The study of analytic solutions for high-dimensional nonlinear wave equations is crucial for comprehending complex nonlinear phenomena in fluid dynamics. In this work, several novel analytic solutions and their dynamical behaviors are investigated by the extended homoclinic test approach and the improved $ (G'/G) $-expansion method for the (3+1)-dimensional extended shallow water wave equation. Based on the Hirota bilinear formulation, various breather wave solutions are constructed via the extended homoclinic test approach. Furthermore, multiple analytic solutions consisting of rational functions, trigonometric functions, and hyperbolic functions are derived using the improved $ (G'/G) $-expansion method. The features of the obtained solutions are illustrated through numerical simulations, including the shape-preserving properties during propagation of the soliton solutions and the divergence characteristics of rogue wave solutions. Furthermore, the relationships between soliton stability and rogue wave generation conditions are given through parameter variation analysis.
Citation: Xixi Wu, Minkun Xiong, Yanfeng Guo. Analytic solutions for a (3+1)-dimensional extended shallow water wave equation in nonlinear phenomena[J]. AIMS Mathematics, 2025, 10(8): 18541-18557. doi: 10.3934/math.2025828
The study of analytic solutions for high-dimensional nonlinear wave equations is crucial for comprehending complex nonlinear phenomena in fluid dynamics. In this work, several novel analytic solutions and their dynamical behaviors are investigated by the extended homoclinic test approach and the improved $ (G'/G) $-expansion method for the (3+1)-dimensional extended shallow water wave equation. Based on the Hirota bilinear formulation, various breather wave solutions are constructed via the extended homoclinic test approach. Furthermore, multiple analytic solutions consisting of rational functions, trigonometric functions, and hyperbolic functions are derived using the improved $ (G'/G) $-expansion method. The features of the obtained solutions are illustrated through numerical simulations, including the shape-preserving properties during propagation of the soliton solutions and the divergence characteristics of rogue wave solutions. Furthermore, the relationships between soliton stability and rogue wave generation conditions are given through parameter variation analysis.
| [1] |
X. Y. Gao, Open-ocean shallow-water dynamics via a (2+1)-dimensional generalized variable-coefficient Hirota-Satsuma-Ito system: Oceanic auto-Bäcklund transformation and oceanic solitons, China Ocean Eng., 39 (2025), 541–547. https://doi.org/10.1007/s13344-025-0057-y doi: 10.1007/s13344-025-0057-y
|
| [2] |
L. A. Dawod, M. Lakestani, J. Manafian, Breather wave solutions for the (3+1)-D generalized shallow water wave equation with variable coefficients, Qual. Theory Dyn. Syst., 22 (2023), 127. https://doi.org/10.1007/s12346-023-00826-8 doi: 10.1007/s12346-023-00826-8
|
| [3] |
S. Tarla, K. K. Ali, A. Yusuf, B. Uzun, S. Salahshour, Exact solutions of the (2+1)-dimensional Konopelchenko–Dubrovsky system using Sardar sub-equation method, Mod. Phys. Lett. B, 39 (2025), 2450485. https://doi.org/10.1142/S0217984924504852 doi: 10.1142/S0217984924504852
|
| [4] |
A. M. Wazwaz, New integrable (2+1)- and (3+1)-dimensional shallow water wave equations: Multiple soliton solutions and lump solutions, Int. J. Numer. Methods H., 32 (2022), 138–149. https://doi.org/10.1108/HFF-01-2021-0019 doi: 10.1108/HFF-01-2021-0019
|
| [5] |
J. F. Lu, Application of variational principle and fractal complex transformation to (3+1)-dimensional fractal potential-YTSF equation, Fractals, 32 (2024), 2450027. https://doi.org/10.1142/S0218348X24500270 doi: 10.1142/S0218348X24500270
|
| [6] |
Y. Feng, S. Bilige, Multiple rogue wave solutions of (2+1)-dimensional YTSF equation via Hirota bilinear method, Waves Random Complex, 34 (2024), 94–110. https://doi.org/10.1080/17455030.2021.1900625 doi: 10.1080/17455030.2021.1900625
|
| [7] |
B. Li, F. Wang, New solutions to a category of nonlinear PDEs, Front. Phys., 13 (2025), 1547245. https://doi.org/10.3389/fphy.2025.1547245 doi: 10.3389/fphy.2025.1547245
|
| [8] |
Y. Zhang, L. Xiao, Breather wave and double-periodic soliton solutions for a (2+1)-dimensional generalized Hirota-Satsuma-Ito equation, Open Phys., 20 (2022), 632–638. https://doi.org/10.1515/phys-2022-0058 doi: 10.1515/phys-2022-0058
|
| [9] |
J. Xiang, G. Yan, Modified linear sampling method for inverse scattering by a partially coated dielectric, J. Inverse Ill-Pose. P., 32 (2024), 713–731. https://doi.org/10.1515/jiip-2022-0096 doi: 10.1515/jiip-2022-0096
|
| [10] |
H. W. A. Riaz, A. Farooq, A (2+1) modified KdV equation with time-dependent coefficients: Exploring soliton solution via Darboux transformation and artificial neural network approach, Nonlinear Dyn., 113 (2025), 3695–3711. https://doi.org/10.1007/s11071-024-10423-2 doi: 10.1007/s11071-024-10423-2
|
| [11] |
J. Pu, Y. Chen, Darboux transformation-based LPNN generating novel localized wave solutions, Physica D, 467 (2024), 134262. https://doi.org/10.1016/j.physd.2024.134262 doi: 10.1016/j.physd.2024.134262
|
| [12] |
Z. Yan, H. Dai, Q. Wang, S. N. Atluri, Harmonic balance methods: A review and recent developments, CMES-Comp. Model. Eng., 137 (2023), 1419–1459. https://doi.org/10.32604/cmes.2023.028198 doi: 10.32604/cmes.2023.028198
|
| [13] |
V. Crismale, L. De Luca, A. Kubin, A. Ninno, M. Ponsiglione, The variational approach to $s$-fractional heat flows and the limit cases $s \to 0^+$ and $s \to 1^-$, J. Funct. Anal., 284 (2023), 109851. https://doi.org/10.1016/j.jfa.2023.109851 doi: 10.1016/j.jfa.2023.109851
|
| [14] |
H. Tao, N. Anjum, Y. J. Yang, The Aboodh transformation-based homotopy perturbation method: New hope for fractional calculus, Front. Phys., 11 (2023), 1168795. https://doi.org/10.3389/fphy.2023.1168795 doi: 10.3389/fphy.2023.1168795
|
| [15] |
C. Pirri, N. Pirri, V. Macchi, D. Guidolin, A. Porzionato, R. De Caro, et al., The value of fractal analysis in ultrasound imaging: Exploring intricate patterns, Appl. Sci., 14 (2024), 9750. https://doi.org/10.3390/app14219750 doi: 10.3390/app14219750
|
| [16] |
D. Tian, Q. T. Ain, N. Anjum, C. H. He, B. Cheng, Fractal N/MEMS: From pull-in instability to pull-in stability, Fractals, 29 (2021), 2150030. https://doi.org/10.1142/S0218348X21500304 doi: 10.1142/S0218348X21500304
|
| [17] |
M. Inc, Ü. Ic, I. E. Inan, J. F. Gomez-Aguilar, Generalized $(G'/G)$-expansion method for some soliton wave solutions of Burgers-like and potential KdV equations, Numer. Meth. Part. D. E., 38 (2022), 422–433. https://doi.org/10.1002/num.22637 doi: 10.1002/num.22637
|
| [18] |
S. K. Mohanty, O. V. Kravchenko, M. K. Deka, A. N. Dev, D. V. Churikov, The exact solutions of the $(2+1)$-dimensional Kadomtsev-Petviashvili equation with variable coefficients by extended generalized $(G'/G)$-expansion method, J. King Saud Univ. Sci., 35 (2023), 541–547. https://doi.org/10.1016/j.jksus.2022.102358 doi: 10.1016/j.jksus.2022.102358
|
| [19] |
B. Mohan, S. Kumar, Generalization and analytic exploration of soliton solutions for nonlinear evolution equations via a novel symbolic approach in fluids and nonlinear sciences, Chinese J. Phys., 92 (2024), 10–21. https://doi.org/10.1016/j.cjph.2024.09.004 doi: 10.1016/j.cjph.2024.09.004
|
| [20] |
Y. Ji, W. Tan, New soliton solutions and trajectory equations of soliton collision to a (2+1)-dimensional shallow water wave model, Qual. Theory Dyn. Syst., 24 (2025), 9. https://doi.org/10.1007/s12346-024-01163-0 doi: 10.1007/s12346-024-01163-0
|
Figures(6)
Xixi Wu, Minkun Xiong, Yanfeng Guo. Analytic solutions for a (3+1)-dimensional extended shallow water wave equation in nonlinear phenomena[J]. AIMS Mathematics, 2025, 10(8): 18541-18557. doi: 10.3934/math.2025828
DownLoad: