This paper studies the analytical and dynamical behavior of a generalized (3+1)-dimensional extended Kadomtsev-Petviashvili (eKP) equation, which includes higher-order nonlinear and dispersive effects. Using the exp$ (-\varphi(\xi)) $-expansion method, exact traveling wave solutions are obtained successfully. The solutions exhibit rich nonlinear wave structures, including localized solitons, periodic waves, and breathing patterns. To further explore the qualitative behavior of the system, a reduction of traveling waves is used, resulting in a planar dynamical system. A bifurcation analysis is carried out in detail by investigating the equilibrium points, Jacobian structures, and their relations to system parameters. The phase portraits show topological transitions such as centers, saddles, and cusp points. Moreover, a perturbed form of the reduced system is considered in order to investigate chaotic dynamics under periodic external forcing. The existence of chaos is verified using bifurcation diagrams, sensitivity analysis, Poincaré maps, time series, and both 2D and 3D phase portraits. These numerical simulations confirm the theoretical predictions and demonstrate the system's complex behavior across varying perturbation parameters. Overall, the outcomes provide a complete picture of the soliton structures and chaotic regimes that the eKP model may exhibit in physico-nonlinear wave propagation in multidimensional contexts.
Citation: Bahadır Kopçasız, Rubayyi T. Alqahtani, Mehmet Şenol. A comprehensive study of solitons and chaotic dynamics in the (3+1)-dimensional extended Kadomtsev-Petviashvili equation[J]. AIMS Mathematics, 2026, 11(3): 6622-6648. doi: 10.3934/math.2026274
This paper studies the analytical and dynamical behavior of a generalized (3+1)-dimensional extended Kadomtsev-Petviashvili (eKP) equation, which includes higher-order nonlinear and dispersive effects. Using the exp$ (-\varphi(\xi)) $-expansion method, exact traveling wave solutions are obtained successfully. The solutions exhibit rich nonlinear wave structures, including localized solitons, periodic waves, and breathing patterns. To further explore the qualitative behavior of the system, a reduction of traveling waves is used, resulting in a planar dynamical system. A bifurcation analysis is carried out in detail by investigating the equilibrium points, Jacobian structures, and their relations to system parameters. The phase portraits show topological transitions such as centers, saddles, and cusp points. Moreover, a perturbed form of the reduced system is considered in order to investigate chaotic dynamics under periodic external forcing. The existence of chaos is verified using bifurcation diagrams, sensitivity analysis, Poincaré maps, time series, and both 2D and 3D phase portraits. These numerical simulations confirm the theoretical predictions and demonstrate the system's complex behavior across varying perturbation parameters. Overall, the outcomes provide a complete picture of the soliton structures and chaotic regimes that the eKP model may exhibit in physico-nonlinear wave propagation in multidimensional contexts.
| [1] |
L. Akinyemi, Solutions of generalized fractional perturbed Zakharov-Kuznetsov equation arising in a magnetized dusty plasma, Rev. Mex. Fís., 67 (2021), 060703. https://doi.org/10.31349/revmexfis.67.060703 doi: 10.31349/revmexfis.67.060703
|
| [2] |
L. Akinyemi, I. Ainomugisha, Stability and solitary wave dynamics of higher-dimensional nonlinear Schrödinger equation with time-dependent potential, Nonlinear Dyn., 113 (2025), 23439–23456. https://doi.org/10.1007/s11071-025-11325-7 doi: 10.1007/s11071-025-11325-7
|
| [3] |
S. Kumar, A. Kukkar, Dynamics of several optical soliton solutions of a (3+1)-dimensional nonlinear Schrödinger equation with parabolic law in optical fibers, Mod. Phys. Lett. B, 39 (2024), 2450453. https://doi.org/10.1142/S0217984924504530 doi: 10.1142/S0217984924504530
|
| [4] |
M. Ozisik, S. A. Durmus, A. Secer, M. Bayram, Pure-cubic optical soliton solutions of the nonlinear Schrödinger equation including parabolic law nonlinearity in the absence of the group velocity dispersion, Int. J. Theor. Phys., 64 (2025), 80. https://doi.org/10.1007/s10773-025-05950-6 doi: 10.1007/s10773-025-05950-6
|
| [5] |
Z. Yang, W. P. Zhong, M. Belić, Dark localized waves in shallow waters: analysis within an extended Boussinesq system, Chin. Phys. Lett., 41 (2024), 044201. https://doi.org/10.1088/0256-307X/41/4/044201 doi: 10.1088/0256-307X/41/4/044201
|
| [6] |
K. J. Wang, X. L. Liu, F. Shi, G. Li, Bifurcation and sensitivity analysis, chaotic behaviors, variational principle, Hamiltonian and diverse wave solutions of the new extended integrable Kadomtsev–Petviashvili equation, Phys. Lett. A, 534 (2025), 130246. https://doi.org/10.1016/j.physleta.2025.130246 doi: 10.1016/j.physleta.2025.130246
|
| [7] |
P. Xu, H. Huang, K. J. Wang, Dynamics of real and complex multi-soliton solutions, novel soliton molecules, asymmetric solitons and diverse wave solutions to the Kadomtsev-Petviashvili equation, Alex. Eng. J., 125 (2025), 537–544. https://doi.org/10.1016/j.aej.2025.04.040 doi: 10.1016/j.aej.2025.04.040
|
| [8] |
Beenish, M. Samreen, F. S. Alshammari, Exploring solitary wave solutions of the generalized integrable Kadomtsev–Petviashvili equation via Lie symmetry and Hirota's bilinear mthod, Symmetry, 17 (2025), 710. https://doi.org/10.3390/sym17050710 doi: 10.3390/sym17050710
|
| [9] |
M. H. Rafiq, J. Lin, Periodic breather waves, stripe-solitons and interaction solutions for the (3+1)-dimensional variable-coefficient Kadomtsev–Petviashvili-like equation, Chaos Solitons Fract., 194 (2025), 116212. https://doi.org/10.1016/j.chaos.2025.116212 doi: 10.1016/j.chaos.2025.116212
|
| [10] |
M. Şenol, M. Ö. Erol, F. M. Çelik, N. Das, Dynamical investigation of the new (3+1)-dimensional Kadomtsev–Petviashvili-Sawada-Kotera-Ramani equation with conformable fractional derivative, Mod. Phys. Lett. B, 39 (2025), 2550110. https://doi.org/10.1142/S0217984925501106 doi: 10.1142/S0217984925501106
|
| [11] |
I. Samir, H. M. Ahmed, H. Emadifar, K. K. Ahmed, Traveling and soliton waves and their characteristics in the extended (3+1)-dimensional Kadomtsev-Petviashvili equation in fluid, Part. Differ. Equations Appl. Math., 14 (2025), 101146. https://doi.org/10.1016/j.padiff.2025.101146 doi: 10.1016/j.padiff.2025.101146
|
| [12] |
J. G. Liu, W. H. Zhu, Y. He, Variable-coefficient symbolic computation approach for finding multiple rogue wave solutions of nonlinear system with variable coefficients, Z. Angew. Math. Phys., 72 (2021), 154. https://doi.org/10.1007/s00033-021-01584-w doi: 10.1007/s00033-021-01584-w
|
| [13] | B. B. Kadomtsev, V. I. Petviashvili, On the stability of solitary waves in weakly dispersive media, Sov. Phys. Dokl., 15 (1970), 539–541. |
| [14] | L. Kaur, A. M. Wazwaz, Dynamical analysis of soliton solutions for space-time fractional Calogero-Degasperis and Sharma-Tasso-Olver equations, Rom. Rep. Phys., 74 (2022), 108. |
| [15] | F. Yuan, Deformed soliton and positon solutions for the (2+1)-dimensional nonlinear Schrödinger equation, Rom. Rep. Phys., 74 (2022), 121. |
| [16] | A. M. Wazwaz, W. Alhejaili, S. A. El-Tantawy, New (3+1)-dimensional Painlevé integrable extensions of Mikhailov-Novikov-Wang equation: variety of multiple soliton solutions, Rom. J. Phys., 67 (2022), 115. |
| [17] |
M. Şenol, B. Ghanbari, Exploration of novel traveling wave solitons to the conformable (3+1)-dimensional Wazwaz-Kaur-Boussinesq equation, Int. J. Theor. Phys., 64 (2025), 151. https://doi.org/10.1007/s10773-025-06018-1 doi: 10.1007/s10773-025-06018-1
|
| [18] |
M. Wang, G. He, T. Xu, N. Li, Localized waves for modified Boussinesq equation and Mikhailov-Lenells equation based on Physics-informed neural networks and Miura transformation, Chaos Solitons Fract., 190 (2025), 115764. https://doi.org/10.1016/j.chaos.2024.115764 doi: 10.1016/j.chaos.2024.115764
|
| [19] |
G. Chen, K. Fang, J. Sun, Z. Liu, P. Wang, H. Jin, A two-layer Boussinesq-type wave model accelerated by graphics processing unit, Phys. Fluids, 37 (2025), 073606. https://doi.org/10.1063/5.0268831 doi: 10.1063/5.0268831
|
| [20] |
Y. Cheng, C. Dong, S. Zheng, W. Hu, Solving nonlinear Boussinesq equation of second-order time derivatives with physics-informed neural networks, Commun. Theor. Phys., 77 (2025), 105001. https://doi.org/10.1088/1572-9494/adcc8e doi: 10.1088/1572-9494/adcc8e
|
| [21] |
Y. L. Ma, A. M. Wazwaz, B. Q. Li, New extended Kadomtsev–Petviashvili equation: multiple soliton solutions, breather, lump and interaction solutions, Nonlinear Dyn., 104 (2021), 1581–1594. https://doi.org/10.1007/s11071-021-06357-8 doi: 10.1007/s11071-021-06357-8
|
| [22] |
Y. L. Ma, A. M. Wazwaz, B. Q. Li, Novel bifurcation solitons for an extended Kadomtsev-Petviashvili equation in fluids, Phys. Lett. A, 413 (2021), 127585. https://doi.org/10.1016/j.physleta.2021.127585 doi: 10.1016/j.physleta.2021.127585
|
| [23] |
A. M. Wazwaz, Study of three integrable extensions of Kadomtsev–Petviashvili, Boussinesq, and Kadomtsev–Petviashvili–Boussinesq equations, Rom. Rep. Phys., 76 (2024), 114. https://doi.org/10.59277/romrepphys.2024.76.114 doi: 10.59277/romrepphys.2024.76.114
|
| [24] | A. M. Wazwaz, Extensions of Kadomtsev–Petviashvili and Kadomtsev–Petviashvili–Boussinesq equations: multiple solitons and lump wave solutions, Roma. Rep. Phys., 77 (2025), 115. |
| [25] |
N. Kadkhoda, H. Jafari, Analytical solutions of the Gerdjikov-Ivanov equation by using exp $(-\varphi(\xi))$-expansion method, Optik, 139 (2017), 72–76. https://doi.org/10.1016/j.ijleo.2017.03.078 doi: 10.1016/j.ijleo.2017.03.078
|
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Bahadır Kopçasız, Rubayyi T. Alqahtani, Mehmet Şenol. A comprehensive study of solitons and chaotic dynamics in the (3+1)-dimensional extended Kadomtsev-Petviashvili equation[J]. AIMS Mathematics, 2026, 11(3): 6622-6648. doi: 10.3934/math.2026274
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