Skip to main content
SpringerLink
Log in

Bright/dark breather-soliton, lump wave-soliton and rogue wave-soliton interactions for a (3+1)-dimensional generalized B-type Kadomtsev-Petviashvili equation in fluid mechanics

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

Abstract

Fluid mechanics is concerned with the mechanics of liquids, plasmas and gases, with the forces on them. Investigated in this paper is a generalized (3+1)-dimensional B-type Kadomtsev-Petviashvili equation for the weakly dispersive waves in fluid mechanics. Breather-soliton, lump wave-soliton and rogue wave-soliton solutions are derived under certain integrable constraints via the Hirota method. We display two types of the interactions between a breather and a soliton. Interactions among a breather and two solitons are shown. We observe the fusion between a dark lump wave and a dark soliton, as well as the fission of a dark soliton. Studying the rogue wave-soliton interactions, we find that a rogue wave appears from one soliton and merges into the other soliton gradually. In addition, effects of \(h_1\), \(h_3\) and \(h_5\) on those waves are observed, where \(h_1\), \(h_3\) and \(h_5\) are the coefficients in that equation.

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

Data Availability Statement

All data generated or analyzed during this study are included in this published article.

Notes

  1. To ensure Constraints (2), when we change \(h_1\), \(h_3\) and \(h_5\), the values of \(h_2\) and \(h_4\) should be changed as well.

  2. Fusion means that two or more nonlinear waves merge into one nonlinear wave [50, 51].

  3. Fission means that one nonlinear wave is divided into two or more nonlinear waves [50, 51].

References

  1. Falkovich, G.: Fluid mechanics. Cambridge Univ. Press, Cambridge (2018)

    MATH  Google Scholar 

  2. Aref, H., Balachandar, S.: A first course in computational fluid dynamics. Cambridge Univ. Press, Cambridge (2018)

    MATH  Google Scholar 

  3. Liu, F.Y., Gao, Y.T., Yu, X., Li, L.Q., Ding, C.C., Wang, D.: Lie group analysis and analytic solutions for a (2+1)-dimensional generalized Bogoyavlensky-Konopelchenko equation in fluid mechanics and plasma physics. Eur. Phys. J. Plus 136, 656 (2021)

    Google Scholar 

  4. Liu, F.Y., Gao, Y.T., Yu, X., Hu, L., Wu, X.H.: Hybrid solutions for the (2+1)-dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation in fluid mechanics. Chaos Solitons Fract. 152, 111355 (2021)

    MathSciNet  Google Scholar 

  5. Li, L.Q., Gao, Y.T., Yu, X., Jia, T.T., Hu, L., Zhang, C.Y.: Bilinear forms, bilinear Bäcklund transformation, soliton and breather interactions of a damped variable-coefficient fifth-order modified Korteweg-de Vries equation for the surface waves in a strait or large channel. Chin. J. Phys. (2022). https://doi.org/10.1016/j.cjph.2021.09.004

    Article  Google Scholar 

  6. Guo, R., Zhao, H.H., Wang, Y.: A higher-order coupled nonlinear Schrödinger system: solitons, breathers, and rogue wave solutions. Nonlinear Dyn. 83, 2475–2484 (2016)

    MATH  Google Scholar 

  7. Li, L.Q., Gao, Y.T., Hu, L., Jia, T.T., Ding, C.C., Feng, Y.J.: Bilinear form, soliton, breather, lump and hybrid solutions for a (2+1)-dimensional Sawada-Kotera equation. Nonlinear Dyn. 100, 2729–2738 (2020)

    Google Scholar 

  8. Xie, X.Y., Meng, G.Q.: Multi-dark soliton solutions for a coupled AB system in the geophysical flows. Appl. Math. Lett. 92, 201–207 (2019)

    MathSciNet  MATH  Google Scholar 

  9. Ablowitz, M.J., Segur, H.: Solitons and the inverse scattering transform. SIAM, Phil (1981)

    MATH  Google Scholar 

  10. Hirota, R.: The direct method in soliton theory. Cambridge Univ. Press, Cambridge (2004)

    MATH  Google Scholar 

  11. Masood, W., Rizvi, H.: Two dimensional nonplanar evolution of electrostatic shock waves in pair-ion plasmas. Phys. Plasmas 19, 012119 (2012)

    Google Scholar 

  12. Kadomtsev, B.B., Petviashvili, V.I.: On the stability of solitary waves in weakly dispersing media. Sov. Phys. Dokl. 15, 539–541 (1970)

    MATH  Google Scholar 

  13. Lü, X., Lin, F.H.: Soliton excitations and shape-changing collisions in alpha helical proteins with interspine coupling at higher order. Commun. Nonlinear Sci. Numer. Simul. 32, 241–261 (2016)

    MathSciNet  MATH  Google Scholar 

  14. Zhao, X.H.: Dark soliton solutions for a coupled nonlinear Schrödinger system. Appl. Math. Lett. 121, 107383 (2021)

    MATH  Google Scholar 

  15. Chen, S.S., Tian, B., Qu, Q.X., Li, H., Sun, Y., Du, X.X.: Alfvén solitons and generalized Darboux transformation for a variable-coefficient derivative nonlinear Schrödinger equation in an inhomogeneous plasma. Chaos Solitons Fract. 148, 111029 (2021)

    Google Scholar 

  16. Wu, X.H., Gao, Y.T., Yu, X., Ding, C.C., Liu, F.Y., Jia, T.T.: Darboux transformation, bright and dark-bright solitons of an N-coupled high-order nonlinear Schrödinger system in an optical fiber. Mod. Phys. Lett. B (2021) in press, Ms. No. MPLB-D-21-00411

  17. Gao, X.Y., Guo, Y.J., Shan, W.R.: Optical waves/modes in a multicomponent inhomogeneous optical fiber via a three-coupled variable-coefficient nonlinear Schrödinger system. Appl. Math. Lett. 120, 107161 (2021)

    MATH  Google Scholar 

  18. Xu, T., Sun, F.W., Zhang, Y., Li, J.: Multi-component Wronskian solution to the Kadomtsev-Petviashvili equation. Comput. Math. Math. Phys. 54, 97–113 (2014)

    MathSciNet  MATH  Google Scholar 

  19. Wang, D., Gao, Y.T., Ding, C.C., Zhang, C.Y.: Solitons and periodic waves for a generalized (3+1)-dimensional Kadomtsev-Petviashvili equation in fluid dynamics and plasma physics. Commun. Theor. Phys. 72, 115004 (2020)

    MathSciNet  Google Scholar 

  20. Zhang, H.Q., Ma, W.Q.: Mixed lump-kink solutions to the KP equation. Comput. Math. Appl. 74, 1399–1405 (2017)

    MathSciNet  MATH  Google Scholar 

  21. Xia, J.W., Zhao, Y.W., Lü, X.: Predictability, fast calculation and simulation for the interaction solution to the cylindrical Kadomtsev-Petviashvili equation. Commun. Nonlinear Sci. Numer. Simul. 90, 105260 (2020)

    MathSciNet  MATH  Google Scholar 

  22. Ruan, H.Y., Chen, Y.X.: Dromion interactions of (2+1)-dimensional nonlinear evolution equations. Phys. Rev. E 62, 5738 (2000)

    Google Scholar 

  23. Zhang, R.F., Li, M.C., Yin, H.M.: Rogue wave solutions and the bright and dark solitons of the (3+1)-dimensional Jimbo-Miwa equation. Nonlinear Dyn. 103, 1071–1079 (2021)

    Google Scholar 

  24. Chen, S.J., Ma, W.X., Lü, X.: Bäcklund transformation, exact solutions and interaction behaviour of the (3+1)-dimensional Hirota-Satsuma-Ito-like equation. Commun. Nonlinear Sci. Numer. Simul. 83, 105135 (2020)

    MathSciNet  MATH  Google Scholar 

  25. Xu, H.N., Ruan, W.Y., Zhang, Y., Lü, X.: Multi-exponential wave solutions to two extended Jimbo-Miwa equations and the resonance behavior. Appl. Math. Lett. 99, 105976 (2020)

    MathSciNet  MATH  Google Scholar 

  26. Zhang, R.F., Bilige, S., Chaolu, T.: Fractal solitons, arbitrary function solutions, exact periodic wave and breathers for a nonlinear partial differential equation by using bilinear neural network method. J. Syst. Sci. Complex. 34, 122–139 (2021)

    MathSciNet  MATH  Google Scholar 

  27. Yin, Y.H., Chen, S.J., Lü, X.: Localized characteristics of lump and interaction solutions to two extended Jimbo-Miwa equations. Chin. Phys. B 29, 120502 (2020)

    Google Scholar 

  28. Zhang, R.F., Li, M.C., Albishari, M., Zheng, F.C., Lan, Z.Z.: Generalized lump solutions, classical lump solutions and rogue waves of the (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada-like equation. Appl. Math. Comput. 403, 126201 (2021)

    MathSciNet  MATH  Google Scholar 

  29. Lü, X., Chen, S.J.: Interaction solutions to nonlinear partial differential equations via Hirota bilinear forms: one-lump-multi-stripe and one-lump-multi-soliton types. Nonlinear Dyn. 103, 947–977 (2021)

    Google Scholar 

  30. Lü, X., Hua, Y.F., Chen, S.J., Tang, X.F.: Integrability characteristics of a novel (2+1)-dimensional nonlinear model: Painlevé analysis, soliton solutions, Bäcklund transformation, Lax pair and infinitely many conservation laws. Commun. Nonlinear Sci. Numer. Simul. 95, 105612 (2021)

    MATH  Google Scholar 

  31. Hu, L., Gao, Y.T., Jia, T.T., Deng, G.F., Li, L.Q.: Higher-order hybrid waves for the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation for an irrotational incompressible fluid via the modified Pfaffian technique. Z. Angew. Math. Phys. 72, 75 (2021)

    MathSciNet  MATH  Google Scholar 

  32. Ma, Y.X., Tian, B., Qu, Q.X., Wei, C.C., Zhao, X.: Bäcklund transformations, kink soliton, breather- and travelling-wave solutions for a (3+1)-dimensional B-type Kadomtsev-Petviashvili equation in fluid dynamics. Chin. J. Phys. 73, 600–612 (2021)

    Google Scholar 

  33. 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)

    MATH  Google Scholar 

  34. Zhang, R.F., Bilige, S.: 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)

    MATH  Google Scholar 

  35. Zhang, R.F., Bilige, S., Liu, J.G., Li, M.C.: Bright-dark solitons and interaction phenomenon for p-gBKP equation by using bilinear neural network method. Phys. Scr. 96, 025224 (2021)

    Google Scholar 

  36. Tu, J.M., Tian, S.F., Xu, M.J., Ma, P.L., Zhang, T.T.: On periodic wave solutions with asymptotic behaviors to a (3+1)-dimensional generalized B-type Kadomtsev-Petviashvili equation in fluid dynamics. Comput. Math. Appl. 72, 2486–2504 (2016)

    MathSciNet  MATH  Google Scholar 

  37. Wu, X.Y., Tian, B., Chai, H.P., Sun, Y.: Rogue waves and lump solutions for a (3+1)-dimensional generalized B-type Kadomtsev-Petviashvili equation in fluid mechanics. Mod. Phys. Lett. B 31, 1750122 (2017)

    MathSciNet  Google Scholar 

  38. Yan, X.W., Tian, S.F., Wang, X.B., Zhang, T.T.: Solitons to rogue waves transition, lump solutions and interaction solutions for the (3+1)-dimensional generalized B-type Kadomtsev-Petviashvili equation in fluid dynamics. Int. J. Comput. Math. 96, 1839–1848 (2019)

    MathSciNet  MATH  Google Scholar 

  39. Wazwaz, A.M.: Two B-type Kadomtsev-Petviashvili equations of (2+1) and (3+1) dimensions: multiple soliton solutions, rational solutions and periodic solutions. Comput. Fluids 86, 357–362 (2013)

    MathSciNet  MATH  Google Scholar 

  40. Hu, W.Q., Gao, Y.T., Jia, S.L., Huang, Q.M., Lan, Z.Z.: Periodic wave, breather wave and travelling wave solutions of a (2+1)-dimensional B-type Kadomtsev-Petviashvili equation in fluids or plasmas. Eur. Phys. J. Plus 131, 390 (2016)

    Google Scholar 

  41. Wu, P.X., Zhang, Y.F., Muhammad, I., Yin, Q.Q.: Lump, periodic lump and interaction lump stripe solutions to the (2+1)-dimensional B-type Kadomtsev-Petviashvili equation. Mod. Phys. Lett. B 32, 1850106 (2018)

    MathSciNet  Google Scholar 

  42. Hu, C.C., Tian, B., Wu, X.Y., Du, Z., Zhao, X.H.: Lump wave-soliton and rogue wave-soliton interactions for a (3+1)-dimensional B-type Kadomtsev-Petviashvili equation in a fluid. Chin. J. Phys. 56, 2395–2403 (2018)

    MathSciNet  Google Scholar 

  43. Ding, C.C., Gao, Y.T., Hu, L., Deng, G.F., Zhang, C.Y.: Vector bright soliton interactions of the two-component AB system in a baroclinic fluid. Chaos Solitons Fract. 142, 110363 (2021)

    MathSciNet  Google Scholar 

  44. Hu, L., Gao, Y.T., Jia, S.L., Su, J.J., Deng, G.F.: Solitons for the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation for an irrotational incompressible fluid via the Pfaffian technique. Mod. Phys. Lett. B 33, 1950376 (2019)

  45. Gao, X.Y., Guo, Y.J., Shan, W.R., Yin, H.M., Du, X.X., Yang, D.Y.: Electromagnetic waves in a ferromagnetic film. Commun. Nonlinear Sci. Numer. Simul. 105, 106066 (2022)

    MathSciNet  MATH  Google Scholar 

  46. Hu, C.C., Tian, B., Zhao, X.: Rogue and lump waves for the (3+1)-dimensional Yu-Toda-Sasa-Fukuyama equation in a liquid or lattice. Int. J. Mod. Phys. B 35, 2150320 (2021)

    Google Scholar 

  47. Feng, Y.J., Gao, Y.T., Jia, T.T., Li, L.Q.: Soliton interactions of a variable-coefficient three-component AB system for the geophysical flows. Mod. Phys. Lett. B 33, 1950354 (2019)

    MathSciNet  Google Scholar 

  48. Ding, C.C., Gao, Y.T., Deng, G.F., Wang, D.: Lax pair, conservation laws, Darboux transformation, breathers and rogue waves for the coupled nonautonomous nonlinear Schrödinger system in an inhomogeneous plasma. Chaos Solitons Fract. 133, 109580 (2020)

    MATH  Google Scholar 

  49. Wang, D., Gao, Y.T., Yu, X., Li, L.Q., Jia, T.T.: Bilinear form, solitons, breathers, lumps and hybrid solutions for a (3+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation. Nonlinear Dyn. 104, 1519–1531 (2021)

    Google Scholar 

  50. 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, 24–31 (2017)

    MathSciNet  MATH  Google Scholar 

  51. Tan, W., Dai, Z.D., Xie, J.L., Qiu, D.Q.: Parameter limit method and its application in the (4+1)-dimensional Fokas equation. Comput. Math. Appl. 75, 4214–4220 (2018)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We express our sincere thanks to the Editors and Reviewers for their valuable comments. This work has been supported by the National Natural Science Foundation of China under Grant Nos. 11772017, 11805020 and 11272023, by the Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), China (IPOC: 2017ZZ05), and by the Fundamental Research Funds for the Central Universities of China under Grant No. 2011BUPTYB02.

Author information

Authors and Affiliations

  1. State Key Laboratory of Information Photonics and Optical Communications, and School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, China

    Cong-Cong Hu, Bo Tian, Xia-Xia Du & Chen-Rong Zhang

Authors
  1. Cong-Cong Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bo Tian
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xia-Xia Du
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Chen-Rong Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bo Tian.

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

Hu, CC., Tian, B., Du, XX. et al. Bright/dark breather-soliton, lump wave-soliton and rogue wave-soliton interactions for a (3+1)-dimensional generalized B-type Kadomtsev-Petviashvili equation in fluid mechanics. Nonlinear Dyn 108, 1585–1598 (2022). https://doi.org/10.1007/s11071-022-07204-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-022-07204-0

Keywords

Search

Navigation