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A new approach to investigate the nonlinear dynamics in a (3 + 1)-dimensional nonlinear evolution equation via Wronskian condition with a free function

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Abstract

This paper proposes a new approach to investigate the nonlinear dynamics in a (3 + 1)-dimensional nonlinear evolution equation via Wronskian condition with a free function. Firstly, a Wronskian condition involving a free function is introduced for the equation. Secondly, by solving the Wronskian condition, some exact solutions are presented. Thirdly, the dynamical behaviors are analyzed by choosing specific functions in the Wronskian condition. In addition, some exact solutions are graphically illustrated by using Mathematica symbolic computations. The dynamical behaviors include stationary y-breather, line-soliton resonance, line-soliton-like phenomenon, parabola–soliton interaction, cubic–parabola–soliton resonance, kink behavior, and singular waves. These results not only illustrate the merits of the proposed method in deriving new exact solutions but also novel dynamical behaviors in the (3 + 1)-dimensional nonlinear evolution equation.

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References

  1. Wazwaz, A.M., El-Tantawy, S.A.: A new (3+1)-dimensional generalized Kadomtsev–Petviashvili equation. Nonlinear Dyn. 84, 1107 (2016)

    Article  MathSciNet  Google Scholar 

  2. Wazwaz, A.M., El-Tantawy, S.A.: New (3+1)-dimensional equations of Burgers type and Sharma–Tasso–Olver type: multiple-soliton solutions. Nonlinear Dyn. 87, 2457 (2017)

    Article  MathSciNet  Google Scholar 

  3. Ma, W.X.: Abundant lumps and their interaction solutions of (3+1)-dimensional linear PDEs. J. Geom. Phys. 133, 10 (2018)

  4. Ma, W.X., Zhou, Y.: Lump solutions to nonlinear partial differential equations via Hirota bilinear forms. J. Differ. Equ. 264, 2633 (2018)

    Article  MathSciNet  Google Scholar 

  5. Geng, X.G.: Algebraic-geometrical solutions of some mutidimentional nonlinear evolution equations. J. Phys. A Math. Gen. 36, 2289 (2003)

  6. Geng, X.G., Ma, Y.L.: \(N\)-soliton solution and its Wronskian form of a (3+1)-dimensional nonlinear evolution equation. Phys. Lett. A 369, 285 (2007)

    Article  MathSciNet  Google Scholar 

  7. Wazwaz, A.M.: A (3+1)-dimensional nonlinear evolution equation with multiple soliton solutions and multiple singular soliton solutions. Appl. Math. Comput. 215, 1548 (2009)

    MathSciNet  MATH  Google Scholar 

  8. Asaad, M.G., Ma, W.X.: Extended Gram-type determinant, wave and rational solutions to two (3+1)-dimensional nonlinear evolution equations. Appl. Math. Comput. 219, 213 (2012)

    MathSciNet  MATH  Google Scholar 

  9. Liu, J.G., You, M.X., Zhou, L., Ai, G.P.: The solitary wave, rogue wave and periodic solutions for the (3+1)-dimensional soliton equation. Z. Angew. Math. Phys. 70, 4 (2019)

    Article  MathSciNet  Google Scholar 

  10. Shi, Y.B., Zhang, Y.: Rogue waves of a (3+1)-dimensional nonlinear evolution equation. Commun. Nonlinear Sci. Numer. Simul. 44, 120 (2017)

    Article  MathSciNet  Google Scholar 

  11. Ma, W.X., Fan, E.G.: Linear superposition principle applying to Hirota bilinear equations. Comput. Math. Appl. 61, 950 (2011)

    Article  MathSciNet  Google Scholar 

  12. Zhang, Y., Liu, Y.P., Tang, X.Y.: \(M\)-lump solutions to a (3+1)-dimensional nonlinear evolution equation. Comput. Math. Appl. 76, 592 (2018)

    Article  MathSciNet  Google Scholar 

  13. Fang, T., Wang, H., Wang, Y.H., Ma, W.X.: High-order lump-type solutions and their interaction solutions to a (3+1)-dimensional nonlinear evolution equation. Commmun. Theor. Phys. 71, 927 (2019)

    Article  MathSciNet  Google Scholar 

  14. Wang, X., Wei, J., Geng, X.G.: Rational solutions for a (3+1)-dimensional nonlinear evolution equation. Commun. Nonlinear Sci. Numer. Simul. 83, 105116 (2020)

    Article  MathSciNet  Google Scholar 

  15. Nimmo, J.J.C., Freeman, N.C.: A method of obtaining the \(N\)-soliton solution of the Boussinesq equation in terms of a Wronskian. Phys. Lett. A 95, 4 (1983)

    Article  MathSciNet  Google Scholar 

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

  17. Ma, W.X., You, Y.C.: Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions. Trans. Am. Math. Soc. 357, 1753 (2005)

    Article  MathSciNet  Google Scholar 

  18. Ma, W.X., Li, C.X., He, J.S.: A second Wronskian formulation of the Boussinesq equation. Nonlinear Anal. 70, 4245 (2009)

    Article  MathSciNet  Google Scholar 

  19. Huang, Q.M., Gao, Y.T.: Wronskian, Pfaffian and periodic wave solutions for a (2+1)-dimensional extended shallow water wave equation. Nonlinear Dyn. 89, 2855 (2017)

    Article  MathSciNet  Google Scholar 

  20. Zhang, Y., Cheng, T.F., Ding, D.J., Dang, X.L.: Wronskian and Grammian solutions for (2+1)-dimensional soliton equation. Commun. Theor. Phys. 55, 20 (2011)

    Article  MathSciNet  Google Scholar 

  21. Cheng, L., Zhang, Y.: Wronskian and linear superposition solutions to generalized KP and BKP equations. Nonlinear Dyn. 90, 355 (2017)

    Article  MathSciNet  Google Scholar 

  22. Chen, D.Y., Zhang, D.J., Bi, J.B.: New double Wronskian solutions of the AKNS equations. Sci. China A 51, 55 (2008)

    Article  MathSciNet  Google Scholar 

  23. Dai, C.Q., Fan, Y., Zhang, N.: Re-observation on localized waves constructed by variable separation solutions of (1+1)-dimensional coupled integrable dispersionless equations via the projective Riccati equation method. Appl. Math. Lett. 96, 20 (2019)

    Article  MathSciNet  Google Scholar 

  24. Wu, G.Z., Dai, C.Q.: Nonautonomous soliton solutions of variable-coefficient fractional nonlinear Schrödinger equation. Appl. Math. Lett. 106, 106365 (2020)

    Article  MathSciNet  Google Scholar 

  25. Dai, C.Q., Fan, Y., Wang, Y.Y.: Three-dimensional optical solitons formed by the balance between different-order nonlinearities and high-order dispersion/diffraction in parity-time symmetric potentials. Nonlinear Dyn. 98, 489 (2019)

    Article  Google Scholar 

  26. Dai, C.Q., Zhang, J.F.: Controlling effect of vector and scalar crossed double-Ma breathers in a partially nonlocal nonlinear medium with a linear potential. Nonlinear Dyn. 100, 1621 (2020)

    Article  Google Scholar 

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Acknowledgements

The author is very grateful to the editor and the anonymous referees for their valuable suggestions. The author would also like to thank the support by the Collaborative Innovation Center for Aviation Economy Development of Henan Province.

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  1. School of Science, Zhengzhou University of Aeronautics, Zhengzhou, 450046, Henan, People’s Republic of China

    Jianping Wu

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  1. Jianping Wu
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Correspondence to Jianping Wu.

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Wu, J. A new approach to investigate the nonlinear dynamics in a (3 + 1)-dimensional nonlinear evolution equation via Wronskian condition with a free function. Nonlinear Dyn 103, 1795–1804 (2021). https://doi.org/10.1007/s11071-020-06155-8

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  • DOI: https://doi.org/10.1007/s11071-020-06155-8

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