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Degeneration of solitons for a (\(3+1\))-dimensional generalized nonlinear evolution equation for shallow water waves

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Abstract

A (\(3+1\))-dimensional generalized shallow water waves equation is investigated with different methods. Based on symbolic computation and Hirota bilinear form, N-soliton solutions are constructed. In the process of degeneration of N-soliton solutions, T-breathers are derived by taking complexication method. Then rogue waves will emerge during the degeneration of breathers by taking the parameter limit method. Through full degeneration of N-soliton, M-lump solutions are derived based on long-wave limit approach. In addition, we also find out that the partial degeneration of N-soliton process can generate the hybrid solutions composed of soliton, breather and lump.

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References

  1. Ablowitz, M.J.: Nonlinear dispersive waves: asymptotic analysis and solitons. Combridge Univ. Press, Combridge (2011)

    Book  MATH  Google Scholar 

  2. Vreugdenhil, C.B.: Numerical methods for shallow-water flow. Springer, London (2013)

    Google Scholar 

  3. Ivanov, S.K., Kamchatnov, A.M.: Evolution of wave pulses in fully nonlinear shallow-water theory. Phys. Fluids. 31, 057102 (2019)

    Article  Google Scholar 

  4. Zabusky, N.J., Kruskal, M.D.: Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15(6), 240–243 (1965)

    Article  MATH  Google Scholar 

  5. Kivshar, Y.S., Malomed, B.A.: Dynamics of solitons in nearly integrable system. Rev. Mod. Phys. 61(4), 763–915 (1989)

    Article  Google Scholar 

  6. Mihalache, D.: Mulitidimensional localized structures in optics and Bose-Einstein condensates: a selection of recent studies. J. Roman. Phys. 59(3), 295–312 (2014)

    Google Scholar 

  7. Forte, S.: Quantum mechanics and field theory with fractional spin and statistics. Appl. Phys. Lett. 64(1), 193–236 (1992)

    MathSciNet  Google Scholar 

  8. Kibler, B., Fatome, J., Finot, C., Millot, G., Dias, F., Genty, G., Akhmediev, N., Dudley, J.M.: The peregrine soliton in nonlinear fibre optics. Nat. Phys. 6(10), 790–795 (2010)

    Article  Google Scholar 

  9. Malomed, B., Torner, L., Wise, F., Mihalache, D.: On multidimensional solitons and their legacy in contemporary atomic, molecular and optical physics. J. Phys. B. At. Mol. Opt. Phys. 49(17), 170502 (2016)

    Article  Google Scholar 

  10. Dai, Z.D., Wang, C.J., Liu, J.: Inclined periodic homoclinic breather and rogue waves for the (1+1)-dimensional Boussinesq equation. Pramana. J. Phys. 83(4), 473–480 (2014)

    Article  Google Scholar 

  11. Wang, C.J., Dai, Z.D., Lin, L.: Exact three-wave solution for higher dimensional KdV-type equation. Appl. Math. Comput. 216(2), 501–505 (2010)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Li, B.Q.: Loop-like kink breather and its transition phenomena for the Vakhnenko equation arising from high-frequency wave propagation in electromagnetic physics. Appl. Math. Lett. 112, 106822 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, B.Q., Ma, Y.L.: Extended generalized Darboux transformation to hybrid rogue wave and breather solutions for a nonlinear Schrodinger equation. Appl. Math. Comput. 386, 125469 (2020)

    MathSciNet  MATH  Google Scholar 

  15. Ma, W.X., Zhou, Y.: Lump solutions to nonlinear partial different equations via Horita bilinear forms. J. Diff. Equ. 264(4), 2633–2659 (2018)

    Article  MATH  Google Scholar 

  16. Ma, C.H., Deng, A.P.: Lump solution of (2+1)-dimensional Boussineaq equation. Commun. Theor. Phys. 65(5), 546–552 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ma, Y.L., Wazwaz, A.M., Li, B.Q.: A new (3+1)-dimensional Kadomtsev-Petviashvili equation and its integrability, multiple-solitons, breathers and lump waves. Math. Comput. Simul. 187, 505–519 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  18. Zhao, Z.L., Chen, Y., Han, B.: Lump soliton, mixed lump stripe and periodic lump solutions of a (2+1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation. Mod. Phys. Lett. B 31(14), 1750157 (2017)

    Article  MathSciNet  Google Scholar 

  19. Wang, C.J.: Spatiotemporal deformation of lump solution to (2+1)-dimensional KdV equation. Nonlinear Dynam. 84(2), 697–702 (2016)

    Article  MathSciNet  Google Scholar 

  20. Peng, W.Q., Tian, S.F., Zhang, T.T.: Analysis on lump, lump off 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 

  21. 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 

  22. 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 

  23. 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 

  24. Zhao, Z.L., He, L.C.: \(M\)-lump and hybrid solutions of a generalized (2+1)-dimensional Hirota-Satsuma-Ito equation. Appl. Math. Lett. 111, 106612 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  25. Guo, H.D., Xia, T.C., Hu, B.B.: Dynamics of abundant solutions to the (3+1)-dimensional generalized Yu-Toda-Sasa-Fukuyama equation. Appl. Math. Lett. 105, 106301 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  26. Tan, W., Dai, Z.D., Xie, J.L., 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 

  27. Ma, W.X., Zhou, Y., Dougherty, R.: Lump-type solutions to nonlinear differential equations derived from generalized bilinear equations. Int. J. Mod. Phys. B 30, 1640018 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zhang, R.F., Li, M.C., Albishari, M., Zheng, F.C., Lan, Z.Z.: Generalized lump solutions, classical lump solutions and rogue waves of the (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada-like equation. App. Math. Compu. 403, 126201 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  29. Ren, B., Lin, J., Lou, Z.M.: Lump and their interaction solutions of a (2+1)-dimensional generalized potential Kadomtsev-Petviashvili equation. J. App. Anal. Comput. 10(3), 935–945 (2020)

    MathSciNet  MATH  Google Scholar 

  30. Guo, Y.F., Dai, Z.D., Guo, C.X.: Lump solutions and interaction solutions for (2+1)-dimensional KPI equation. Front. Math. China 9, 1–12 (2021). https://doi.org/10.1007/s11464-021-0973-y

    Article  Google Scholar 

  31. Wang, H., Tian, S.F., Chen, Y., Zhang, T.T.: Dynamics of kink solitary waves and lump waves with interaction phenomena in a generalized (3+1)-dimensional Kadomtsev-Petviashvili-Boussinesq equation. Int. J. Comput. Math. 97(11), 2178–2190 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  32. Verma, P., Kaur, L.: Integrability, bilinearization and analytic study of new form of (3+1)-dimensional B-type Kadomtsev-Petviashvili(BPK)-Boussinesq equation. Appl. Math. Comput. 346, 879–886 (2019)

    MathSciNet  MATH  Google Scholar 

  33. Wang, C.J., Fang, H., Tang, X.X.: State transition of lump-type waves for the (2+1)-dimensional generalized KdV equation. Nonlinear Dyn. 95, 2943–2961 (2019)

    Article  MATH  Google Scholar 

  34. Tan, W.: Some new dynamical behaviour of double breathers and lump-\(N\)-solitons for the Ito equation. Int. J. Comput. Math. 98(5), 961–974 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  35. Tan, W., Dai, Z.D.: Dynamics of kinky wave for (3+1)-dimensional potential Yu-Tada-Sasa-Fukuyama equation. Nonlinear Dyn. 85(2), 817–823 (2016)

    Article  MATH  Google Scholar 

  36. Tan, W., Dai, Z.D., Xie, J.L., Qiu, D.Q.: Parameter limit method and its application in the (4+1)-dimensional Fokas equation. Compu. Math. App. 75(12), 4214–4220 (2018)

    MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  38. 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 

  39. Li, B.Q., Ma, Y.L.: Interaction dynamics of hybrid solitons and breathers for extend generalization Vakhnenko equation. Nonlinear Dyn. 102, 1787–1799 (2020)

    Article  Google Scholar 

  40. Ma, Y.L., Wazwaz, A.M., Li, B.Q.: New extend Kadomtsev-Petviashvili equation: multiple soliton solutions, breather, lump and interaction solutions. Nonlinear Dyn. 104, 1581–1594 (2021)

    Article  Google Scholar 

  41. Zhang, R.F., Bilige, S., Liu, J.G., Li, M.C.: Bright-dark solitons and interaction phenomenon for p-gBKP equation by using bilinear neural network method. Phys. Scr. 96, 025224 (2021)

    Article  Google Scholar 

  42. Hirota, R.: Direct Methods in Soliton Theory. Springer, Berlin (1980)

    Book  Google Scholar 

  43. Feng, Y.J., Gao, Y.T., Li, L.Q., Jia, T.T.: Bilinear form and solutions of a (3+1)-dimensional generalized nonlinear evolution equation for the shallow-water waves. Appl. Anal. 378, 1–13 (2019)

    Google Scholar 

  44. Shen, Y., Tian, B.: Bilinear auto-Bäcklund transformations and soliton solutions of a (3+1)-dimensional generalized nonlinear evolution equation for the shallow-water waves. Appl. Math. Lett. 21, 107301 (2021)

    Article  MATH  Google Scholar 

  45. Shen, Y., Tian, B., Liu, S.H.: Solitonic fusion and fission for a (3+1)-dimensional generalized nonlinear evolution equation arising in the shallow water waves. Phys. Lett. A 405, 127429 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  46. Hirota, R., Satsuma, J.: \(N\)-soliton solutions of model equation for shallow water waves. J. Phys. Soc. Japan 40(2), 611–612 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  47. Tan, W., Zhang, W., Zhang, J.: Evolutionary behaviour of breathers and interaction solutions with \(M\)-solitons for (2+1)-dimensional KdV system. Appl. Math. Lett. 101(C), 106063 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  48. Tian, Y., Dai, Z.D.: Rogue waves and new multi-wave solutions of the (2+1)-dimensional Ito equation. Z. Naturforsch. A 70(6), 437–443 (2015)

    Article  Google Scholar 

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Acknowledgements

The authors would like to express their sincere thanks to referees for their enthusiastic guidance and help. 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, Basic research projects of Yunnan, China, Grant No. 202101AT070057 and 2022 Joint Special Youth Project of Yunnan Provincial Colleges and Universities (Study on space-time dynamics of solutions of high dimensional nonlinear evolution equations), For the record No.112081620042.

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

    Long-Xing Li

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Li, LX. Degeneration of solitons for a (\(3+1\))-dimensional generalized nonlinear evolution equation for shallow water waves. Nonlinear Dyn 108, 1627–1640 (2022). https://doi.org/10.1007/s11071-022-07270-4

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