The Phi-Four (also embodied as $ \phi^4 $) equation (PFE) is one of the most significant models in nonlinear physics, that emerges in particle physics, condensed matter physics and cosmic theory. In this study, propagating soliton solutions for the PFE were obtained by employing the extended direct algebraic method (EDAM). This transformational method reformulated the model into an assortment of nonlinear algebraic equations using a series-form solution. These equations were then solved with the aid of Maple software, producing a large number of soliton solutions. New families of soliton solutions, including exponential, rational, hyperbolic, and trigonometric functions, are included in these solutions. Using 3D, 2D, and contour graphs, the shape, amplitude, and propagation behaviour of some solitons were visualized which revealed the existence of kink, shock, bright-dark, hump, lump-type, dromion, and periodic solitons in the context of PFE. The study was groundbreaking as it extended the suggested strategy to the PFE that was being aimed at, yielding a significant amount of soliton wave solutions while providing new insights into the behavioral characteristics of soliton. This approach surpassed previous approaches by offering a systematic approach to solving nonlinear problems in analogous challenging situations. Furthermore, the results also showed that the suggested method worked well for building families of propagating soliton solutions for intricate models such as the PFE.
Citation: Mohammed Aldandani, Abdulhadi A. Altherwi, Mastoor M. Abushaega. Propagation patterns of dromion and other solitons in nonlinear Phi-Four ($ \phi^4 $) equation[J]. AIMS Mathematics, 2024, 9(7): 19786-19811. doi: 10.3934/math.2024966
The Phi-Four (also embodied as $ \phi^4 $) equation (PFE) is one of the most significant models in nonlinear physics, that emerges in particle physics, condensed matter physics and cosmic theory. In this study, propagating soliton solutions for the PFE were obtained by employing the extended direct algebraic method (EDAM). This transformational method reformulated the model into an assortment of nonlinear algebraic equations using a series-form solution. These equations were then solved with the aid of Maple software, producing a large number of soliton solutions. New families of soliton solutions, including exponential, rational, hyperbolic, and trigonometric functions, are included in these solutions. Using 3D, 2D, and contour graphs, the shape, amplitude, and propagation behaviour of some solitons were visualized which revealed the existence of kink, shock, bright-dark, hump, lump-type, dromion, and periodic solitons in the context of PFE. The study was groundbreaking as it extended the suggested strategy to the PFE that was being aimed at, yielding a significant amount of soliton wave solutions while providing new insights into the behavioral characteristics of soliton. This approach surpassed previous approaches by offering a systematic approach to solving nonlinear problems in analogous challenging situations. Furthermore, the results also showed that the suggested method worked well for building families of propagating soliton solutions for intricate models such as the PFE.
| [1] |
W. Gao, H. Rezazadeh, Z. Pinar, H. Baskonus, S. Sarwar, G. Yel, Novel explicit solutions for the nonlinear Zoomeron equation by using newly extended direct algebraic technique, Opt. Quant. Electron., 52 (2020), 52. http://dx.doi.org/10.1007/s11082-019-2162-8 doi: 10.1007/s11082-019-2162-8
|
| [2] | M. Khater, Solitary wave solutions for the generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony nonlinear evolution equation, Global J. Sci. Front. Res. Phys. Space Sci., 16 (2016), 37–41. |
| [3] |
H. Bulut, T. Sulaiman, H. Baskonus, H. Rezazadeh, M. Eslami, M. Mirzazadeh, Optical solitons and other solutions to the conformable space time fractional Fokas Lenells equation, Optik, 172 (2018), 20–27. http://dx.doi.org/10.1016/j.ijleo.2018.06.108 doi: 10.1016/j.ijleo.2018.06.108
|
| [4] |
S. Vlase, M. Marin, A. Öchsner, M. Scutaru, Motion equation for a flexible one-dimensional element used in the dynamical analysis of a multibody system, Continuum Mech. Thermodyn., 31 (2019), 715–724. http://dx.doi.org/10.1007/s00161-018-0722-y doi: 10.1007/s00161-018-0722-y
|
| [5] |
M. Khater, A. Seadawy, D. Lu, Bifurcations of solitary wave solutions for (two and three)-dimensional nonlinear partial differential equation in quantum and magnetized plasma by using two different methods, Results Phys., 9 (2018), 142–150. http://dx.doi.org/10.1016/j.rinp.2018.02.010 doi: 10.1016/j.rinp.2018.02.010
|
| [6] |
V. Senthil Kumar, H. Rezazadeh, M. Eslami, F. Izadi, M. Osman, Jacobi elliptic function expansion method for solving KdV equation with conformable derivative and dual-power law nonlinearity, Int. J. Appl. Comput. Math., 5 (2019), 127. http://dx.doi.org/10.1007/s40819-019-0710-3 doi: 10.1007/s40819-019-0710-3
|
| [7] |
M. Khater, A. Seadawy, D. Lu, Dispersive solitary wave solutions of new coupled Konno-Oono, Higgs field and Maccari equations and their applications, J. King Saud Univ. Sci., 30 (2018), 417–423. http://dx.doi.org/10.1016/j.jksus.2017.11.003 doi: 10.1016/j.jksus.2017.11.003
|
| [8] |
M. Ghasemi, High order approximations using spline-based differential quadrature method: implementation to the multi-dimensional PDEs, Appl. Math. Model., 46 (2017), 63–80. http://dx.doi.org/10.1016/j.apm.2017.01.052 doi: 10.1016/j.apm.2017.01.052
|
| [9] |
S. Noor, W. Albalawi, R. Shah, M. Mossa Al-Sawalha, S. Ismaeel, S. El-Tantawy, On the approximations to fractional nonlinear damped Burger's-type equations that arise in fluids and plasmas using Aboodh residual power series and Aboodh transform iteration methods, Front. Phys., 12 (2024), 1374481. http://dx.doi.org/10.3389/fphy.2024.1374481 doi: 10.3389/fphy.2024.1374481
|
| [10] |
N. Perrone, R. Kao, A general finite difference method for arbitrary meshes, Comput. Struct., 5 (1975), 45–57. http://dx.doi.org/10.1016/0045-7949(75)90018-8 doi: 10.1016/0045-7949(75)90018-8
|
| [11] |
M. Abdou, A. Soliman, New applications of variational iteration method, Physica D, 211 (2005), 1–8. http://dx.doi.org/10.1016/j.physd.2005.08.002 doi: 10.1016/j.physd.2005.08.002
|
| [12] |
M. Hammad, R. Shah, B. Alotaibi, M. Alotiby, C. Tiofack, A. Alrowaily, et al., On the modified versions of $\frac{G'}{G} $-expansion technique for analyzing the fractional coupled Higgs system, AIP Adv., 13 (2023), 105131. http://dx.doi.org/10.1063/5.0167916 doi: 10.1063/5.0167916
|
| [13] |
E. Yusufoğlu, A. Bekir, Solitons and periodic solutions of coupled nonlinear evolution equations by using the sine cosine method, Int. J. Comput. Math., 83 (2006), 915–924. http://dx.doi.org/10.1080/00207160601138756 doi: 10.1080/00207160601138756
|
| [14] |
Y. Chen, B. Li, H. Zhang, Generalized Riccati equation expansion method and its application to the Bogoyavlenskii's generalized breaking soliton equation, Chinese Phys., 12 (2003), 940. http://dx.doi.org/10.1088/1009-1963/12/9/303 doi: 10.1088/1009-1963/12/9/303
|
| [15] |
H. Liu, T. Zhang, A note on the improved $\tan (\phi (\xi)/2)$-expansion method, Optik, 131 (2017), 273–278. http://dx.doi.org/10.1016/j.ijleo.2016.11.029 doi: 10.1016/j.ijleo.2016.11.029
|
| [16] |
M. Guo, H. Dong, J. Liu, H. Yang, The time-fractional mZK equation for gravity solitary waves and solutions using sech-tanh and radial basic function method, Nonlinear Anal.-Model., 24 (2018), 1–19. http://dx.doi.org/10.15388/NA.2019.1.1 doi: 10.15388/NA.2019.1.1
|
| [17] | M. Kaplan, A. Bekir, A. Akbulut, E. Aksoy, The modified simple equation method for nonlinear fractional differential equations, Rom. J. Phys., 60 (2015), 1374–1383. |
| [18] |
K. L. Wang, K. J. Wang, C. He, Physical insight of local fractional calculus and its application to fractional Kdv-Burgers-Kuramoto equation, Fractals, 27 (2019), 1950122. http://dx.doi.org/10.1142/S0218348X19501226 doi: 10.1142/S0218348X19501226
|
| [19] |
K. L. Wang, K. J. Wang, A modification of the reduced differential transform method for fractional calculus, Therm. Sci., 22 (2018), 1871–1875. http://dx.doi.org/10.2298/TSCI1804871W doi: 10.2298/TSCI1804871W
|
| [20] |
K. J. Wang, On a high-pass filter described by local fractional derivative, Fractals, 28 (2020), 2050031. http://dx.doi.org/10.1142/S0218348X20500310 doi: 10.1142/S0218348X20500310
|
| [21] |
R. Ali, Z. Zhang, H. Ahmad, Exploring soliton solutions in nonlinear spatiotemporal fractional quantum mechanics equations: an analytical study, Opt. Quant. Electron., 56 (2024), 838. http://dx.doi.org/10.1007/s11082-024-06370-2 doi: 10.1007/s11082-024-06370-2
|
| [22] |
A. Iftikhar, A. Ghafoor, T. Zubair, S. Firdous, S. Mohyud-Din, (G'/G, 1/G)-expansion method for traveling wave solutions of (2+1) dimensional generalized KdV, Sin Gordon and Landau-Ginzburg-Higgs equations, Sci. Res. Essays, 8 (2013), 1349–1359. http://dx.doi.org/10.5897/SRE2013.5555 doi: 10.5897/SRE2013.5555
|
| [23] |
R. Ali, S. Barak, A. Altalbe, Analytical study of soliton dynamics in the realm of fractional extended shallow water wave equations, Phys. Scr., 99 (2024), 065235. http://dx.doi.org/10.1088/1402-4896/ad4784 doi: 10.1088/1402-4896/ad4784
|
| [24] |
M. Bhatti, D. Lu, An application of Nwogu Boussinesq model to analyze the head-on collision process between hydroelastic solitary waves, Open Phys., 17 (2019), 177–191. http://dx.doi.org/10.1515/phys-2019-0018 doi: 10.1515/phys-2019-0018
|
| [25] |
S. Behera, N. Aljahdaly, Nonlinear evolution equations and their traveling wave solutions in fluid media by modified analytical method, Pramana, 97 (2023), 130. http://dx.doi.org/10.1007/s12043-023-02602-4 doi: 10.1007/s12043-023-02602-4
|
| [26] |
H. Khan, S. Barak, P. Kumam, M. Arif, Analytical solutions of fractional Klein-Gordon and gas dynamics equations, via the (G'/G)-expansion method, Symmetry, 11 (2019), 566. http://dx.doi.org/10.3390/sym11040566 doi: 10.3390/sym11040566
|
| [27] |
J. He, X. Wu, Exp-function method for nonlinear wave equations, Chaos Soliton. Fract., 30 (2006), 700–708. http://dx.doi.org/10.1016/j.chaos.2006.03.020 doi: 10.1016/j.chaos.2006.03.020
|
| [28] | A. Alharbi, M. Almatrafi, Riccati-Bernoulli sub-ODE approach on the partial differential equations and applications, Int. J. Math. Comput. Sci., 15 (2020), 367–388. |
| [29] | W. Thadee, A. Chankaew, S. Phoosree, Effects of wave solutions on shallow-water equation, optical-fibre equation and electric-circuit equation, Maejo Int. J. Sci. Tech., 16 (2022), 262–274. |
| [30] |
J. Alzaidy, Fractional sub-equation method and its applications to the space-time fractional differential equations in mathematical physics, British Journal of Mathematics and Computer Science, 3 (2013), 153–163. http://dx.doi.org/10.9734/BJMCS/2013/2908 doi: 10.9734/BJMCS/2013/2908
|
| [31] |
M. Cinar, A. Secer, M. Ozisik, M. Bayram, Derivation of optical solitons of dimensionless Fokas-Lenells equation with perturbation term using Sardar sub-equation method, Opt. Quant. Electron., 54 (2022), 402. http://dx.doi.org/10.1007/s11082-022-03819-0 doi: 10.1007/s11082-022-03819-0
|
| [32] |
K. J. Wang, F. Shi, Multi-soliton solutions and soliton molecules of the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation for the incompressible fluid, EPL, 145 (2024), 42001. http://dx.doi.org/10.1209/0295-5075/ad219d doi: 10.1209/0295-5075/ad219d
|
| [33] |
M. Alqhtani, K. Saad, R. Shah, W. Hamanah, Discovering novel soliton solutions for (3+1)-modified fractional Zakharov-Kuznetsov equation in electrical engineering through an analytical approach, Opt. Quant. Electron., 55 (2023), 1149. http://dx.doi.org/10.1007/s11082-023-05407-2 doi: 10.1007/s11082-023-05407-2
|
| [34] |
H. Yasmin, N. Aljahdaly, A. Saeed, R. Shah, Probing families of optical soliton solutions in fractional perturbed Radhakrishnan Kundu Lakshmanan model with improved versions of extended direct algebraic method, Fractal Fract., 7 (2023), 512. http://dx.doi.org/10.3390/fractalfract7070512 doi: 10.3390/fractalfract7070512
|
| [35] |
M. Mossa Al-Sawalha, H. Yasmin, R. Shah, A. Ganie, K. Moaddy, Unraveling the dynamics of singular stochastic solitons in stochastic fractional Kuramoto-Sivashinsky equation, Fractal Fract., 7 (2023), 753. http://dx.doi.org/10.3390/fractalfract7100753 doi: 10.3390/fractalfract7100753
|
| [36] |
H. Yasmin, N. Aljahdaly, A. Saeed, R. Shah, Investigating families of soliton solutions for the complex structured coupled fractional Biswas-Arshed model in birefringent fibers using a novel analytical technique, Fractal Fract., 7 (2023), 491. http://dx.doi.org/10.3390/fractalfract7070491 doi: 10.3390/fractalfract7070491
|
| [37] |
W. Gao, P. Veeresha, D. Prakasha, H. Baskonus, G. Yel, New numerical results for the time-fractional Phi-four equation using a novel analytical approach, Symmetry, 12 (2020), 478. http://dx.doi.org/10.3390/sym12030478 doi: 10.3390/sym12030478
|
| [38] |
H. Rezazadeh, H. Tariq, M. Eslami, M. Mirzazadeh, Q. Zhou, New exact solutions of nonlinear conformable time-fractional Phi-4 equation, Chinese J. Phys., 56 (2018), 2805–2816. http://dx.doi.org/10.1016/j.cjph.2018.08.001 doi: 10.1016/j.cjph.2018.08.001
|
| [39] |
M. Khater, A. Mousa, M. El-Shorbagy, R. Attia, Analytical and semi-analytical solutions for Phi-four equation through three recent schemes, Results Phys., 22 (2021), 103954. http://dx.doi.org/10.1016/j.rinp.2021.103954 doi: 10.1016/j.rinp.2021.103954
|
| [40] |
Z. Li, T. Han, C. Huang, Bifurcation and new exact traveling wave solutions for time-space fractional Phi-4 equation, AIP Adv., 10 (2020), 115113. http://dx.doi.org/10.1063/5.0029159 doi: 10.1063/5.0029159
|
| [41] |
S. Bibi, N. Ahmed, U. Khan, S. Mohyud-Din, Auxiliary equation method for ill-posed Boussinesq equation, Phys. Scr., 94 (2019), 085213. http://dx.doi.org/10.1088/1402-4896/ab1951 doi: 10.1088/1402-4896/ab1951
|
| [42] |
M. Abdelrahman, H. Alkhidhr, Closed-form solutions to the conformable space-time fractional simplified MCH equation and time fractional Phi-4 equation, Results Phys., 18 (2020), 103294. http://dx.doi.org/10.1016/j.rinp.2020.103294 doi: 10.1016/j.rinp.2020.103294
|
| [43] |
F. Mahmud, M. Samsuzzoha, M. Ali 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. http://dx.doi.org/10.1016/j.rinp.2017.10.049 doi: 10.1016/j.rinp.2017.10.049
|
| [44] | M. Younis, A. Zafar, The modified simple equation method for solving nonlinear Phi-Four equation, International Journal of Innovation and Applied Studies, 2 (2013), 661–664. |
| [45] |
P. Sunthrayuth, N. Aljahdaly, A. Ali, R. Shah, I. Mahariq, A. Tchalla, $\phi$-Haar wavelet operational matrix method for fractional relaxation-oscillation equations containing $\phi$-Caputo fractional derivative, J. Funct. Space., 2021 (2021), 7117064. http://dx.doi.org/10.1155/2021/7117064 doi: 10.1155/2021/7117064
|
| [46] |
S. Noor, H. Alyousef, A. Shafee, R. Shah, S. El-Tantawy, A novel analytical technique for analyzing the (3+1)-dimensional fractional calogero-bogoyavlenskii-schiff equation: investigating solitary/shock waves and many others physical phenomena, Phys. Scr., 99 (2024), 065257. http://dx.doi.org/10.1088/1402-4896/ad49d9 doi: 10.1088/1402-4896/ad49d9
|
| [47] |
S. Noor, A. Alshehry, A. Shafee, R. Shah, Families of propagating soliton solutions for (3+1)-fractional Wazwaz-BenjaminBona-Mahony equation through a novel modification of modified extended direct algebraic method, Phys. Scr., 99 (2024), 045230. http://dx.doi.org/10.1088/1402-4896/ad23b0 doi: 10.1088/1402-4896/ad23b0
|
| [48] |
H. Yasmin, A. Alshehry, A. Ganie, A. Mahnashi, R. Shah, Perturbed Gerdjikov-Ivanov equation: soliton solutions via Backlund transformation, Optik, 298 (2024), 171576. http://dx.doi.org/10.1016/j.ijleo.2023.171576 doi: 10.1016/j.ijleo.2023.171576
|
| [49] |
S. El-Tantawy, H. Alyousef, R. Matoog, R. Shah, On the optical soliton solutions to the fractional complex structured (1+1)-dimensional perturbed gerdjikov-ivanov equation, Phys. Scr., 99 (2024), 035249. http://dx.doi.org/10.1088/1402-4896/ad241b doi: 10.1088/1402-4896/ad241b
|
| [50] |
S. Alshammari, K. Moaddy, R. Shah, M. Alshammari, Z. Alsheekhhussain, M. Mossa Al-sawalha, et al., Analysis of solitary wave solutions in the fractional-order Kundu-Eckhaus system, Sci. Rep., 14 (2024), 3688. http://dx.doi.org/10.1038/s41598-024-53330-7 doi: 10.1038/s41598-024-53330-7
|
| [51] |
H. Yasmin, A. Alshehry, A. Ganie, A. Shafee, R. Shah, Noise effect on soliton phenomena in fractional stochastic Kraenkel-Manna-Merle system arising in ferromagnetic materials, Sci. Rep., 14 (2024), 1810. http://dx.doi.org/10.1038/s41598-024-52211-3 doi: 10.1038/s41598-024-52211-3
|
Figures(10)
Mohammed Aldandani, Abdulhadi A. Altherwi, Mastoor M. Abushaega. Propagation patterns of dromion and other solitons in nonlinear Phi-Four ($ \phi^4 $) equation[J]. AIMS Mathematics, 2024, 9(7): 19786-19811. doi: 10.3934/math.2024966
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