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Bilinear form, soliton, breather, hybrid and periodic-wave solutions for a (3+1)-dimensional Korteweg–de Vries equation in a fluid

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Fluids are studied in such disciplines as atmospheric science, oceanography and astrophysics. In this paper, we investigate a (3+1)-dimensional Korteweg–de Vries equation in a fluid. Bilinear form and N-soliton solutions are obtained, where N is a positive integer. Via the N-soliton solutions, we derive the higher-order breather solutions. We observe the interaction between the two perpendicular first-order breathers on the \(x-y\) and \(x-z\) planes and the interaction between the periodic line wave and the first-order breather on the \(y-z\) plane, where x, y and z are the independent variables in the equation. We discuss the effects of \(\alpha \), \(\beta \), \(\gamma \) and \(\delta \) on the amplitude of the second-order breather, where \(\alpha \), \(\beta \), \(\gamma \) and \(\delta \) are the constant coefficients in the equation: Amplitude of the second-order breather decreases as \(\alpha \) increases; amplitude of the second-order breather increases as \(\beta \) increases; amplitude of the second-order breather keeps invariant as \(\gamma \) or \(\delta \) increases. Via the N-soliton solutions, hybrid solutions comprising the breathers and solitons are derived. Based on the Riemann theta function, we obtain the periodic-wave solutions, and find that the periodic-wave solutions approach to the one-soliton solutions under a limiting condition.

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References

  1. Beron-Vera, F.J.: Nonlinear dynamics of inertial particles in the ocean: from drifters and floats to marine debris and Sargassum. Nonlinear Dyn. 103, 1–26 (2021)

    Article  Google Scholar 

  2. Ghommem, M., Najar, F., Arabi, M., Abdel-Rahman, E., Yavuz, M.: A unified model for electrostatic sensors in fluid media. Nonlinear Dyn. 101, 271–291 (2020)

    Article  Google Scholar 

  3. Romatschke, P.: Relativistic fluid dynamics far from local equilibrium. Phys. Rev. Lett. 120, 012301 (2018)

    Article  Google Scholar 

  4. Falkovich, G.: Fluid Mechanics. Cambridge Univ. Press, Cambridge (2018)

  5. Deng, G.F., Gao, Y.T., Su, J.J., Ding, C.C., Jia, T.T.: Solitons and periodic waves for the (2+1)-dimensional generalized Caudrey-Dodd-Gibbon-Kotera-Sawada equation in fluid mechanics. Nonlinear Dyn. 99, 1039–1052 (2020)

    Article  Google Scholar 

  6. Ding, C.C., Gao, Y.T., Deng, G.F.: Breather and hybrid solutions for a generalized (3+1)-dimensional B-type Kadomtsev-Petviashvili equation for the water waves. Nonlinear Dyn. 97, 2023–2040 (2019)

    Article  Google Scholar 

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

    Article  Google Scholar 

  8. Feng, Y.J., Gao, Y.T., Li, L.Q., Jia, T.T.: Bilinear form, solitons, breathers and lumps of a (3+1)-dimensional generalized Konopelchenko-Dubrovsky-Kaup-Kupershmidt equation in ocean dynamics, fluid mechanics and plasma physics. Eur. Phys. J. Plus 135, 272 (2020)

    Article  Google Scholar 

  9. Lan, Z.Z., Guo, B.L.: Nonlinear waves behaviors for a coupled generalized nonlinear Schrödinger-Boussinesq system in ahomogeneous magnetized plasma. Nonlinear Dyn. 100, 3771–3784 (2020)

    Article  Google Scholar 

  10. Su, J.J., Gao, Y.T., Deng, G.F., Jia, T.T.: Solitary waves, breathers, and rogue waves modulated by long waves for a model of a baroclinic shear flow. Phys. Rev. E 100, 042210 (2019)

    Article  MathSciNet  Google Scholar 

  11. Zhang, R.F., Bilige, S.D.: Bilinear neural network method to obtain the exact analytical solutions of nonlinear partial differential equations and its application to p-gBKP equation. Nonlinear Dyn. 95, 3041–3048 (2020)

    Article  Google Scholar 

  12. 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 

  13. Zhang, R.F., Li, M.C., Yin, H.M.: Rogue wave solutions and the bright and dark solitons of the (3+1)-dimensional Jimbo-Miwa equation. Nonlinear Dyn. 103, 1071–1079 (2021)

    Article  Google Scholar 

  14. Jia, T.T., Gao, Y.T., Deng, G.F., Hu, L.: Quintic time-dependent-coefficient derivative nonlinear Schrödinger equation in hydrodynamics or fiber optics: bilinear forms and dark/anti-dark/gray solitons. Nonlinear Dyn. 98, 269–282 (2019)

    Article  Google Scholar 

  15. Pan, Q., Chung, W.C., Chow, K.W.: The coupled Hirota system as an example displaying discrete breathers: Rogue waves, modulation instability and varying cross-phase modulations. AIP Adv. 8, 095303 (2018)

    Article  Google Scholar 

  16. Deng, G.F., Gao, Y.T., Ding, C.C., Su, J.J.: Solitons and breather waves for the generalized Konopelchenko-Dubrovsky-Kaup-Kupershmidt system in fluid mechanics, ocean dynamics and plasma physics. Chaos Solitons Fract. 140, 110085 (2020)

    Article  MathSciNet  Google Scholar 

  17. Pan, Q., Grimshaw, R.H.J., Chow, K.W.: Modulation instability and rogue waves for shear flows with a free surface. Phys. Rev. Fluids 4, 084803 (2019)

    Article  Google Scholar 

  18. Du, Z., Ma, Y.P.: Beak-shaped rogue waves for a higher-order coupled nonlinear Schrödinger system with 4\(\times \)4 Lax pair. Appl. Math. Lett. 116, 106999 (2021)

    Article  Google Scholar 

  19. Zhang, R.F., Bilige, S.D., Chaolu, T.: Fractal solitons, arbitrary function solutions, exact periodic wave and breathers for a nonlinear partial differential equation by using bilinear neural network method. J. Syst. Sci. Complex 34, 122–139 (2021)

    Article  MathSciNet  Google Scholar 

  20. Hu, L., Gao, Y.T., Jia, T.T., Deng, G.F., Li, L.Q.: 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)

  21. Seadawy, A.R., Lu, D., Nasreen, N., Nasreen, S.: Structure of optical solitons of resonant Schrödinger equation with quadratic cubic nonlinearity and modulation instability analysis. Phys. A 534, 122155 (2019)

    Article  MathSciNet  Google Scholar 

  22. Abdou, M.A., Owyed, S., Abdel-Aty, A., Raffah, B.M., Abdel-Khalek, S.: Optical soliton solutions for a space-time fractional perturbed nonlinear Schrödinger equation arising in quantum physics. Results Phys. 16, 102895 (2020)

    Article  Google Scholar 

  23. Peng, W.Q., Tian, S.F., Zhang, T.T.: Dynamics of the soliton waves, breather waves, and rogue waves to the cylindrical Kadomtsev-Petviashvili equation in pair-ion-electron plasma. Phys. Fluids 31, 102107 (2019)

    Article  Google Scholar 

  24. Feng, Y.J., Gao, Y.T., Jia, T.T., Li, L.Q.: Soliton interactions of a variable-coefficient three-component AB system for the geophysical flows. Modern Phys. Lett. B 33, 1950354 (2019)

    Article  MathSciNet  Google Scholar 

  25. Zhang, R.F., Bilige, S.D., 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 

  26. Jia, T.T., Gao, Y.T., Yu, X., Li, L.Q.: Lax pairs, Darboux transformation, bilinear forms and solitonic interactions for a combined Calogero-Bogoyavlenskii-Schiff-type equation. Appl. Math. Lett. 114, 106702 (2021)

    Article  Google Scholar 

  27. Pan, Q., Peng, N.N., Chan, H.N., Chow, K.W.: Coupled triads in the dynamics of internal waves: case study using a linearly stratified fluid. Phys. Rev. Fluids 6, 024802 (2021)

    Article  Google Scholar 

  28. Li, L.Q., Gao, Y.T., Hu, L., Jia, T.T., Ding, C.C., Feng, Y.J.: Bilinear form, soliton, breather, lump and hybrid solutions for a (2+1)-dimensional Sawada-Kotera equation. Nonlinear Dyn. 100, 2729–2738 (2020)

    Article  Google Scholar 

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

    Book  Google Scholar 

  30. Guo, B.L., Ling, L.M., Liu, Q.P.: Nonlinear Schrödinger equation: Generalized Darboux transformation and rogue wave solutions. Phys. Rev. E 85, 026607 (2012)

    Article  Google Scholar 

  31. Liu, F.Y., Gao, Y.T., Yu, X., Ding, C.C., Deng, G.F., Jia, T.T.: Painlevé analysis, Lie group analysis and soliton-cnoidal, resonant, hyperbolic function and rational solutions for the modified Korteweg-de Vries-Calogero-Bogoyavlenskii-Schiff equation in fluid mechanics. Chaos Solitons Fract. 144, 110559 (2021)

    Article  Google Scholar 

  32. Liu, F.Y., Gao, Y.T., Yu, X., Li, L.Q., Ding, C.C., Wang, D.: Lie group analysis and analytic solutions for a (2+1)-dimensional generalized Bogoyavlensky-Konopelchenko equation in fluid mechanics and plasma physics. Eur. Phys. J. Plus 136, 656 (2021)

  33. 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)

    Article  MathSciNet  Google Scholar 

  34. 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)

    Article  Google Scholar 

  35. Wahlquist, H.D., Estabrook, F.B.: Bäcklund transformation for solutions of the Korteweg-de Vries equation. Phys. Rev. Lett. 31, 1386 (1973)

    Article  MathSciNet  Google Scholar 

  36. Su, J.J., Gao, Y.T., Ding, C.C.: Darboux transformations and rogue wave solutions of a generalized AB system for the geophysical flows. Appl. Math. Lett. 88, 201–208 (2019)

    Article  MathSciNet  Google Scholar 

  37. Ding, C.C., Gao, Y.T., Deng, G.F., Wang, D.: Lax pair, conservation laws, Darboux transformation, breathers and rogue waves for the coupled nonautonomous nonlinear Schrodinger system in an inhomogeneous plasma. Chaos Solitons Fract. 133, 109580 (2020)

    Article  MathSciNet  Google Scholar 

  38. Wazwaz, A.M.: Painlevé analysis for Boiti-Leon-Manna-Pempinelli equation of higher dimensions with time-dependent coefficients: multiple soliton solutions. Phys. Rev. A 384, 126310 (2020)

    Google Scholar 

  39. Wang, X.B., Zhang, T.T., Dong, M.J.: Dynamics of the breathers and rogue waves in the higher-order nonlinear Schrödinger equation. Appl. Math. Lett. 86, 298–304 (2018)

    Article  MathSciNet  Google Scholar 

  40. Tu, J.M., Tian, S.F., Xu, M.J., Ma, P.L., Zhang, T.T.: On periodic wave solutions with asymptotic behaviors to a (3+1)-dimensional generalized B-type Kadomtsev-Petviashvili equation in fluid dynamics. Comput. Math. Appl. 72, 2486–2504 (2016)

    Article  MathSciNet  Google Scholar 

  41. Wazwaz, A.M.: Two new Painlevé-integrable (2+1) and (3+1)-dimensional KdV equations with constant and time-dependent coefficients. Nucl. Phys. B 954, 115009 (2020)

  42. Korteweg, D.J., de Vries, G.: 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)

  43. Hirota, R.: Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192 (1971)

    Article  Google Scholar 

  44. Nakamura, Y., Tsukabayashi, I.: Modified Korteweg-de Vries ion-acoustic solitons in a plasma. J. Plasma Phys. 34, 401–415 (1985)

    Article  Google Scholar 

  45. Darvishi, M.T., Najafi, M., Kavitha, L., Venkatesh, M.: Stair and step soliton solutions of the integrable (2+1) and (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equations. Commun. Theor. Phys. 58, 785–794 (2012)

  46. Guo, H.D., Xia, T.C.: Lump and Lump-Kink Soliton solutions of an extended Boiti-Leon-Manna-Pempinelli equation. Int. J. Nonlinear Sci. Numer. Simul. 21, 371–377 (2020)

  47. Sun, B.N., Wazwaz, A.M.: General high-order breathers and rogue waves in the (3+1)-dimensional KP-Boussinesq equation. Commun. Nonlinear Sci. Numer. Simul. 64, 1–13 (2018)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We express our sincere thanks to the Editors and Reviewers for their valuable comments. This work has been supported by the National Natural Science Foundation of China under Grant Nos. 11772017, 11272023 and 11471050, by the Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), China (IPOC: 2017ZZ05), and by the Fundamental Research Funds for the Central Universities of China under Grant No. 2011BUPTYB02.

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  1. State Key Laboratory of Information Photonics and Optical Communications, and School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, China

    Chong-Dong Cheng, Bo Tian, Chen-Rong Zhang & Xin Zhao

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  1. Chong-Dong Cheng
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Correspondence to Bo Tian.

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Cheng, CD., Tian, B., Zhang, CR. et al. 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). https://doi.org/10.1007/s11071-021-06540-x

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