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Higher-order hybrid waves for the (2 + 1)-dimensional Boiti–Leon–Manna–Pempinelli equation for an irrotational incompressible fluid via the modified Pfaffian technique

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Abstract

Fluids, as a phase of matter including liquids, gases and plasmas, are seen to be common in nature, the study of which helps the design in the related industries. In this paper, we optimize the Pfaffian technique and investigated the Boiti–Leon–Manna–Pempinelli equation for an irrotational incompressible fluid. Higher-order hybrid solutions consisting of the L lumps, M breathers and N solitons are constructed with L, M and N being positive integers. Relative extrema of the breather and lump are presented, respectively. Breather is found to be localized along the curve \(a_1 x+b_1 \varphi (y)+\omega _1 t+\xi _1=0\) and periodic along the curve \(\alpha _1 x+\beta _1 \varphi (y)+\gamma _1 t+\theta _1=0\). Under the lump existence condition, higher-order rogue wave solutions do not exist. Hybrid solutions composed of breathers, lumps and solitons are illustrated graphically. It can be found that when certain parameters are chosen, the breather, lump and soliton included in the hybrid solutions possess the same properties as those of the breather and lump solutions.

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References

  1. Gibis, T., Wenzel, C., Kloker, M., Rist, U.: Self-similar compressible turbulent boundary layers with pressure gradients. Part 2. Self-similarity analysis of the outer layer. J. Fluid Mech. 880, 284 (2019)

    Article  MathSciNet  Google Scholar 

  2. Downer, M.C., Zgadzaj, R., Debus, A., Schramm, U., Kaluza, M.C.: Diagnostics for plasma-based electron accelerators. Rev. Mod. Phys. 90, 035002 (2018)

    Article  MathSciNet  Google Scholar 

  3. Wang, W., Kim, H.H., Van Laer, K., Bogaerts, A.: Streamer propagation in a packed bed plasma reactor for plasma catalysis applications. Chem. Eng. J. 334, 2467 (2018)

    Article  Google Scholar 

  4. Yao, J., Hussain, F.: Supersonic turbulent boundary layer drag control using spanwise wall oscillation. J. Fluid Mech. 880, 388 (2019)

    Article  MathSciNet  Google Scholar 

  5. Dou, S., Tao, L., Wang, R., Hankari, S.E., Chen, R., Wang, S.: Plasma-assisted synthesis and surface modification of electrode materials for renewable energy. Adv. Mater. 30, 1705850 (2018)

    Article  Google Scholar 

  6. Estkvez, P.G., Leblet, S.: A wave equation in 2+1: Painlevé analysis and solutions. Inverse Probl. 11, 925 (1995)

    Article  Google Scholar 

  7. Li, B.Q., Ma, Y.L.: Multiple-lump waves for a (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation arising from incompressible fluid. Comput. Math. Appl. 76, 204 (2018)

    Article  MathSciNet  Google Scholar 

  8. Lou, S.Y.: Conformal invariance and integrable models. J. Phys. A: Math. Gen. 30, 4803 (1997)

    Article  MathSciNet  Google Scholar 

  9. Lou, S.Y.: Generalized dromion solutions of the (2+1)-dimensional KdV equation. J. Phys. A: Math. Gen. 28, 7227 (1995)

    Article  MathSciNet  Google Scholar 

  10. Tang, X.Y.: What will happen when a dromion meets with a ghoston? Phys. Lett. A 314, 286 (2003)

    Article  MathSciNet  Google Scholar 

  11. Ma, S.H., Fang, J.P.: Multi Dromion–Solitoff and fractal excitations for (2+1)-dimensional Boiti–Leon–Manna–Pempinelli system. Commun. Theor. Phys. 52, 641 (2009)

    Article  Google Scholar 

  12. Luo, L.: Quasi-periodic waves and asymptotic property for Boiti–Leon–Manna–Pempinelli equation. Commun. Theor. Phys. 54, 208 (2010)

    Article  MathSciNet  Google Scholar 

  13. Darvishi, M., Najafi, M.: Stair and step soliton solutions of the integrable (2+1) and (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equations. Commun. Theor. Phys. 58, 785 (2012)

    Article  Google Scholar 

  14. Luo, L.: New exact solutions and Bäcklund transformation for Boiti–Leon–Manna–Pempinelli equation. Phys. Lett. A 375, 1059 (2011)

    Article  MathSciNet  Google Scholar 

  15. Tang, Y., Zai, W.: New periodic-wave solutions for (2+1)- and (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equations. Nonlinear Dyn. 81, 249 (2015)

    Article  MathSciNet  Google Scholar 

  16. Delisle, L., Mosaddeghi, M.: Classical and SUSY solutions of the Boiti–Leon–Manna–Pempinelli equation. J. Phys. A: Math. Theor. 46, 115203 (2013)

    Article  MathSciNet  Google Scholar 

  17. Wang, C.: Spatiotemporal deformation of lump solution to (2+1)-dimensional KdV equation. Nonlinear Dyn. 84, 697 (2016)

    Article  MathSciNet  Google Scholar 

  18. Li, Y., Li, D.: New exact solutions for the (2+1)-dimensional Boiti–Leon–Manna–Pempinelli equation. Appl. Math. Sci. 6, 579 (2012)

    MathSciNet  MATH  Google Scholar 

  19. Kaplan, M., Akbulut, A., Bekir, A.: The Auto-Bäcklund transformations for the (2+1)-dimensional Boiti–Leon–Manna–Pempinelli equation. AIP Conf. Pro. 1798, 020071 (2017)

    Article  Google Scholar 

  20. Najafi, M., Najafi, M., Arbabi, S.: Wronskian determinant solutions of the (2+1)-dimensional Boiti–Leon–Manna–Pempinelli equation. Int. J. Adv. Math. Sci. 1, 8 (2013)

    Google Scholar 

  21. Kaplan, M.: Two different systematic techniques to find analytical solutions of the (2+1)-dimensional Boiti–Leon–Manna–Pempinelli equation. Chin. J. Phys. 56, 2523 (2018)

    Article  MathSciNet  Google Scholar 

  22. 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)

    Article  MathSciNet  Google Scholar 

  23. Ma, W.X., Abdeljabbar, A., Asaad, M.G.: Wronskian and Grammian solutions to a (3+1)-dimensional generalized KP equation. Appl. Math. Comput. 217, 10016 (2011)

    MathSciNet  MATH  Google Scholar 

  24. Chen, S.T., Ma, W.X.: Lump solutions of a generalized Calogero–Bogoyavlenskii–Schiff equation. Comput. Math. Appl. 76, 1680 (2018)

    Article  MathSciNet  Google Scholar 

  25. Ma, W.X.: \(N\)-soliton solutions and the Hirota conditions in (2+1)-dimensions. Opt. Quantum Electron. 52, 511 (2020)

    Article  Google Scholar 

  26. Ma, W.X.: Symbolic computation of lump solutions to a combined equation involving three types of nonlinear terms. East Asian J. Appl. Math. 10, 732 (2020)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  28. Gao, X.Y., Guo, Y.J., Shan, W.R.: Water-wave symbolic computation for the Earth, Enceladus and Titan: the higher-order Boussinesq-Burgers system, auto- and non-auto-Bäcklund transformations. Appl. Math. Lett. 104, 106170 (2020)

  29. Gao, X.Y., Guo, Y.J., Shan, W.R.: Shallow water in an open sea or a wide channel: auto- and non-auto-Bäcklund transformations with solitons for a generalized (2+1)-dimensional dispersive long-wave system. Chaos Solitons Fract. 138, 109950 (2020)

  30. Zhang, C.R., Tian, B., Qu, Q.X., Liu, L., Tian, H.Y.: Vector bright solitons and their interactions of the couple Fokas-Lenells system in a birefringent optical fiber. Z. Angew. Math. Phys. 71, 18 (2020)

  31. Zhang, C.R., Tian, B., Sun, Y., Yin, H.M.: Binary Darboux transformation and vector-soliton-pair interactions with the negatively coherent coupling in a weakly birefringent fiber. EPL 127, 40003 (2019)

  32. Du, X.X., Tian, B., Yuan, Y. Q., Du, Z.: Symmetry reductions, group-invariant solutions, and conservation laws of a (2+1)-dimensional nonlinear Schrodinger equation in a Heisenberg ferromagnetic spin chain. Ann. Phys. (Berlin) 531, 1900198 (2019)

  33. Du, X.X., Tian, B., Qu, Q.X., Yuan, Y.Q., Zhao, X.H.: Lie group analysis, solitons, self-adjointness and conservation laws of the modified Zakharov-Kuznetsov equation in an electron-positron-ion magnetoplasma. Chaos Solitons Fract. 134, 109709 (2020)

  34. Chen, S.S., Tian, B., Chai, J., Wu, X.Y., Du, Z.: Lax pair, binary Darboux transformations and dark-soliton interaction of a fifth-order defocusing nonlinear Schrodinger equation for the attosecond pulses in the optical fiber communication. Wave. Random Complex 30, 389–402 (2020)

  35. Chen, S.S., Tian, B., Sun, Y., Zhang, C.R.: Generalized Darboux Transformations, Rogue Waves, and Modulation Instability for the Coherently Coupled Nonlinear Schrodinger Equations in Nonlinear Optics. Ann. Phys. (Berlin) 531, 1900011 (2019)

  36. Hu, C.C., Tian, B., Yin, H.M., Zhang, C.R., Zhang, Z.: Dark breather waves, dark lump waves and lump wave-soliton interactions for a (3+1)-dimensional generalized Kadomtsev-Petviashvili equation in a fluid. Comput. Math. Appl. 78, 166–177 (2019)

    Article  MathSciNet  Google Scholar 

  37. Tian, H.Y., Tian, B., Yuan, Y.Q., Zhang, C.R.: Superregular solutions for a coupled nonlinear Schrödinger system in a two-mode nonlinear fiber. Phys. Scr. 96, 045213 (2021)

  38. Wang, M., Tian, B., Sun, Y., Zhang, Z.: Lump, mixed lump-stripe and rogue wave-stripe solutions of a (3+1)-dimensional nonlinear wave equation for a liquid with gas bubbles. Comput. Math. Appl. 79, 576 (2020)

  39. Wang, M., Tian, B., Hu, C.C., Liu, S.H.: Generalized Darboux transformation, solitonic interactions and bound states for a coupled fourth-order nonlinear Schrodinger system in a birefringent optical fiber. Appl. Math. Lett. (2021). https://doi.org/10.1016/j.aml.2020.106936

  40. Yang, D.Y., Tian, B., Qu, Q. X., Zhang, C.R., Chen, S.S., Wei, C.C.: Lax pair, conservation laws, Darboux transformation and localized waves of a variable-coefficient coupled Hirota system in an inhomogeneous optical fiber. Chaos Solitons Fract. (2021). https://doi.org/10.1016/j.chaos.2020.110487

  41. Yang, D.Y., Tian, B., Qu, Q.X., Li, H., Zhao, X.H., Chen, S.S., Wei, C.C.: Darboux-dressing transformation, semi-rational solutions, breathers and modulation instability for the cubic-quintic nonlinear Schrodiger system with variable coefficients in a non-Kerr medium, twin-core nonlinear optical fiber or waveguide. Phys. Scr. 96, 045210 (2021)

  42. Zhao, X., Tian, B., Qu, Q.X., Yuan, Y.Q., Du, X.X., Chu, M.X.: Dark-dark solitons for the coupled spatially modulated Gross-Pitaevskii system in the Bose-Einstein condensation. Mod. Phys. Lett. B 34, 2050282 (2020)

  43. Zhao, X., Tian, B., Tian, H.Y., Yang, D.Y.: Bilinear Backlund transformation, Lax pair and interactions of nonlinear waves for a generalized (2+1)-dimensional nonlinear wave equation in nonlinear optics/fluid mechanics/plasma physics. Nonlinear Dyn. 103, 1785–1794 (2021)

  44. Chen,Y. Q., Tian, B., Qu, Q.X., Li, H., Zhao, X.H., Tian, H.Y., Wang, M.: Ablowitz-Kaup-Newell-Segur system, conservation laws and Backlund transformation of a variable-coefficient Korteweg-de Vries equation in plasma physics, fluid dynamics or atmospheric science. Int. J. Mod. Phys. B 34, 2050226 (2020)

  45. Chen, Y. Q., Tian, B., Qu, Q.X., Li, H., Zhao, X.H., Tian, H.Y., Wang, M.: Reduction and analytic solutions of a variable-coefficient Korteweg-de Vries equation in a fluid, crystal or plasma, Mod. Phys. Lett. B 34, 2050287 (2020)

  46. Hirota, R.: The Direct Method in Soliton Theory. Cambridge University Press, Cambridge (2004)

    Book  Google Scholar 

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Acknowledgements

The authors express their sincere thanks to the members of their discussion group for their valuable suggestions. This work has been supported by the National Natural Science Foundation of China under Grant No. 11772017, and by the Fundamental Research Funds for the Central Universities under Grant No. 50100002016105010.

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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

    Lei Hu, Yi-Tian Gao, Ting-Ting Jia, Gao-Fu Deng & Liu-Qing Li

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  1. Lei Hu
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Hu, L., Gao, YT., Jia, TT. 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). https://doi.org/10.1007/s00033-021-01482-1

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