The soliton solution and periodic lump solution of the (3+1)-dimensional generalized Bogoyavlensky-Konopelchenko equation with variable coefficients

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Abstract

This paper investigates the soliton solution and periodic lump solution of the (3 + 1)- dimensional generalized Bogoyavlensky–Konopelchenko equation with variable coefficients. First, by introducing specific constraint conditions, a novel bilinear expression for the aforementioned equation is derived. Subsequently, with the aid of symbolic computation, the corresponding soliton and periodic lump solution are successfully constructed. Finally, three-dimensional plots, density plots, and x-axis projection diagrams are presented to analyze the interaction behaviors of the obtained solutions, which provides intuitive visual evidence for the physical characteristics of these solutions.

Keywords

the (3+1)-dimensional generalized Bogoyavlensky-Konopelchenko equation with variable coefficient / periodic lump solution / soliton solution

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References

[1] X. Y. Gao, In plasma physics and fluid dynamics: Symbolic computation on a (2+1)- dimensional variable-coefficient Sawada-Kotera system, Appl. Math. Lett., 159(2025), 109262. https://doi.org/10.1016/j.aml.2024.109262

[2] S. F. Tian, H. Q. Zhang, On the integrability of a generalized variable-coefficient Kadomtsev- Petviashvili equation, J. Phys. A: Math. Theor., 45(2012), 055203. https://doi.org/10.1088/1751-8113/45/5/055203

[3] X. T. Gao, B. Tian, Similarity reductions on a (2+1)-dimensional variable-coefficient modified Kadomtsev-Petviashvili system describing certain electromagnetic waves in a thin film, Int. J. Theor. Phys., 63(2024), 99. https://doi.org/10.1007/s10773-024-05629-4

[4] S. Kumar, M. Niwas, Analyzing multi-peak and lump solutions of the variable- coefficient Boiti-Leon-Manna-Pempinelli equation: A comparative study of the Lie classical method and unified method with applications, Nonlinear Dyn., 111(2023), 22457-22475. https://doi.org/10.1007/s11071-023-09012-6

[5] Z. Z. Lan, Multi-soliton solutions for a (2+1)-dimensional variable-coefficient nonlinear Schrödinger equation, Appl. Math. Lett., 86(2018), 243-248. https://doi.org/10.1016/j.aml.2018.05.014

[6] M. Shakeel, S. T. Mohyud-Din, Improved (G’/G)-expansion and extended tanh methods for (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff equation, Alex. Eng. J., 54(2015), 27-33. https://doi.org/10.1016/j.aej.2014.11.003

[7] W. B. Rabie, H. M. Ahmed, A. Darwish, H. H. Hussein, Construction of new solitons and other wave solutions for a concatenation model using modified extended tanh-function method, Alex. Eng. J., 74(2023), 445-451. https://doi.org/10.1016/j.aej.2023.05.046

[8] M. S. Ghayad, H. M. Ahmed, N. M. Badra, H. Rezazadeh, M. A. Hosseinzadeh, W. B. Rabie, New analytical wave structures for generalized B-type Kadomtsev-Petviashvili equation by improved modified extended tanh function method, Phys. Scripta, 99(2024), 125224. https://doi.org/10.1088/1402-4896/ad8aa4

[9] W. X. Ma, A. Abdeljabbar, A bilinear Bäcklund transformation of a (3+1)-dimensional generalized KP equation, Appl. Math. Lett., 25(2012), 1500-1504. https://doi.org/10.1016/j.aml.2012.01.003

[10] Y. H. Yin, X. Lü W. X. Ma, Bäcklund transformation, exact solutions and diverse interaction phenomena to a (3+1)-dimensional nonlinear evolution equation, Nonlinear Dyn., 108(2022), 4181-4194. https://doi.org/10.1007/s11071-021-06531-y

[11] L. N. Gao, Y. Y. Zi, Y. H. Yin, W. X. Ma, X. Lü, Bäcklund transformation, multiple wave solutions and lump solutions to a (3+1)-dimensional nonlinear evolution equation, Nonlinear Dyn., 89(2017), 2233-2240. https://doi.org/10.1007/s11071-017-3581-3

[12] S. Sahoo, G. Garai, S. Saha Ray, Lie symmetry analysis for similarity reduction and exact solutions of modified KdV-Zakharov-Kuznetsov equation, Nonlinear Dyn., 87(2017), 1995-2000. https://doi.org/10.1007/s11071-016-3169-3

[13] V. Jadaun, S. Kumar, Lie symmetry analysis and invariant solutions of (3+1)-dimensional Calogero-Bogoyavlenskii-Schiff equation, Nonlinear Dyn., 93(2018), 349-360. https://doi.org/10.1007/s11071-018-4196-z

[14] P. F. Han, T. Bao, Novel hybrid-type solutions for the (3+1)-dimensional generalized Bogoyavlensky-Konopelchenko equation with time-dependent coefficients, Nonlinear Dyn., 107(2022), 1163-1177. https://doi.org/10.1007/s11071-021-07019-5

[15] K. K. Ahmed, H. H. Hussein, H. M. Ahmed, W. B. Rabie, W. Alexan, Analysis of the dynamical behaviors for the generalized Bogoyavlensky-Konopelchenko equation and its analytical solutions occurring in mathematical physics, Ain Shams Eng. J., 15(2024), 103000. https://doi.org/10.1016/j.asej.2024.103000

[16] W. Razzaq, A. Zafar, Bilinearization of generalized Bogoyavlensky-Konopelchenko equation with neural networks: Painlevé analysis, Nonlinear Dyn., 113(2025), 25083-25096. https://doi.org/10.1007/s11071-025-11384-w

[17] W. X. Ma, Lump waves in a generalized Bogoyavlensky-Konopelchenko model with spatially balanced derivatives, Math. Eng., 8(2026), 140-149. https://doi.org/10.3934/mine.2026005

[18] G. M. Moatimid, R. M. El-Shiekh, A. G. A. Al-Nowehy, Exact solutions for Calogero- Bogoyavlenskii-Schiff equation using symmetry method,Appl. Math. Comput., 220(2013), 455-462. https://doi.org/10.1016/j.amc.2013.06.034

[19] B. Al Alwan, M. Abu Bakar, W. A. Faridi, A. C. Turcu, A. Akgül, M. Sallah, The propagating exact solitary waves formation of generalized Calogero-Bogoyavlenskii- Schiff equation with robust computational approaches, Fractal. Fract., 7(2023), 191. https://doi.org/10.3390/fractalfract7020191

[20] A. M. Wazwaz, Painlevé analysis for new (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equations with constant and time-dependent coefficients, Int. J. Numer. Method. H., 30(2020), 4259-4266. https://doi.org/10.1108/HFF-10-2019-0760

[21] R. Hirota, The Direct Method in Soliton Theory , Cambridge University Press, 2004. https://doi.org/10.1017/CBO9780511543043