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Pfaffian, soliton, breather and hybrid solutions for a (2+1)-dimensional combined potential Kadomtsev-Petviashvili-B-type Kadomtsev-Petviashvili equation in fluid mechanics

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Abstract

In this paper, we study a (2+1)-dimensional combined potential Kadomtsev-Petviashvili-B-type Kadomtsev-Petviashvili equation in fluid mechanics. The Nth-order Pfaffian solutions are constructed and proved via the modified Pfaffian technique, where N is a positive integer. Soliton, breather and hybrid solutions are worked out based on the Nth-order Pfaffian solutions. Interaction between the two solitons is classified into the elastic interaction, inelastic interaction and soliton resonance via the asymptotic analysis. For the two solitons, elastic interactions, inelastic interactions and soliton resonances are discussed graphically. Breather, lump and hybrid solutions composed of the breathers, lumps and solitons are illustrated graphically.

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References

  1. Tamang, J., Saha, A.: Bifurcations of small-amplitude supernonlinear waves of the mKdV and modified Gardner equations in a three-component electron-ion plasma. Phys. Plasmas 27, 012105 (2020)

    Article  Google Scholar 

  2. Yang, D.Y., Tian, B., Tian, H.Y., et al.: Darboux transformation, localized waves and conservation laws for an M-coupled variable-coefficient nonlinear Schrödinger system in an inhomogeneous optical fiber. Chaos Solitons Fract. 156, 111719 (2022)

    Article  Google Scholar 

  3. Moleleki, L.D., Simbanefayi, I., Khalique, C.M.: Symmetry solutions and conservation laws of a (3+1)-dimensional generalized KP-Boussinesq equation in fluid mechanics. Chin. J. Phys. 68, 940–949 (2020)

    Article  MathSciNet  Google Scholar 

  4. Cheng, C.D., Tian, B., Ma, Y.X., et al.: Pfaffian, breather and hybrid solutions for a (2+1)-dimensional generalized nonlinear system in fluid mechanics and plasma physics. Phys. Fluids 34, 115132 (2022)

  5. Shen, Y., Tian, B., Liu, S.H., et al.: Studies on certain bilinear form, N-soliton, higher-order breather, periodic-wave and hybrid solutions to a (3+1)-dimensional shallow water wave equation with time-dependent coefficients. Nonlinear Dyn. 108, 2447–2460 (2022)

    Article  Google Scholar 

  6. Gao, X.Y., Guo, Y.J., Shan, W.R.: Reflecting upon some electromagnetic waves in a ferromagnetic film via a variable-coefficient modified Kadomtsev-Petviashvili system. Appl. Math. Lett. 132, 108189 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  7. Liu, F.Y., Gao, Y.T., Yu, X., et al.: Wronskian, Gramian, Pfaffian and periodic-wave solutions for a (3+1)-dimensional generalized nonlinear evolution equation arising in the shallow water waves. Nonlinear Dyn. 108, 1599–1616 (2022)

  8. Gao, X.T., Tian, B., Shen, Y., et al.: Considering the shallow water of a wide channel or an open sea through a generalized (2+1)-dimensional dispersive long-wave system. Qual. Theory Dyn. Syst. 21, 104 (2022)

  9. Yang, D.Y., Tian, B., Hu, C.C., et al.: Conservation laws and breather-to-soliton transition for a variable-coefficient modified Hirota equation in an inhomogeneous optical fiber. Wave. Random Complex (2022) in press, https://doi.org/10.1080/17455030.2021.1983237

  10. Shen, Y., Tian, B., Zhou, T.Y., et al.: N-fold Darboux transformation and solitonic interactions for the Kraenkel-Manna-Merle system in a saturated ferromagnetic material. Nonlinear Dyn. (2022) in press, https://doi.org/10.1007/s11071-022-07959-6

  11. Wu, X.H., Gao, Y.T., Yu, X., et al.: Modified generalized Darboux transformation, degenerate and bound-state solitons for a Laksmanan-Porsezian-Daniel equation. Chaos Solitons Fract. 162, 112399 (2022)

  12. Zhou, T.Y., Tian, B.: Auto-Bäcklund transformations, Lax pair, bilinear forms and bright solitons for an extended (3+1)-dimensional nonlinear Schrödinger equation in an optical fiber. Appl. Math. Lett. 133, 108280 (2022)

  13. Wang, M., Tian, B., Zhou, T.Y.: Darboux transformation, generalized Darboux transformation and vector breathers for a matrix Lakshmanan-Porsezian-Daniel equation in a Heisenberg ferromagnetic spin chain. Chaos Solitons Fract. 152, 111411 (2021)

  14. Zhou, T.Y., Tian, B., Chen, Y.Q., et al.: Studies on certain bilinear form, N-soliton, higher-order breather, periodic-wave and hybrid solutions to a (3+1)-dimensional shallow water wave equation with time-dependent coefficients Nonlinear Dyn. 108, 2417-2428 (2022)

  15. Wazwaz, A.M., Albalawi, W., El-Tantawy, S.A.: Optical envelope soliton solutions for coupled nonlinear Schrödinger equations applicable to high birefringence fibers. Optik 255, 168673 (2022)

    Article  Google Scholar 

  16. Zhang, R.F., Li, M.C.: Bilinear residual network method for solving the exactly explicit solutions of nonlinear evolution equations. Nonlinear Dyn. 108, 521–531 (2022)

    Article  Google Scholar 

  17. Zhang, R.F., Bilige, S., Liu, J.G., et al.: Bright-dark solitons and interaction phenomenon for p-gBKP equation by using bilinear neural network method. Phys. Scr. 96, 025224 (2020)

    Article  Google Scholar 

  18. Qiao, J.M., Zhang, R.F., Yue, R.X., et al.: Three types of periodic solutions of new (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation via bilinear neural network method. Math. Meth. Appl. Sci. 45, 5612–5621 (2022)

    Article  MathSciNet  Google Scholar 

  19. 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)

    Article  Google Scholar 

  20. 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)

    Article  MATH  Google Scholar 

  21. Gao, X.Y., Guo, Y.J., Shan, W.R.: Regarding the shallow water in an ocean via a Whitham-Broer-Kaup-like system: hetero-Bäcklund transformations, bilinear forms and M solitons. Chaos Solitons Fract. 162, 112486 (2022)

  22. Helal, M.A.: Soliton solution of some nonlinear partial differential equations and its applications in fluid mechanics. Chaos Solitons Fract. 13, 1917–1929 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Gao, X.T., Tian, B., Feng, C.H.: In oceanography, acoustics and hydrodynamics: investigations on an extended coupled (2+1)-dimensional Burgers system. Chin. J. Phys. 77, 2818–2824 (2022)

    Article  MathSciNet  Google Scholar 

  24. Yang, D.Y., Tian, B., Hu, C.C., et al.: The generalized Darboux transformation and higher-order rogue waves for a coupled nonlinear Schrödinger system with the four-wave mixing terms in a birefringent fiber. Eur. Phys. J. Plus 137, 1213 (2022)

  25. Gao, X.Y., Guo, Y.J., Shan, W.R.: Bilinear auto-Bäcklund transformations and similarity reductions for a (3+1)-dimensional generalized Yu-Toda-Sasa-Fukuyama system in fluid mechanics and lattice dynamics. Qual. Theory Dyn. Syst. 21, 95 (2022)

  26. Wu, X.H., Gao, Y.T., Yu, X., et al.: N-fold generalized Darboux transformation and soliton interactions for a three-wave resonant interaction system in a weakly nonlinear dispersive medium. Chaos Solitons Fract. 165, 112786 (2022)

  27. Shen, Y., Tian, B., Zhou, T.Y., et al.: Nonlinear differential-difference hierarchy relevant to the Ablowitz-Ladik equation: Lax pair, conservation laws, N-fold Darboux transformation and explicit exact solutions. Chaos Silotons Fract. 164, 112460 (2022)

  28. Wu, X.H., Gao, Y.T., Yu, X., et al.: Binary Darboux transformation, solitons, periodic waves and modulation instability for a nonlocal Lakshmanan-Porsezian-Daniel equation. Wave Motion 114, 103036 (2022)

  29. Zhou, T.Y., Tian, B., Zhang, C.R., Liu, S.H.: Auto-Bäcklund transformations, bilinear forms, multiple-soliton, quasi-soliton and hybrid solutions of a (3+1)-dimensional modified Korteweg-de Vries-Zakharov-Kuznetsov equation in an electron-positron plasma. Eur. Phys. J. Plus 137, 912 (2022)

  30. Gao, X.T., Tian, B.: Water-wave studies on a (2+1)-dimensional generalized variable-coefficient Boiti-Leon-Pempinelli system. Appl. Math. Lett. 128, 107858 (2022)

  31. Guo, R., Hao, H.Q., Zhang, L.L.: Dynamic behaviors of the breather solutions for the AB system in fluid mechanics. Nonlinear Dyn. 74, 701–709 (2013)

    Article  MathSciNet  Google Scholar 

  32. Wazwaz, A.M.: New (3+1)-dimensional Painlevé integrable fifth-order equation with third-order temporal dispersion. Nonlinear Dyn. 106, 891–897 (2021)

    Article  Google Scholar 

  33. Wazwaz, A.M., El-Tantawy, S.A.: Bright and dark optical solitons for (3+1)-dimensional hyperbolic nonlinear Schrödinger equation using a variety of distinct schemes. Optik 270, 170043 (2022)

    Article  Google Scholar 

  34. Zhang, R.F., Li, M.C., Gan, J.Y., et al.: Novel trial functions and rogue waves of generalized breaking soliton equation via bilinear neural network method. Chaos Solitons Fract. 154, 111692 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zhang, R.F., Li, M.C., Albishari, M., et al.: 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 

  36. Zhang, R.F., Bilige, S., Fang, T., et al.: New periodic wave, cross-kink wave and the interaction phenomenon for the Jimbo-Miwa-like equation. Comput. Math. Appl. 78, 754–764 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  37. Ma, W.X.: \(N\)-soliton solution of a combined pKP-BKP equation. J. Geom. Phys. 165, 104191 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  38. Ma, Z.Y., Fei, J.X., Cao, W.P., et al.: The explicit solution and its soliton molecules in the (2+1)-dimensional pKP-BKP equation. Results Phys. 35, 105363 (2022)

    Article  Google Scholar 

  39. Feng, Y.Y., Bilige, S.: Resonant multi-soliton, \(M\)-breather, \(M\)-lump and hybrid solutions of a combined pKP-BKP equation. J. Geom. Phys. 169, 104322 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  40. Liang, Y.Q., Wei, G.M., Li, X.N.: Painlevé integrability, similarity reductions, new soliton and soliton-like similarity solutions for the (2+1)-dimensional BKP equation. Nonlinear Dyn. 62, 195–202 (2021)

    Article  MATH  Google Scholar 

  41. Hu, W.Q., Gao, Y.T., Jia, S.L.: 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)

    Article  Google Scholar 

  42. Peng, W.Q., Tian, S.F., Zhang, T.T.: Analysis on lump, lumpoff and rogue waves with predictability to the (2+1)-dimensional B-type Kadomtsev-Petviashvili equation. Phys. Lett. A 382, 2701–2708 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  43. Hirota, R.: The Direct Method in Soliton Theory. Cambridge Univ. Press, New York (2004)

    Book  MATH  Google Scholar 

  44. Wei, G.M., Gao, Y.T., Xu, T., et al.: Painlevé Property and New Analytic Solutions for a Variable-Coecient Kadomtsev-Petviashvili Equation with Symbolic Computation. Chin. Phys. Lett. 25, 1599–1602 (2008)

    Article  Google Scholar 

  45. Peng, W.Q., Tian, S.F., Zhang, T.T.: On the breather waves, rogue waves and solitary waves to a generalized (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada equation. Filomat 32, 4959–4969 (2018)

  46. Gupta, A.K., Saha Ray, S.: Numerical treatment for the solution of fractional fifth-order Sawada-Kotera equation using second kind Chebyshev wavelet method. Appl. Math. Model. 39, 5121–5130 (2015)

  47. Wazwaz, A.M.: Abundant solitons solutions for several forms of the fifth-order KdV equation by using the tanh method. Appl. Math. Comput. 182, 283–300 (2006)

  48. Wazwaz, A.M.: Construction of solitary wave solutions and rational solutions for the KdV equation by Adomian decomposition method. Chaos Solitons Fract. 12, 2283–2293 (2001)

  49. Hu, L., Gao, Y.T., Jia, T.T., et al.: 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)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We express our sincere thanks to the Editors, Reviewers and members of our discussion group for their valuable suggestions.

Funding

This project was funded by the National Natural Science Foundation of China under Grant No. 11772017.

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Authors and Affiliations

  1. Ministry-of-Education Key Laboratory of Fluid Mechanics and National Laboratory for Computational Fluid Dynamics, Beijing University of Aeronautics and Astronautics, Beijing, 100191, China

    Fei-Yan Liu, Yi-Tian Gao & Xin Yu

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  1. Fei-Yan Liu
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Liu, FY., Gao, YT. & Yu, X. Pfaffian, soliton, breather and hybrid solutions for a (2+1)-dimensional combined potential Kadomtsev-Petviashvili-B-type Kadomtsev-Petviashvili equation in fluid mechanics. Nonlinear Dyn 111, 5681–5692 (2023). https://doi.org/10.1007/s11071-022-08111-0

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