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General high-order localized waves and hybrid solutions of the extend (3+1)-dimensional Ito equation

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Abstract

Under investigation in this work is an extend (3+1)-dimensional Ito equation. On the basis of Hirota bilinear method and symbolic computation, we aim to derive a series of localized wave solutions composed of the N-soliton solution, resonance Y-type soliton solution, high-order kink breather solution, high-order kink lump solution and abundant hybrid solution. Firstly, kink exponentially N-soliton solutions are constructed, resonance Y-type soliton solutions are degenerated from N-soliton solutions with new constraint. It is shown that the fission and fusion phenomena exists in exponentially localized wave solutions. What is more, high-order kink breather solutions are obtained by adding the complex conjugate relations of the parameters in N-solitons solution. Secondly, high-order kink lump solutions are also presented by means of long wave limit approach. Finally, the hybrid solutions including Y-type solitons, high-order breather and high-order lump are derived respectively, which could be considered as the high-order mixed localized wave solutions. Plentiful dynamical behaviors are contained in these general high-order localized waves. These new results greatly extend the localized wave solution of (3+1)-dimensional Ito equation already available in the literature and provide novel ideas for investigating the dynamical behaviors of fluid mechanic, soliton and so on.

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The authors declare that data supporting the findings of this study are available within the article, the figures are concrete expression.

References

  1. Ablowitz, M.J., Clarkson, P.A.: Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge (1991)

    Book  MATH  Google Scholar 

  2. Müller, P., Garrett, C., Osborne, A.: Rogue waves. Oceanography 8, 66–75 (2005)

    Article  Google Scholar 

  3. Kharif, C., Pelinovsky, E., Slunyaev, A.: Rogue Waves in the Ocean, Advances in Geophysical and Environmental Mechanics and Mathematics. Springer-Verlag, Berlin (2009)

    MATH  Google Scholar 

  4. Guo, B.L., Pang, X.F., Wang, Y.F., Liu, N.: Solitons. Walter de Gruyter GmbH, Berlin (2018)

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  6. Manakov, M.Q., Zakharov, V.E., Bordag, L.A.: Analysis on lump, Two-dimensional solitons of the Kadomtsev Petviashvili equation and their interaction. Phys. Lett. A 63(3), 205–206 (1977)

    Article  Google Scholar 

  7. Ablowitz, M.J., Satsuma, J.: Solitons and rational solutions of nonlinear evolution equations. J. Math. Phys. 19(10), 2180–2186 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  8. Satsum, J., Ablowitz, M.J.: Two-dimensional lumps in nonlinear dispersive system. J. Math. Phys. 20(7), 1496–1503 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  9. Yang, J., Zhang, Y.: Higher-order rogue wave solutions of a general coupled nonlinear Fokas-Lenells system. Nonlinear Dyn. 93, 585–597 (2018)

    Article  MATH  Google Scholar 

  10. Ye, R.S., Zhang, Y., Zhang, Q.Y., Chen, X.T.: Vector rational and semi-rational rogue wave solutions in the coupled complex modified Korteweg-de Vries equations. Wave Motion 92, 102425 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ali, A.M., Wazwaz, A.M., Mahmud, F., Baleanu, D., Ripan, R., Barman, H.K., Mahmoud, W., Mohammed, A., Sharif, A., Osman, M.S.: Dynamical behavior of solitons of the perturbed nonlinear Schrödinger equation and microtubules through the generalized Kudryashov scheme. Result Phys. 43, 106079 (2022)

    Article  Google Scholar 

  12. Guo, B.L., Liu, N., Wang, Y.F.: A Riemann-Hilbert approach to a new type coupled nonlinear Schrödinger equations. J. Math. Anal. Appl. 459, 145–158 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  13. Tian, S.F.: Riemann–Hilbert problem to a generalized derivative nonlinear Schrödinger equation: Long-time asymptotic behavior (in Chinese). Sci. Sin. Math. 52, 505–542 (2022)

    Article  Google Scholar 

  14. Chen, S.J., Lü, X.: Lump and lump-multi-kink solutions in the (3+1)-dimensions. Commun. in Nonlinear Sci. and Numer. Simul. 109, 106103 (2022)

  15. Wang, C.J., Fang, H.: General high-order localized waves to the Bogoyavlenskii-Kadomtsev-Petviashvili equation. Nonlinear Dyn. 100, 583–599 (2020)

    Article  MATH  Google Scholar 

  16. Tan, W., Dai, Z.D., Yin, Z.Y.: Dynamics of multi-breathers,\(N\)-solitons and \(M\)-lump solutions in the (2+1)-dimensional KdV equation. Nonlinear Dyn. 96, 1605–1614 (2019)

    Article  MATH  Google Scholar 

  17. Li, L.X.: Degeneration of solitons for a (3+1)-dimensional generalized nonlinear evolution equation for shallow water waves. Nonlinear Dyn. 108, 1627–1640 (2022)

    Article  Google Scholar 

  18. Guo, H.D., Xia, T.C., Hu, B.B.: High-order lumps, high-order breathers and hybrid solutions for an extend (3+1)-dimensional Jimbo-Miwa equation in fluid dynamics. Nonlinear Dyn. 100(1), 601–614 (2020)

    Article  MATH  Google Scholar 

  19. Zhang, R.F., Li, M.C., Gan, J.Y., Li, Q., Lan, Z.Z.: Novel trial functions and rogue waves of generalized breaking soliton equation via bilinear neural network method. Chaos, Solitons Fractals 403, 111692 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  20. Park, C., Nuruddeen, R.I., Ali, K.K., Muhammad, L., Osman, M.S., Baleanu, D.: Novel hyperbolic and exponential ansatz methods to the fractional fifth-order Korteweg-de Vries equations. Adv. Differ. Equ. 2020(1), 627 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wazwaz, A.M.: Multiple-soliton solutions for extended (3+1)-dimensional Jimbo–Miwa equations. Appl. Math. Lett. 64, 21–26 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ohta, Y., Yang, J.: General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation. Proc. R.Soc. A 468, 1716-1740 (2012)

  23. Guo, L.J., Chabchoub, A., He, J.S.: Higher-order rogue wave solutions to the Kadomtsev-Petviashvili 1 equation. Physica D 426, 132990 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  24. 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  MATH  Google Scholar 

  25. Zhang, Z., Li, B., Wazwaz, A.M., Guo, Q.: Lump molecules in fluid systems: Kadomtsev-Petviashvili I case. Phys. Lett. A 424, 127848 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  26. Saliou, Y., Abbagari, S., Houwe, A., Osman, M.S., Inc, M.: W-shape bright and several other solutions to the (3+1)-dimensional nonlinear evolution equations. Mod. Phys. Lett. B 35(30), 2150468 (2021)

    Article  MathSciNet  Google Scholar 

  27. Ito, M.: An extension of nonlinear evolution equations of the KdV (mKdV) type to higher order. J. Phys. Soc. Jpn. 49(2), 771 (1980)

    Article  MATH  Google Scholar 

  28. Zhang, Y., Chen, D.Y.: \(N\)-soliton-like solution of Ito equation. Commun. Theor. Phys. 42, 641 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wazwaz, A.M.: Multiple-soliton solutions for the generalized (1+1)-dimensional and the generalized (2+1)-dimensional Ito equations. Appl. Math. Comput. 202, 840 (2008)

    MathSciNet  MATH  Google Scholar 

  30. Wang, X.B., Tian, S.F., Qin, C.Y.: Dynamics of the breathers, rogue waves and solitary waves in the (2+1)-dimensional Ito equation. App. Math. Lett. 68, 40–47 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  31. Yang, J.Y., Ma, W.X., Qin, Z.Y.: Lump and lunmp-soliton solutions to the (2+1)-dimensional Ito equation. Analy. Math. Phys. 8, 427–436 (2018)

    Article  MATH  Google Scholar 

  32. Wazwaz, A.M.: Integrable (3+1)-dimensional Ito equation: variety of lump solutions and multiple-soliton solutions. Nonlinear Dyn. 109, 1929–1934 (2022)

    Article  Google Scholar 

  33. Chen, A.H., Wang, F.F.: Fission wave solutions, lump solutions and interactional solutions for the (2+1)-dimensional Sawada-Kotera equation. Phys. Scr. 94, 055206 (2019)

    Article  Google Scholar 

  34. Li, J.H., Chen, Q.Q., Li, B.: Resonance \(Y\)-type soliton solutions and some new types of hybrid solutions in the (2+1)-dimensional Sawada-Kotera equation. Commun. Theor. Phys. 73, 045006 (2021)

    Article  MathSciNet  Google Scholar 

  35. Ma, H.C., Gao, Y.D., Deng, A.P.: Fission and fusion solutions of the (2+1)-dimensional generalized Konopelchenko-Dubrovsky-Kaup-Kupershmidt equation: case of fluid mecheanics amd plasma physics. Nonlinear Dyn. 108, 4123–4137 (2022)

    Article  Google Scholar 

  36. Tan, W., Dai, H.P., Dai, Z.D., Zhong, W.Y.: Emergence and space-time structure of lump solution to the (2+1)-dimensional generalized KP equation. Pramana J. Phys. 89(5), 77–83 (2017)

    Article  Google Scholar 

  37. Tan, W., Dai, Z.D.: Dynamics of kinky wave for (3+1)-dimensional potential Yu-Toda-Sasa-Fukuyama equation. Nonlinear Dyn. 85, 817–823 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  38. Yuan, F., Cheng, Y., He, J.S.: Degeneration of breathers in the Kadomtsev–Petviashvili I equation. Commu. Nonlinear Sci. Numer. Simul. 83, 105027 (2019)

    Article  MATH  Google Scholar 

  39. Cao, Y.L., Cheng, Y., He, J.S., Chen, Y.R.: High-order breather, \(M\)-kink lump and semi-rational solutions of potential Kadomtsev-Petviashvili equation. Commun. Theor. Phys. 73, 035004 (2021)

    Article  MathSciNet  Google Scholar 

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Funding

This work was supported by National Natural Science Foundation of China, Grant No.11901345, Scientific and Technological Innovation Team of Nonlinear Analysis and Algebra with Their Applications in Universities of Yunnan Province, China, Grant No.2020CXTD25, Yunnan Fundamental Research Projects, China, Grant No.202101AT070057, 2022 Joint Special Youth Project of Yunnan Provincial Colleges and Universities (Study on space-time dynamics of solutions of high dimensional nonlinear evolution equations), Grant No.202101BA070001-280, Scientific Research Fund Project of Education Department of Yunnan Province, Grant No.2023J1031 and Technological Planning Project of Yunnan Province, China, Grant No. 202305AC160005.

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Authors and Affiliations

  1. College of Mathematics and Statistics, Qujing Normal University, Qujing, Yunnan, 655000, China

    Long-Xing Li & Bi-Tao Cheng

  2. Key Laboratory of Analytical Mathematics and Intelligent Computing for Yunnan Provincial Department of Education, Qujing, Yunnan, 655000, China

    Long-Xing Li & Bi-Tao Cheng

  3. School of Mathematics and Statistics, Yunnan University, Kunming, 650091, China

    Zheng-De Dai

Authors
  1. Long-Xing Li
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  2. Zheng-De Dai
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Contributions

LXL: Writing-original draft, Investigation. ZDD: Supervision, Conceptualization. BT C: Software, Formal analysis.

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Correspondence to Long-Xing Li.

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Li, LX., Dai, ZD. & Cheng, BT. General high-order localized waves and hybrid solutions of the extend (3+1)-dimensional Ito equation. Nonlinear Dyn 111, 13357–13373 (2023). https://doi.org/10.1007/s11071-023-08551-2

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