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Rogue-wave, rational and semi-rational solutions for a generalized (3+1)-dimensional Yu-Toda-Sasa-Fukuyama equation in a two-layer fluid

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Abstract

Two-layer-fluid models are used to describe certain nonlinear phenomena in medical science and fluid mechanics. Under investigation in this paper is a generalized (3+1)-dimensional Yu-Toda-Sasa-Fukuyama equation for the interfacial waves in a two-layer fluid. Rogue-wave, rational and semi-rational solutions are given via the Kadomtsev-Petviashvili hierarchy reduction. We discuss the influence of the coefficients in that equation on the semi-rational solutions. For the first-order semi-rational solutions, we derive that: (1) when \(h_{0}>0\), the lump catches up with the soliton, and then the lump merges into the soliton; when \(h_{0}<0\), the lump appears from the soliton and then separates from the soliton; (2) the amplitudes of the soliton and lump decrease with \(h_1\) decreasing; (3) the amplitudes of the soliton and lump decrease with \(h_2\) increasing; (4) the lump becomes narrower with \(h_4\) decreasing, where \(h_{0},~h_{1},~h_{2}\) and \(h_{4}\) are the constant coefficients in that equation.

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Data Availability Statement

Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.

References

  1. Abd Elmaboud, Y., Abdelsalam, S.I., Mekheimer, K.S., et al.: Electromagnetic flow for two-layer immiscible fluids. Eng. Sci. Technol. 22, 237–248 (2019)

    Google Scholar 

  2. Sudhakar, S., Weibel, J.A., Zhou, F., et al.: Area-scalable high-heat-flux dissipation at low thermal resistance using a capillary-fed two-layer evaporator wick. Int. J. Heat Mass Transfer 135, 1346–1356 (2019)

    Google Scholar 

  3. 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)

    MATH  Google Scholar 

  4. 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)

    Google Scholar 

  5. Ma, H.C., Yue, S.P., Deng, A.P.: Nonlinear superposition between lump and other waves of the (2+1)-dimensional generalized Caudrey-Dodd-Gibbon-Kotera-Sawada equation in fluid dynamics. Nonlinear Dyn. 109, 1969–1983 (2022)

    Google Scholar 

  6. Liu, F.Y., Gao, Y.T., Yu, X., Ding, C.C.: 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)

    Google Scholar 

  7. Han, P.F., Bao, T.: New periodic solitary wave solutions for the (3+1)-dimensional generalized shallow water equation. Nonlinear Dyn. 108, 2513–2530 (2022)

    Google Scholar 

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

    Article  Google Scholar 

  9. Kumar, S., Kumar, D., Wazwaz, A.M.: Lie symmetries, optimal system, group-invariant solutions and dynamical behaviors of solitary wave solutions for a (3+1)-dimensional KdV-type equation. Eur. Phys. J. Plus 136, 531 (2021)

    Google Scholar 

  10. Kumar, S., Niwasby, M., Wazwaz, A.M.: Lie symmetry analysis, exact analytical solutions and dynamics of solitons for (2+1)-dimensional NNV equations. Phys. Scr. 95, 095204 (2020)

    Google Scholar 

  11. Kumar, S., Kumar, D., Wazwaz, A.M.: Group invariant solutions of (3+1)-dimensional generalized B-type Kadomstsev Petviashvili equation using optimal system of Lie subalgebra. Phys. Scr. 94, 065204 (2019)

    Google Scholar 

  12. Kumar, S., Wazwaz, A.M., Kumar, D., et al.: Group invariant solutions of (2+1)-dimensional rdDym equation using optimal system of Lie subalgebra. Phys. Scr. 94, 115202 (2019)

    Google Scholar 

  13. Kumar, S., Mohan, B., Kumar, R.: Lump, soliton, and interaction solutions to a generalized two-mode higher-order nonlinear evolution equation in plasma physics. Nonlinear Dyn. 110, 693–704 (2022)

    Google Scholar 

  14. Kumar, S., Mohan, B., Kumar, A.: Generalized fifth-order nonlinear evolution equation for the Sawada-Kotera, Lax, and Caudrey-Dodd-Gibbon equations in plasma physics: Painlevé analysis and multi-soliton solutions. Phys. Scr. 97, 035201 (2022)

    Google Scholar 

  15. Kumar, S., Rani, S.: Invariance analysis, optimal system, closed-form solutions and dynamical wave structures of a (2+1)-dimensional dissipative long wave system. Phys. Scr. 96, 125202 (2021)

    Google Scholar 

  16. 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)

    MATH  Google Scholar 

  17. Kumar, S., Rani, S.: Symmetries of optimal system, various closed-form solutions, and propagation of different wave profiles for the Boussinesq-Burgers system in ocean waves. Phys. Fluids 34, 037109 (2022)

    Google Scholar 

  18. Wu, X.H., Gao, Y.T., Yu, X., Ding, C.C.: 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)

    Google Scholar 

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

  20. 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)

  21. Gao, X.T., Tian, B., Shen, Y., Feng, C.H.: 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)

    MATH  Google Scholar 

  22. 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)

    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)

    Google Scholar 

  24. Kumar, S., Kumar, A., Kharbanda, H.: Lie symmetry analysis and generalized invariant solutions of (2+1)-dimensional dispersive long wave (DLW) equations. Phys. Scr. 95, 065207 (2020)

    Google Scholar 

  25. Kumar, S., Kumar, A., Mohan, B.: Evolutionary dynamics of solitary wave profiles and abundant analytical solutions to a (3+1)-dimensional burgers system in ocean physics and hydrodynamics. J. Ocean Eng. Sci. (2021). https://doi.org/10.1016/j.joes.2021.11.002

    Article  Google Scholar 

  26. Gao, X.Y., Guo, Y.J., Shan, W.R.: Auto-Bäcklund transformation, similarity reductions and solitons of an extended (2+1)-dimensional coupled Burgers system in fluid mechanics. Qual. Theory Dyn. Syst. 21, 60 (2022)

    MATH  Google Scholar 

  27. Kumar, S., Kumar, A.: Lie symmetry reductions and group invariant solutions of (2+1)-dimensional modified Veronese web equation. Nonlinear Dyn. 98, 1891–1903 (2019)

    MATH  Google Scholar 

  28. 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)

    MATH  Google Scholar 

  29. Kumar, S., Kumar, A., Wazwaz, A.M.: New exact solitary wave solutions of the strain wave equation in microstructured solids via the generalized exponential rational function method. Eur. Phys. J. Plus 135, 870 (2020)

    Google Scholar 

  30. Kumar, S., Kumar, D., Kumar, A.: Lie symmetry analysis for obtaining the abundant exact solutions, optimal system and dynamics of solitons for a higher-dimensional Fokas equation. Chaos Solitons Fract. 142, 110507 (2021)

    MATH  Google Scholar 

  31. Yang, D.Y., Tian, B., Tian, H.Y., Wei, C.C., Shan, W.R., Jiang, Y.: 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)

    Google Scholar 

  32. Feng, B.F., Luo, X.D., Ablowitz, M.J., et al.: General soliton solution to a nonlocal nonlinear Schrödinger equation with zero and nonzero boundary conditions. Nonlinearity 31, 5385–5409 (2018)

    MATH  Google Scholar 

  33. Zhou, T.Y., Tian, B., Chen, Y.Q., et al.: Painlevé analysis, auto-Bäcklund transformation and analytic solutions of a (2+1)-dimensional generalized Burgers system with the variable coefficients in a fluid. Nonlinear Dyn. 108, 2417–2428 (2022)

    Google Scholar 

  34. Chen, J., Feng, B.F., Chen, Y.: Bilinear Bäcklund transformation, Lax pair and multi-soliton solution for a vector Ramani equation. Mod. Phys. Lett. B 31, 1750133 (2017)

    Google Scholar 

  35. Hosseini, K., Mirzazadeh, M., Aligoli, M., et al.: Rational wave solutions to a generalized (2+1)-dimensional Hirota bilinear equation. Math. Model. Nat. Phenom. 15, 61 (2020)

    MATH  Google Scholar 

  36. Wu, X.H., Gao, Y.T., Yu, X., Ding, C.C., Li, L.Q.: Modified generalized Darboux transformation, degenerate and bound-state solitons for a Laksmanan-Porsezian-Daniel equation. Chaos Solitons Fract. 162, 112399 (2022)

    Google Scholar 

  37. Shen, Y., Tian, B., Zhou, T.Y., Gao, X.T.: 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)

    Google Scholar 

  38. Liu, F.Y., Gao, Y.T.: Lie group analysis for a higher-order Boussinesq-Burgers system. Appl. Math. Lett. 132, 108094 (2022)

    MATH  Google Scholar 

  39. Wu, X.H., Gao, Y.T., Yu, X., Ding, C.C., Hu, L., Li, L.Q.: Binary Darboux transformation, solitons, periodic waves and modulation instability for a nonlocal Lakshmanan-Porsezian-Daniel equation. Wave Motion 114, 103036 (2022)

    Google Scholar 

  40. Cheng, C.D., Tian, B., Zhang, C.R., Zhao, X.: Bilinear form, soliton, breather, hybrid and periodic-wave solutions for a (3+1)-dimensional Korteweg-de Vries equation in a fluid. Nonlinear Dyn. 105, 2525–2538 (2021)

    Google Scholar 

  41. 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)

    MATH  Google Scholar 

  42. Hunag, S.T., Wu, C.F., Qi, C.: Rational and semi-rational solutions of the modified Kadomtsev-Petviashvili equation and the (2+1)-dimensional Konopelchenko-Dubrovsky equation. Nonlinear Dyn. 97, 2829–2841 (2019)

    MATH  Google Scholar 

  43. Jia, H.X., Liu, Y.J., Wang, Y.N.: Rogue-wave interaction of a nonlinear Schrödinger model for the Alpha Helical Protein. Z. Naturforsch. A 71, 27–32 (2015)

    Google Scholar 

  44. Shats, M., Punzmann, H., Xia, H.: Capillary rogue waves. Phys. Rev. Lett. 104, 104503 (2010)

    Google Scholar 

  45. Yang, D.Y., Tian, B., Hu, C.C., Liu, S.H., Shan, W.R., Jiang, Y.: Conservation laws and breather-to-soliton transition for a variable-coefficient modified Hirota equation in an inhomogeneous optical fiber. Wave. Random Complex (2022). https://doi.org/10.1080/17455030.2021.1983237

    Article  Google Scholar 

  46. Yang, D.Y., Tian, B., Hu, C.C., Zhou, T.Y.: 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)

  47. Satsuma, J., Ablowitz, M.J.: Two-dimensional lumps in nonlinear dispersive systems. J. Math. Phys. 20, 1496–1503 (1979)

    MATH  Google Scholar 

  48. Ma, W.X.: Lump solutions to the Kadomtsev-Petviashvili equation. Phys. Lett. A 379, 1975–1978 (2015)

    MATH  Google Scholar 

  49. Pelinovsky, D.E., Stepanyants, Y.A., Kivshar, Y.S.: Self-focusing of plane dark solitons in nonlinear defocusing media. Phys. Rev. E 51, 5016–5026 (1995)

    Google Scholar 

  50. Falcon, E., Laroche, C., Fauve, S.: Observation of depression solitary surface waves on a thin fluid layer. Phys. Rev. Lett. 89, 204501 (2002)

    Google Scholar 

  51. Sun, B., Wazwaz, A.M.: Interaction of lumps and dark solitons in the Melnikov equation. Nonliear Dyn. 92, 2049–2059 (2018)

    MATH  Google Scholar 

  52. Rao, J.G., Porsezian, K., He, J.S.: Semi-rational solutions of the third-type Davey-Stewartson equation. Chaos 27, 083115 (2017)

    MATH  Google Scholar 

  53. Hu, W.Q., Gao, Y.T., Zhao, C., et al.: 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 (2016)

    Google Scholar 

  54. Deng, G.F., Gao, Y.T., Su, J.J., et al.: Multi-breather wave solutions for a generalized (3+1)-dimensional Yu-Toda-Sasa-Fukayama equation in a two-layer liquid. Appl. Math. Lett. 98, 177–183 (2019)

    MATH  Google Scholar 

  55. Yin, H.M., Tian, B., Chai, J., et al.: Solitons and bilinear Bäcklund transformations for a (3+1)-dimensional Yu-Toda-Sasa-Fukuyama equation in a liquid or lattice. Appl. Math. Lett. 58, 178–183 (2016)

  56. Chai, J., Tian, B., Sun, W.R., et al.: Solitons and rouge waves for a generalized (3+1)-dimensional variable-coefficient Kadomtsev-Petviashvili equation in fluid mechanics. Comput. Math. Appl. 71, 2060–2068 (2016)

    MATH  Google Scholar 

  57. Korteweg, D.J., de Vries, G.: XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos. Mag. 39, 422–443 (1895)

  58. Zabusky, N.J., Kruskal, M.D.: Interaction of “soliton” in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, 240–243 (1965)

  59. Gardner, C.S., Greene, J.M., Kruskal, M.D., et al.: Method for solving the Korteweg-deVries equation. Phys. Rev. Lett. 19, 1095 (1967)

    MATH  Google Scholar 

  60. Ablowitz, M.J., Clarkson, P.A.: Solitons. Nonlinear Evolution Equations and Inverse Scattering. Cambridge Univ. Press, New York (1991)

    MATH  Google Scholar 

  61. Jeffrey, A., Kakutani, T.: Weak nonlinear dispersive waves: a discussion centered around the Korteweg-de Vries equation. Siam Rev. 14, 582–643 (1972)

    MATH  Google Scholar 

  62. Miura, R.M.: The Korteweg-de Vries equation: a survey of results. SIAM Rev. 18, 412–459 (1976)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  64. Jimbo, M., Miwa, T.: Solitons and infinite dimensional Lie algebras. Publ. Res. Inst. Math. Sci. 19, 943–1001 (1983)

    MATH  Google Scholar 

  65. Ohta, Y., Yang, J.: Rogue waves in the Davey-Stewartson I equation. Phys. Rev. E 86, 036604 (2012)

    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. This work has been supported by the National Natural Science Foundation of China under Grant No. 11772017.

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  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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Liu, FY., Gao, YT. & Yu, X. Rogue-wave, rational and semi-rational solutions for a generalized (3+1)-dimensional Yu-Toda-Sasa-Fukuyama equation in a two-layer fluid. Nonlinear Dyn 111, 3713–3723 (2023). https://doi.org/10.1007/s11071-022-08017-x

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