Nonlinear evolution equations (NLEEs) are widely used to model wave propagation and many other phenomena in physics and engineering. Obtaining exact analytical solutions of these equations remains important for understanding their mathematical structure and physical behavior. In this paper, the Sardar subequation method (SSM) is applied to derive new traveling-wave solutions for the third extended fifth-order nonlinear equation and the nonlinear electrical transmission-line model. By introducing a traveling-wave transformation, the governing partial differential equations are reduced to nonlinear ordinary differential equations, which are then solved using the proposed method. Several families of exact solutions are obtained in hyperbolic, trigonometric, and rational forms. To illustrate their physical characteristics, selected solutions are presented through three-dimensional (3D) surface and contour plots. The results show that the derived solutions exhibit different wave structures and provide additional analytical solutions for the models under consideration. Compared with many existing methods, the SSM offers a straightforward approach for constructing exact solutions without requiring restrictive assumptions. The findings presented in this work extend the known solution sets for these nonlinear models and provide useful analytical results for future studies on nonlinear wave propagation and related problems in mathematical physics.
Citation: Osama Alkhazaleh, Osama Ala'yed, Abeer M. M. Jaradat. Analytical solutions for the third extended fifth-order nonlinear equation and the nonlinear electrical transmission line model using the Sardar subequation method[J]. AIMS Mathematics, 2026, 11(8): 24007-24032. doi: 10.3934/math.2026968
Nonlinear evolution equations (NLEEs) are widely used to model wave propagation and many other phenomena in physics and engineering. Obtaining exact analytical solutions of these equations remains important for understanding their mathematical structure and physical behavior. In this paper, the Sardar subequation method (SSM) is applied to derive new traveling-wave solutions for the third extended fifth-order nonlinear equation and the nonlinear electrical transmission-line model. By introducing a traveling-wave transformation, the governing partial differential equations are reduced to nonlinear ordinary differential equations, which are then solved using the proposed method. Several families of exact solutions are obtained in hyperbolic, trigonometric, and rational forms. To illustrate their physical characteristics, selected solutions are presented through three-dimensional (3D) surface and contour plots. The results show that the derived solutions exhibit different wave structures and provide additional analytical solutions for the models under consideration. Compared with many existing methods, the SSM offers a straightforward approach for constructing exact solutions without requiring restrictive assumptions. The findings presented in this work extend the known solution sets for these nonlinear models and provide useful analytical results for future studies on nonlinear wave propagation and related problems in mathematical physics.
| [1] |
W. Malfliet, Solitary wave solutions of nonlinear wave equations, Am. J. Phys., 60 (1992), 650–654. https://doi.org/10.1119/1.17120 doi: 10.1119/1.17120
|
| [2] |
O. González-Gaxiola, J. R. de Chávez, Traveling wave solutions of the generalized scale-invariant analog of the KdV equation by tanh–coth method, Nonlinear Eng., 12 (2023), 20220325. https://doi.org/10.1515/nleng-2022-0325 doi: 10.1515/nleng-2022-0325
|
| [3] |
O. Ala'yed, New traveling wave solutions of modified Burgers' equations, Partial Differ. Equ. Appl. Math., 13 (2025), 101265. https://doi.org/10.1016/j.padiff.2025.101265 doi: 10.1016/j.padiff.2025.101265
|
| [4] |
M. A. Kayum, M. A. Akbar, M. S. Osman, Stable soliton solutions to the shallow water waves and ion-acoustic waves in a plasma, Waves Random Complex Media, 32 (2022), 1672–1693. http://doi.org/10.1080/17455030.2020.1831711 doi: 10.1080/17455030.2020.1831711
|
| [5] |
M. A. Kayum, S. Ara, M. S. Osman, M. A. Akbar, K. A. Gepreel, Onset of the broad-ranging general stable soliton solutions of nonlinear equations in physics and gas dynamics, Results Phys., 20 (2020), 103762. http://doi.org/10.1016/j.rinp.2020.103762 doi: 10.1016/j.rinp.2020.103762
|
| [6] |
M. E. Islam, H. K. Barman, M. A. Akbar, Search for interactions of phenomena described by the coupled Higgs field equation through analytical solutions, Opt. Quant. Electron., 52 (2020), 468. https://doi.org/10.1007/s11082-020-02583-3 doi: 10.1007/s11082-020-02583-3
|
| [7] |
R. Roy, M. A. Akbar, A. M. Wazwaz, Exact wave solutions for the time fractional Sharma–Tasso–Olver equation and the fractional Klein–Gordon equation in mathematical physics, Opt. Quant. Electron., 50 (2018), 25. https://doi.org/10.1007/s11082-017-1296-9 doi: 10.1007/s11082-017-1296-9
|
| [8] |
E. C. Aslan, M. Inc, Optical soliton solutions of the NLSE with quadratic-cubic-Hamiltonian perturbations and modulation instability analysis, Optik, 196 (2019), 162661. https://doi.org/10.1016/j.ijleo.2019.04.008 doi: 10.1016/j.ijleo.2019.04.008
|
| [9] |
R. Roy, M. A. Akbar, A. R. Seadawy, D. Baleanu, Search for adequate closed form wave solutions to space-time fractional nonlinear equations, Partial Differ. Equ. Appl. Math., 3 (2021), 100025. https://doi.org/10.1016/j.padiff.2021.100025 doi: 10.1016/j.padiff.2021.100025
|
| [10] |
W. Y. Guan, B. Q. Li, Mixed structures of optical breather and rogue wave for a variable coefficient inhomogeneous fiber system, Opt. Quant. Electron., 51 (2019), 352. https://doi.org/10.1007/s11082-019-2060-0 doi: 10.1007/s11082-019-2060-0
|
| [11] |
B. Q. Li, Y. L. Ma, N-order rogue waves and their novel colliding dynamics for a transient stimulated Raman scattering system arising from nonlinear optics, Nonlinear Dyn., 101 (2020), 2449–2461. https://doi.org/10.1007/s11071-020-05906-x doi: 10.1007/s11071-020-05906-x
|
| [12] |
Y. L. Ma, Interaction and energy transition between the breather and rogue wave for a generalized nonlinear Schrödinger system with two higher-order dispersion operators in optical fibers, Nonlinear Dyn., 97 (2019), 95–105. https://doi.org/10.1007/s11071-019-04956-0 doi: 10.1007/s11071-019-04956-0
|
| [13] |
H. M. Baskonus, M. N. Raihen, M. Kayalar, On the extraction of complex behavior of generalized higher-order nonlinear Boussinesq dynamical wave equation and (1+1)-dimensional Van der Waals gas system, AIMS Mathematics, 9 (2024), 28379–28399. https://doi.org/10.3934/math.20241377 doi: 10.3934/math.20241377
|
| [14] |
H. U. Rehman, I. Iqbal, S. S. Aiadi, N. Mlaiki, M. S. Saleem, Soliton solutions of the Klein–Fock–Gordon equation using the Sardar subequation method, Mathematics, 10 (2022), 3377. https://doi.org/10.3390/math10183377 doi: 10.3390/math10183377
|
| [15] |
M. Wang, X. Li, J. Zhang, The $(G'/G)$-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Phys. Lett. A, 372 (2008), 417–423. https://doi.org/10.1016/j.physleta.2007.07.051 doi: 10.1016/j.physleta.2007.07.051
|
| [16] |
F. Mahmud, M. Samsuzzoha, M. A. Akbar, The generalized Kudryashov method to obtain exact traveling wave solutions of the PHI-four equation and the Fisher equation, Results Phys., 7 (2017), 4296–4302. https://doi.org/10.1016/j.rinp.2017.10.049 doi: 10.1016/j.rinp.2017.10.049
|
| [17] |
H. K. Barman, R. Roy, F. Mahmud, M. A. Akbar, M. S. Osman, Harmonizing wave solutions to the Fokas-Lenells model through the generalized Kudryashov method, Optik, 229 (2021), 166294. http://doi.org/10.1016/j.ijleo.2021.166294 doi: 10.1016/j.ijleo.2021.166294
|
| [18] |
J. Shen, J. Muhammad, U. Younas, Investigating the wave dynamics and interaction structures: Exploring the Benney–Roskes/Zakharov–Rubenchik model in the oceanic atmosphere, Ocean Eng., 354 (2026), 124969. http://doi.org/10.1016/j.oceaneng.2026.124969 doi: 10.1016/j.oceaneng.2026.124969
|
| [19] |
I. Siddique, K. B. Mehdi, M. M. M. Jaradat, A. Zafar, M. E. Elbrolosy, A. A. Elmandouh, et al., Bifurcation of some new traveling wave solutions for the time-space M-fractional MEW equation via three altered methods, Results Phys., 41 (2022), 105896. http://dx.doi.org/10.1016/j.rinp.2022.105896 doi: 10.1016/j.rinp.2022.105896
|
| [20] |
K. J. Wang, Exploring exact wave solutions of the Cahn–Allen equation via a novel Bernoulli sub-equation neural networks method, Modern Phys. Lett. B, 40 (2026), 2650062. http://doi.org/10.1142/S0217984926500624 doi: 10.1142/S0217984926500624
|
| [21] |
O. Ozer, H. M. M. Baskonus, H. Bulut, I. Amirali, G. Yel, A new survey to the nonlinear electrical transmission line model, Int. J. Cogn. Comput. Eng., 2 (2021), 208–214. https://doi.org/10.1016/j.ijcce.2021.11.002 doi: 10.1016/j.ijcce.2021.11.002
|
| [22] |
S. P. Joseph, New traveling wave exact solutions to the coupled Klein–Gordon system of equations, Partial Differ. Equ. Appl. Math., 5 (2022), 100208. https://doi.org/10.1016/j.padiff.2021.100208. doi: 10.1016/j.padiff.2021.100208
|
| [23] |
E. Yusufoglu, A. Bekir, Exact solutions of coupled nonlinear Klein–Gordon equations, Math. Comput. Modellin, 48 (2008), 1694–1700. https://doi.org/10.1016/j.mcm.2008.02.007 doi: 10.1016/j.mcm.2008.02.007
|
| [24] |
S. Akram, M. ur Rahman, L. A. AL-Essa, Bifurcation analysis, chaos, sensitivity, and diverse soliton solutions with propagation insights in a nonlinear spatiotemporal fractional quantum mechanics system, High Energy Density Phys., 57 (2025), 101234. http://doi.org/10.1016/j.hedp.2025.101234 doi: 10.1016/j.hedp.2025.101234
|
| [25] | P. Guo, X. Wu, L. B. Wang, Multiple soliton solutions for the variant Boussinesq equations, Adv. Differ. Equ., 2015 (2015), 371. |
| [26] |
A. M. Wazwaz, The extended tanh method for new compact and noncompact solutions for the KP–BBM and the ZK–BBM equations, Chaos Solitons Fractals, 38 (2008), 1505–1516. https://doi.org/10.1016/j.chaos.2007.01.135 doi: 10.1016/j.chaos.2007.01.135
|
| [27] |
F. B. Pelap, M. M. Faye, Solitonlike excitations in a one-dimensional electrical transmission line, J. Math. Phys., 46 (2005), 033502. https://doi.org/10.1063/1.1843272 doi: 10.1063/1.1843272
|
| [28] |
A. Biswas, C. Zony, E. Zerrad, Soliton perturbation theory for the quadratic nonlinear Klein–Gordon equation, Appl. Math. Comput., 203 (2008), 153–156. http://dx.doi.org/10.1016/j.amc.2008.04.013 doi: 10.1016/j.amc.2008.04.013
|
| [29] |
A. M. Wazwaz, Kink solutions for three new fifth-order nonlinear equations, Appl. Math. Model., 38 (2014), 110–118. http://doi.org/10.1016/j.apm.2013.06.009 doi: 10.1016/j.apm.2013.06.009
|
| [30] |
A. R. Seadawy, D. Yaro, D. Lu, Computational wave solutions of generalized higher-order nonlinear Boussinesq dynamical wave equation, Modern Phys. Lett. A, 34 (2019), 1950338. http://doi.org/10.1142/S0217732319503383 doi: 10.1142/S0217732319503383
|
| [31] |
K. K. Ali, M. S. Mehanna, On some new analytical solutions to the (2+1)-dimensional nonlinear electrical transmission line model, Eur. Phys. J. Plus, 137 (2022), 280. http://doi.org/10.1140/epjp/s13360-022-02481-5 doi: 10.1140/epjp/s13360-022-02481-5
|
| [32] |
E. Tala-Tebue, E. M. E. Zayed, New Jacobi elliptic function solutions, solitons and other solutions for the (2+1)-dimensional nonlinear electrical transmission line equation, Eur. Phys. J. Plus, 133 (2018), 314. http://doi.org/10.1140/epjp/i2018-12118-7 doi: 10.1140/epjp/i2018-12118-7
|
| [33] |
W. M. Taha, M. S. M. Noorani, I. Hashim, New application of the (G$'$/G)-expansion method for thin film equations, Abstr. Appl. Anal., 2013 (2013), 535138. http://doi.org/10.1155/2013/535138 doi: 10.1155/2013/535138
|
Figures(8) / Tables(1)
Osama Alkhazaleh, Osama Ala'yed, Abeer M. M. Jaradat. Analytical solutions for the third extended fifth-order nonlinear equation and the nonlinear electrical transmission line model using the Sardar subequation method[J]. AIMS Mathematics, 2026, 11(8): 24007-24032. doi: 10.3934/math.2026968
DownLoad: