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Painlevé Analysis, Bäcklund Transformation, Lax Pair, Periodic- and Travelling-Wave Solutions for a Generalized (2+1)-Dimensional Hirota–Satsuma–Ito Equation in Fluid Mechanics

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Abstract

In this paper, under investigation is a generalized (2+1)-dimensional Hirota–Satsuma–Ito (HSI) equation in fluid mechanics. Motivated by its application in simulating the propagation of small-amplitude surface waves and shallow water waves, we focus on the Painlevé integrability, commonly used transformation forms and analytical solutions of the HSI equation. Via the Painlevé analysis, it is found that the HSI equation is Painlevé integrable under certain condition. Bilinear form, Bell-polynomial-type Bäcklund transformation and Lax pair are constructed with the binary Bell polynomials. One-periodic-wave solutions are derived via the Hirota–Riemann method and displayed graphically. Through the polynomial-expansion method, travelling-wave solutions are obtained.

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References

  1. Yusuf, A., Sulaiman, T.A.: Dynamics of Lump-periodic, breather and two-wave solutions with the long wave in shallow water under gravity and 2D nonlinear lattice. Commun. Nonlinear Sci. Numer. Simul. 99, 105846 (2021)

    Article  MathSciNet  Google Scholar 

  2. Baals, C., Moreno, A.G., Jiang, J., Benary, J., Ott, H.: Stability analysis and attractor dynamics of three-dimensional dark solitons with localized dissipation. Phys. Rev. A 103, 043304 (2021)

    Article  Google Scholar 

  3. Kruglov, V.I., Triki, H.: Periodic and solitary waves in an inhomogeneous optical waveguide with third-order dispersion and self-steepening nonlinearity. Phys. Rev. A 103, 013521 (2021)

    Article  MathSciNet  Google Scholar 

  4. Zhao, Z.L., He, L.C., Wazwaz, A.M.: Dynamics of lump chains for the BKP equation describing propagation of nonlinear waves. Chin. Phys. B 32, 040501 (2023)

    Article  Google Scholar 

  5. Masnadi, N., Duncan, J.H.: Observations of gravity-capillary lump interactions. J. Fluid Mech. 814, R1 (2017)

    Article  Google Scholar 

  6. Diorio, J., Cho, Y., Duncan, J.H., Akylas, T.R.: Gravity-capillary lumps generated by a moving pressure source. Phys. Rev. Lett. 103, 214502 (2009)

    Article  Google Scholar 

  7. Khan, Y., Faraz, N., Sulaimani, H. A.I.-: Two core optical fibers coupled nonlinear model in the framework of Hausdorff fractal derivative. Results Phys. 24, 104103 (2021)

    Article  Google Scholar 

  8. Houwe, A., Yakada, S., Abbagari, S., Saliou, Y., Inc, M., Doka, S.Y.: Survey of third- and fourth-order dispersions including ellipticity angle in birefringent fibers on W-shaped soliton solutions and modulation instability analysis. Eur. Phys. J. Plus 136, 357 (2021)

    Article  Google Scholar 

  9. Abbas, G., Kevrekidis, P.G., Allen, J.E., Koukouloyannis, V., Frantzeskakis, D.J., Karachalios, N.: Propagation of periodic wave trains along the magnetic field in a collision-free plasma. J. Phys. A 53, 425701 (2020)

    Article  MathSciNet  Google Scholar 

  10. Dudley, J.M., Genty, G., Mussot, A., Chabchoub, A., Dias, F.: Rogue waves and analogies in optics and oceanography. Nat. Rev. Phys. 1, 675–689 (2019)

    Article  Google Scholar 

  11. Cousins, W., Onorato, M., Chabchoub, A., Sapsis, T.P.: Predicting ocean rogue waves from point measurements: an experimental study for unidirectional waves. Phys. Rev. E 99, 032201 (2019)

    Article  Google Scholar 

  12. Dematteis, G., Grafke, T., Onorato, M., Vanden-Eijnden, E.: Experimental evidence of hydrodynamic instantons: the universal route to rogue waves. Phys. Rev. X 9, 041057 (2019)

    Google Scholar 

  13. Gao, X.Y.: Mathematical view with observational/experimental consideration on certain (2+1)-dimensional waves in the cosmic/laboratory dusty plasmas. Appl. Math. Lett. 91, 165–172 (2019)

    Article  MathSciNet  Google Scholar 

  14. Mir, A.A., Tiwari, S.K., Goree, J., Sen, A., Crabtree, C., Ganguli, G.: A forced Korteweg–de Vries model for nonlinear mixing of oscillations in a dusty plasma. Phys. Plasmas 27, 113701 (2020)

    Article  Google Scholar 

  15. Beji, S.: Kadomtsev–Petviashvili type equation for entire range of relative water depths. Coast. Eng. J. 60, 60–68 (2018)

    Article  Google Scholar 

  16. Chen, S.S., Tian, B., Liu, L., Yuan, Y.Q., Zhang, C.R.: Conservation laws, binary Darboux transformations and solitons for a higher-order nonlinear Schrödinger system. Chaos Soliton. Fract. 118, 337–346 (2019)

    Article  Google Scholar 

  17. Chekhovskoy, I., Medvedev, S.B., Vaseva, I.A., Sedov, E.V., Fedoruk, M.P.: Introducing phase jump tracking—a fast method for eigenvalue evaluation of the direct Zakharov–Shabat problem. Commun. Nonlinear Sci. Numer. Simul. 96, 105718 (2021)

    Article  MathSciNet  Google Scholar 

  18. Wazwaz, A.M., Kaur, L.: A new nonlinear integrable fifth-order equation: multiple soliton solutions with unusual phase shifts. Phys. Scr. 93, 115201 (2018)

    Article  Google Scholar 

  19. 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 Soliton. Fract. 134, 109709 (2020)

    Article  MathSciNet  Google Scholar 

  20. 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)

    Article  MathSciNet  Google Scholar 

  21. Zhao, Z.L., Yue, J., He, L.C.: New type of multiple lump and rogue wave solutions of the (2+1)-dimensional Bogoyavlenskii–Kadomtsev–Petviashvili equation. Appl. Math. Lett. 133, 108294 (2022)

    Article  MathSciNet  Google Scholar 

  22. Long, F., Alsallami, S.A.M., Rezaei, S., Nonlaopon, K., Khalil, E.M.: New interaction solutions to the (2+1)-dimensional Hirota–Satsuma–Ito equation. Z. Results Phys. 37, 105475 (2022)

    Article  Google Scholar 

  23. Ma, W.X., Li, J., Khalique, C.M.: A study on lump solutions to a generalized Hirota–Satsuma–Ito equation in (2+1)-dimensions. Complexity 2018, 905958 (2018)

    Article  Google Scholar 

  24. Kuo, C.K., Ma, W.X.: A study on resonant multi-soliton solutions to the (2+1)-dimensional Hirota–Satsuma–Ito equations via the linear superposition principle. Nonlinear Anal. 190, 111592 (2020)

    Article  MathSciNet  Google Scholar 

  25. Zhao, X., Tian, B., Du, X.X., Hu, C.C., Liu, S.H.: Bilinear Bäcklund transformation, kink and breather-wave solutions for a generalized (2+1)-dimensional Hirota–Satsuma–Ito equation in fluid mechanics. Eur. Phys. J. Plus 136, 159 (2021)

    Article  Google Scholar 

  26. Wang, M., Tian, B., Liu, S.H., Shan, W.R., Jiang, Y.: Soliton, multiple-lump and hybrid solutions of a (2+1)-dimensional generalized Hirota–Satsuma–Ito equation for the water waves. Eur. Phys. J. Plus 136, 635 (2021)

    Article  Google Scholar 

  27. Aliyu, A.I., Li, Y.: Bell polynomials and lump-type solutions to the Hirota–Satsuma–Ito equation under general and positive quadratic polynomial functions. Eur. Phys. J. Plus 135, 119 (2020)

    Article  Google Scholar 

  28. Chen, X., Liu, Y.Q., Zhuang, J.H.: Soliton solutions and their degenerations in the (2+1)-dimensional Hirota–Satsuma–Ito equations with time-dependent linear phase speed. Nonlinear Dyn. 111, 10367–10380 (2023)

    Article  Google Scholar 

  29. Liu, W., Wazwaz, A.M., Zheng, X.: High-order breathers, lumps, and semi-rational solutions to the (2+1)-dimensional Hirota–Satsuma–Ito equation. Phys. Scr. 94, 075203 (2019)

    Article  Google Scholar 

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

    Book  Google Scholar 

  31. Ablowitz, M.J., Segur, H.: Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38, 1103 (1977)

    Article  MathSciNet  Google Scholar 

  32. Xu, G.Q., Liu, Y.P., Cui, W.Y.: Painlevé analysis, integrability property and multiwave interaction solutions for a new (4+1)-dimensional KdV–Calogero–Bogoyavlenkskii–Schiff equation. Appl. Math. Lett. 132, 108184 (2022)

    Article  Google Scholar 

  33. Huang, Q.M., Gao, Y.T., Jia, S.L., Wang, Y.L., Deng, G.F.: Bilinear Bäcklund transformation, soliton and periodic wave solutions for a (3+1)-dimensional variable-coefficient generalized shallow water wave equation. Nonlinear Dyn. 87, 2529–2540 (2017)

    Article  Google Scholar 

  34. Xu, G.Q., Wazwaz, A.M.: A new (n+1)-dimensional generalized Kadomtsev–Petviashvili equation: integrability characteristics and localized solutions. Nonlinear Dyn. 111, 9495–9507 (2023)

    Article  Google Scholar 

  35. Qin, B., Tian, B., Wang, Y.F., Shen, Y.J., Wang, M.: Bell-polynomial approach and Wronskian determinant solutions for three sets of differential-difference nonlinear evolution equations with symbolic computation. Z. Angew. Math. Phys. 68(5), 111 (2017)

    Article  MathSciNet  Google Scholar 

  36. Tian, S., Zhang, H.: Riemann theta functions periodic wave solutions and rational characteristics for the nonlinear equations. J. Math. Anal. Appl. 371(2), 585–608 (2010)

    Article  MathSciNet  Google Scholar 

  37. Shen, Y., Tian, B., Zhang, C.R., Tian, H.Y., Liu, S.H.: Breather-wave, periodic-wave and traveling-wave solutions for a (2+1)-dimensional extended Boiti–Leon–Manna–Pempinelli equation for an incompressible fluid. Mod. Phys. Lett. B 35(15), 2150261 (2021)

    Article  MathSciNet  Google Scholar 

  38. Zhao, Z.L., He, L.C.: Bäcklund transformations and Riemann–Bäcklund method to a (3+1)-dimensional generalized breaking soliton equation. Eur. Phys. J. Plus 135, 639 (2020)

    Article  Google Scholar 

  39. Hu, W.Q., Gao, Y.T., Zhao, C., Jia, S.L., Lan, Z.Z.: Breathers, quasi-periodic and travelling waves for a generalized (3+1)-dimensional Yu–Toda–Sasa–Fukayama equation in fluids. Wave. Random Complex 27, 458–481 (2017)

    Article  Google Scholar 

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Acknowledgements

We express our sincere thanks to all the members of our discussion group for their valuable comments. This work has been supported by the National Natural Science Foundation of China under Grant No. 11272023, and by the Fundamental Research Funds for the Central Universities.

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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

    Dong Wang, Yi-Tian Gao, Xin Yu, Gao-Fu Deng & Fei-Yan Liu

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  1. Dong Wang
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  2. Yi-Tian Gao
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  3. Xin Yu
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Correspondence to Yi-Tian Gao or Xin Yu.

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Wang, D., Gao, YT., Yu, X. et al. Painlevé Analysis, Bäcklund Transformation, Lax Pair, Periodic- and Travelling-Wave Solutions for a Generalized (2+1)-Dimensional Hirota–Satsuma–Ito Equation in Fluid Mechanics. Qual. Theory Dyn. Syst. 23, 12 (2024). https://doi.org/10.1007/s12346-023-00850-8

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