. Ma, Exact solutions to a generalized Bogoyavlensky–Konopelchenko equation via maple symbolic computations. Complexity 8787460, 1–6 (2019)

    MATH  Google Scholar 

  31. J. Manafian, O.A. Ilhan, A. Alizadeh, S.A. Mohammed, Multiple rogue wave and solitary solutions for the generalized BK equation via Hirota bilinear and SIVP schemes arising in fluid mechanics. Commun. Theor. Phys. 72, 075002 (13pp) (2020)

    Article  ADS  MathSciNet  Google Scholar 

  32. L.T. Gai, W.X. Ma, M.C. Li, Lump-type solution and breather lump-kink interaction phenomena to a (3+1)-dimensional GBK equation based on trilinear form. Nonlinear Dyn. 100, 2715–2727 (2020)

    Article  Google Scholar 

  33. W.X. Ma, Bilinear equations, Bell polynomials and linear superposition principle. J. Phys. Conf. Ser. 411, 012021 (2013)

    Article  Google Scholar 

Download references

Acknowledgements

This paper is supported by the Research Affairs Office of University of Tabriz, Tabriz, Iran, under Grant No. S-7532, as part of a postdoctoral research project.

Funding

There is no fund for this manuscript.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran

    Jalil Manafian

Authors
  1. Jalil Manafian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jalil Manafian.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Manafian, J. Variety interaction solutions comprising lump solitons for a generalized BK equation by trilinear analysis. Eur. Phys. J. Plus 136, 1097 (2021). https://doi.org/10.1140/epjp/s13360-021-02065-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/s13360-021-02065-9

Keywords

Search

Navigation