Skip to main content
SpringerLink
Log in

Lump-type solution and breather lump–kink interaction phenomena to a (\(\mathbf{{3{\varvec{+}}1}}\))-dimensional GBK equation based on trilinear form

  • Original paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In this paper, the multivariate trilinear operators in the (\(3+1\))-dimensional space are applied to a (\(3+1\))-dimensional GBK equation. The resulting trilinear form is used to study its wave dynamics. Particularly, we generate a type of new interaction solutions between breather lump-type solitons and other multi-kink solitons, thereby formulating a kind of breather lump–kink solitons. By setting time constants, we change the coordinates of kink solitons to make them collide with the breather lump-type soliton, during which breather lump-type soliton is swallowed eventually by those kink solitons. The evolution behaviours of the breather lump–kink solitons are depicted by plotting 3-D and density graphs from the perspective of wave characteristics.

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

Similar content being viewed by others

References

  1. Hirota, R.: Exact solution of the Korteweg–De Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192–1194 (1971)

    Article  Google Scholar 

  2. Liu, J.G., He, Y.: Abundant lump and lump–kink solutions for the new (\(3+1\))-dimensional generalized Kadomtsev–Petviashvili equation. Nonlinear Dyn. 92, 1103–1108 (2018)

    Article  Google Scholar 

  3. Wang, C.J., Fang, H., Tang, X.X.: State transition of lump-type waves for the (\(2+1\))-dimensional generalized KdV equation. Nonlinear Dyn. 95, 2943–2961 (2019)

    Article  Google Scholar 

  4. Ren, B., Ma, W.X., Yu, J.: Characteristics and interactions of solitary and lump waves of a (\(2+1\))-dimensional coupled nonlinear partial differential equation. Nonlinear Dyn. 96, 717–727 (2019)

    Article  Google Scholar 

  5. Ma, W.X., Zhou, Y.: Lump solutions to nonlinear partial differential equations via Hirota bilinear forms. J. Differ. Equ. 264, 2633–2659 (2018)

    Article  MathSciNet  Google Scholar 

  6. 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)

    Article  MathSciNet  Google Scholar 

  7. Ma, W.X.: Abundant lumps and their interaction solutions of (\(3+1\))-dimensional linear PDEs. J. Geom. Phys. 133, 10–16 (2018)

    Article  MathSciNet  Google Scholar 

  8. Ma, W.X.: Lump and interaction solutions of linear PDEs in (\(3+1\))-dimensions. East Asian J. Appl. Math. 9, 185–194 (2019)

    Article  MathSciNet  Google Scholar 

  9. Wang, M., Tian, B., Sun, Y., Yin, H.M., Zhang, Z.: Mixed lump-stripe, bright rogue wave-stripe, dark rogue wavestripe and dark rogue wave solutions of a generalized Kadomtsev–Petviashvili equation in fluid mechanics. Chin. J. Phys. 60, 440–449 (2019)

    Article  Google Scholar 

  10. Mao, J.J., Tian, S.F., Zou, L., Zhang, T.T., Yan, X.J.: Bilinear formalism, lump solution, lumpoff and instanton/rogue wave solution of a (\(3+1\))-dimensional B-type Kadomtsev–Petviashvili equation. Nonlinear Dyn. 95, 3005–3017 (2019)

    Article  Google Scholar 

  11. Xu, H., Ma, Z.Y., Fei, J.X., Zhu, Q.Y.: Novel characteristics of lump and lump-soliton interaction solutions to the generalized variable-coefficient Kadomtsev–Petviashvili equation. Nonlinear Dyn. 98, 551–560 (2019)

    Article  Google Scholar 

  12. Manukure, S., Zhou, Y., Ma, W.X.: Lump solutions to a (\(2+1\))-dimensional extended KP equation. Comput. Math. Appl. 75, 2414–2419 (2018)

    Article  MathSciNet  Google Scholar 

  13. Gai, L.T., Ma, W.X., Li, M.C.: Lump-type solutions, rogue wave type solutions and periodic lump–stripe interaction phenomena to a (\(3+1\))-dimensional generalized breaking soliton equation. Phys. Lett. A 384, 126178 (2020)

    Article  MathSciNet  Google Scholar 

  14. Geng, X.G., Ma, Y.L.: \(N\)-soliton solution and its Wronskian form of a (\(3+1\))-dimensional nonlinear evolution equation. Phys. Lett. A 369, 285–289 (2007)

    Article  MathSciNet  Google Scholar 

  15. Wazwaz, A.M.: Multiple soliton solutions for the Bogoyavlenskii’s generalized breaking soliton equations and its extension form. Appl. Math. Comput. 217, 4282–4288 (2010)

    MathSciNet  MATH  Google Scholar 

  16. Zhou, Y., Manukure, S., Ma, W.X.: Lump and lump-soliton solutions to the Hirota–Satsuma–Ito equation. Commun. Nonlinear Sci. Numer. Simul. 68, 56–62 (2019)

    Article  MathSciNet  Google Scholar 

  17. He, C.H., Tang, Y.N., Ma, W.X., Ma, J.L.: Interaction phenomena between a lump and other multi-solitons for the (\(2+1\))-dimensional BLMP and Ito equations. Nonlinear Dyn. 95, 29–42 (2019)

    Article  Google Scholar 

  18. Zhang, Y., Liu, Y.P., Tang, X.Y.: M-lump and interactive solutions to a (\(3+1\))-dimensional nonlinear system. Nonlinear Dyn. 93, 2533–2541 (2018)

    Article  Google Scholar 

  19. Zhang, Y., Liu, Y.P., Tang, X.Y.: M-lump solutions to a (\(3+1\))-dimensional nonlinear evolution equation. Comput. Math. Appl. 76, 592–601 (2018)

    Article  MathSciNet  Google Scholar 

  20. Zhao, Z.L., He, L.C.: Multiple lump solutions of the (\(3+1\))-dimensional potential Yu–Toda–Sasa–Fukuyama equation. Appl. Math. Lett. 95, 114–121 (2019)

    Article  MathSciNet  Google Scholar 

  21. Ren, B., Ma, W.X.: Rational solutions and their interaction solutions of the (\(2+1\))-dimensional modified dispersive water wave equation. Comput. Math. Appl. 77, 2086–2095 (2019)

    Article  MathSciNet  Google Scholar 

  22. Xie, J.J., Yang, X.: Rogue waves, breather waves and solitary waves for a (\(3+1\))-dimensional nonlinear evolution equation. Appl. Math. Lett. 97, 6–13 (2019)

    Article  MathSciNet  Google Scholar 

  23. Liu, Y.Q., Wen, X.Y.: Novel interaction phenomena of localized waves in the generalized (\(3+1\))-dimensional KP equation. Comput. Math. Appl. 78, 1–19 (2019)

    Article  MathSciNet  Google Scholar 

  24. Lin, F.H., Wang, J.P., Zhou, X.W., Ma, W.X., Lü, X.: Observation of interaction phenomena for two dimensionally reduced nonlinear models. Nonlinear Dyn. 94, 2643–2654 (2018)

    Article  Google Scholar 

  25. Ma, W.X., Fan, E.G.: Linear superposition principle applying to Hirota bilinear equations. Comput. Math. Appl. 61, 950–959 (2011)

    Article  MathSciNet  Google Scholar 

  26. Ma, W.X.: Generalized bilinear differential equations. Stud. Nonlinear Sci. 2, 140–144 (2011)

    Google Scholar 

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

    MathSciNet  Google Scholar 

  28. Zhang, R.F., Bilige, S.D.: Bilinear neural network method to obtain the exact analytical solutions of nonlinear partial differential equations and its application to p-gBKP equation. Nonlinear Dyn. 95, 3041–3048 (2019)

    Article  Google Scholar 

  29. Jia, T.T., Gao, Y.T., Deng, G.F., Hu, L.: Quintic time-dependent-coefficient derivative nonlinear Schrödinger equation in hydrodynamics or fiber optics: bilinear forms and dark/anti-dark/gray solitons. Nonlinear Dyn. 98, 269–282 (2019)

    Article  Google Scholar 

  30. Wang, Y.H., Wang, H., Dong, H.H., Zhang, H.S., Temuer, C.L.: Interaction solutions for a reduced extended (\(3+1\))-dimensional Jimbo–Miwa equation. Nonlinear Dyn. 92, 487–497 (2018)

    Article  Google Scholar 

  31. Deng, G.F., Gao, Y.T., Su, J.J., Ding, C.C., Jia, T.T.: Solitons and periodic waves for the (\(2+1\))-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada equation in fluid mechanics. Nonlinear Dyn. 99, 1039–1052 (2020)

    Article  Google Scholar 

  32. Liu, X.Y., Liu, W.Y., Triki, H., Zhou, Q., Biswas, A.: Periodic attenuating oscillation between soliton interactions for higher-order variable coefficient nonlinear Schrödinger equation. Nonlinear Dyn. 96, 801–809 (2019)

    Article  Google Scholar 

  33. Kuo, C.K., Ghanbari, B.: Resonant multi-soliton solutions to new (\(3+1\))-dimensional Jimbo–Miwa equations by applying the linear superposition principle. Nonlinear Dyn. 96, 459–464 (2019)

    Article  Google Scholar 

  34. Ma, W.X.: Trilinear equations, Bell polynomials, and resonant solutions. Front. Math. China 8, 1139–1156 (2013)

    Article  MathSciNet  Google Scholar 

  35. Ma, W.X., Zhang, Y., Tang, Y.N., Tu, J.Y.: Hirota bilinear equations with linear subspaces of solutions. Appl. Math. Comput. 218, 7174–7183 (2012)

    MathSciNet  MATH  Google Scholar 

  36. Chen, S.T., Ma, W.X.: Exact solutions to a generalized Bogoyavlensky–Konopelchenko equation via Maple symbolic computations. Complexity 2019, 8787460 (2019)

    MATH  Google Scholar 

Download references

Acknowledgements

The work was supported in part by NSFC (61572095, 61877007, 11371326, 11301331, 11371086, 11571079 and 51771083), NSF under the Grant (DMS-1664561), Natural Science Fund for Colleges and Universities of Jiangsu Province under the Grant (17KJB110020), Emphasis Foundation of Special Science Research on Subject Frontiers of CUMT under Grant (No. 2017XKZD11).

Author information

Authors and Affiliations

  1. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, Zhejiang, China

    Wen-Xiu Ma

  2. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, Liaoning, China

    Litao Gai

  3. Department of Mathematics and Statistics, University of South Florida, Tampa, FL, 33620-5700, USA

    Wen-Xiu Ma

  4. School of Mathematics, South China University of Technology, Guangzhou, 510640, Guangdong, China

    Wen-Xiu Ma

  5. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, Shandong, China

    Wen-Xiu Ma

  6. Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X2046, Mmabatho, 2735, South Africa

    Wen-Xiu Ma

  7. School of Software, Dalian University of Technology, Dalian, 116620, Liaoning, China

    Litao Gai & Mingchu Li

Authors
  1. Litao Gai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wen-Xiu Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mingchu Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Litao Gai or Wen-Xiu Ma.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gai, L., Ma, WX. & Li, M. Lump-type solution and breather lump–kink interaction phenomena to a (\(\mathbf{{3{\varvec{+}}1}}\))-dimensional GBK equation based on trilinear form. Nonlinear Dyn 100, 2715–2727 (2020). https://doi.org/10.1007/s11071-020-05554-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-020-05554-1

Keywords

Search

Navigation