Analysis of peakon-like soliton solutions: (3+1)-dimensional Fractional Klein-Gordon equation

Waleed Hamali; Abdullah A. Zaagan; Hamad Zogan; Waleed Hamali; Abdullah A. Zaagan; Hamad Zogan

doi:10.3934/math.2024722

AIMS Mathematics

2024, Volume 9, Issue 6: 14913-14931. doi: 10.3934/math.2024722

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Analysis of peakon-like soliton solutions: (3+1)-dimensional Fractional Klein-Gordon equation

1.
Department of Mathematics, Faculty of Science, Jazan University, P.O. Box 2097, Jazan 45142, kingdom of Saudi Arabia
2.
Department of Computer Science, College of Engineering and Computer Science, Jazan University, Jazan, Kingdom of Saudi Arabia

Received: 13 March 2024 Revised: 12 April 2024 Accepted: 17 April 2024 Published: 25 April 2024
MSC : 26A33, 34A08

In this study, we investigate the fundamental properties of ($ 3+1 $)-$ D $ Fractional Klein-Gordon equation using the sophisticated techniques of Riccatti-Bornoulli sub-ODE approach with Backlund transformation. Using a more stringent criterion, our study reveals new soliton solutions that have peakon-like properties and unique cusp features. This research provides significant understanding of the dynamic behaviours and odd events related to these solutions. This work is important because it helps to elucidate the complex dynamics that exist within physical systems, which will benefit many different scientific fields. Our method is used to examine the existence and stability of compactons and kinks in the context of actual physical systems. Under a double-well on-site potential, these structures are made up of a network of connected nonlinear pendulums. Both $ 2D $ and contour plots produced by parameter changes provide as clear examples of the efficiency, simplicity, and conciseness of the computational method used. The results highlight how flexible this approach is, and demonstrate how symbolic calculations broaden its application to more complex events. This work offers a useful framework and studying intricate physical systems, as well as a flexible computational tool that may be used in a variety of scientific fields.
- Fractional Klein-Gordon equation,
- backlund transformation,
- solitary Wave solution,
- analytical method
Citation: Waleed Hamali, Abdullah A. Zaagan, Hamad Zogan. Analysis of peakon-like soliton solutions: (3+1)-dimensional Fractional Klein-Gordon equation[J]. AIMS Mathematics, 2024, 9(6): 14913-14931. doi: 10.3934/math.2024722

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Abstract

In this study, we investigate the fundamental properties of ($ 3+1 $)-$ D $ Fractional Klein-Gordon equation using the sophisticated techniques of Riccatti-Bornoulli sub-ODE approach with Backlund transformation. Using a more stringent criterion, our study reveals new soliton solutions that have peakon-like properties and unique cusp features. This research provides significant understanding of the dynamic behaviours and odd events related to these solutions. This work is important because it helps to elucidate the complex dynamics that exist within physical systems, which will benefit many different scientific fields. Our method is used to examine the existence and stability of compactons and kinks in the context of actual physical systems. Under a double-well on-site potential, these structures are made up of a network of connected nonlinear pendulums. Both $ 2D $ and contour plots produced by parameter changes provide as clear examples of the efficiency, simplicity, and conciseness of the computational method used. The results highlight how flexible this approach is, and demonstrate how symbolic calculations broaden its application to more complex events. This work offers a useful framework and studying intricate physical systems, as well as a flexible computational tool that may be used in a variety of scientific fields.

References

[1]	L. Debnath, A brief historical introduction to solitons and the inverse scattering transforma vision of Scott Russell, Int. J. Math. Educ. Sci. Technol., 38 (2007), 1003–1028. https://doi.org/10.1080/00207390600597849 doi: 10.1080/00207390600597849
[2]	H. Yasmin, N. H. Aljahdaly, A. M. Saeed, Probing families of optical soliton solutions in fractional perturbed RadhakrishnanKunduLakshmanan model with improved versions of extended direct algebraic method, Fractal Fract., 7 (2023), 512. https://doi.org/10.3390/fractalfract7070512 doi: 10.3390/fractalfract7070512
[3]	H. Yasmin, N. H. Aljahdaly, A. M. Saeed, Investigating families of soliton solutions for the complex structured coupled fractional biswasarshed model in birefringent fibers using a novel analytical technique, Fractal Fract., 7 (2023), 491. https://doi.org/10.3390/fractalfract7070491 doi: 10.3390/fractalfract7070491
[4]	A. Saad Alshehry, M. Imran, A. Khan, W. Weera, Fractional view analysis of KuramotoSivashinsky equations with non-singular kernel operators. Symmetry, 14 (2022), 1463. https://doi.org/10.3390/sym14071463
[5]	P. Sunthrayuth, A. M. Zidan, S. W. Yao, R. Shah, M. Inc, The comparative study for solving fractional-order FornbergWhitham equation via $\rho$-Laplace transform, Symmetry, 13 (2021), 784. https://doi.org/10.3390/sym13050784 doi: 10.3390/sym13050784
[6]	H. M. Srivastava, H. Khan, M. Arif, Some analytical and numerical investigation of a family of fractional-order Helmholtz equations in two space dimensions, Math. Meth. Appl. Sci., 43 (2020), 199–212. https://doi.org/10.1002/mma.5846 doi: 10.1002/mma.5846
[7]	R. Almeida, N. R. O. Bastos, M. Teresa, T. Monteiro, Modeling some real phenomena by fractional differential equations, Math. Methods Appl. Sci., 39 (2016), 4846–4855. https://doi.org/10.1002/mma.3818 doi: 10.1002/mma.3818
[8]	H. Sun, Y. Zhang, D. Baleanu, W. Chen, Y. Chen, A new collection of real world applications of fractional calculus in science and engineering. Commun. Nonlinear Sci. Numer. Simul., 64 (2018), 213–231. https://doi.org/10.1016/j.cnsns.2018.04.019
[9]	V. Tarasov, Generalized memory: Fractional calculus approach, Fractals, 2 (2018), 23. https://doi.org/10.3390/fractalfract2040023 doi: 10.3390/fractalfract2040023
[10]	Y. Hu, B. Oksendal, Fractional white noise calculus and applications to finance, Inf. Dimens. Anal. Quantum Probab. Relat. Top., 6 (2003), 1–32. https://doi.org/10.1142/S0219025703001110 doi: 10.1142/S0219025703001110
[11]	W. Zhang, J. Li, Y. Yang, A fractional diffusion-wave equation with non-local regularization for image denoising, Signal Process, 103 (2014), 6–15. https://doi.org/10.1016/j.sigpro.2013.10.028 doi: 10.1016/j.sigpro.2013.10.028
[12]	J. Wu, A wavelet operational method for solving fractional partial differential equations numerically, Appl. Math. Comput., 214 (2009), 31–40. https://doi.org/10.1016/j.amc.2009.03.066 doi: 10.1016/j.amc.2009.03.066
[13]	S. Mukhtar, M. Sohaib, I. Ahmad, A numerical approach to solve volume-based batch crystallization model with fines dissolution unit, Processes, 7 (2019), 453. https://doi.org/10.3390/pr7070453 doi: 10.3390/pr7070453
[14]	R. Garrappa, Numerical solution of fractional differential equations: A survey and a software tutorial, Mathematics, 6 (2018), 16. https://doi.org/10.3390/math6020016 doi: 10.3390/math6020016
[15]	J. Jiang, Y. Feng, S. Li, Exact Solutions to the Fractional Differential Equations with Mixed Partial Derivatives, Axioms, 7 (2018), 10. https://doi.org/10.3390/axioms7010010 doi: 10.3390/axioms7010010
[16]	Y. Xie, I. Ahmad, T. I. Ikpe, E. F. Sofia, H. Seno, What Influence Could the Acceptance of Visitors Cause on the Epidemic Dynamics of a Reinfectious Disease?: A Mathematical Model, Acta Biotheor., 72 (2024), 3. https://doi.org/10.3390/axioms7010010 doi: 10.3390/axioms7010010
[17]	J. Duan, R. Rach, D. Baleanu, A. Wazwaz, A review of the Adomian decomposition method and its applications to fractional differential equations, Commun. Fract. Calc., 3 (2012), 73–99.
[18]	S. Mahmood, R. Shah, M. Arif, Laplace adomian decomposition method for multi dimensional time fractional model of Navier-Stokes equation, Symmetry, 11 (2019), 149. https://doi.org/10.3390/sym11020149 doi: 10.3390/sym11020149
[19]	Q. Wang, Homotopy perturbation method for fractional KdV-Burgers equation, Chaos Soliton. Fract., 35, (2008), 843–850. https://doi.org/10.1016/j.chaos.2006.05.074
[20]	K. R. Raslan, K. K. Ali, M. A. Shallal, The modified extended tanh method with the Riccati equation for solving the space-time fractional EW and MEW equations, Chaos Soliton. Fract., 103 (2017), 404–409. https://doi.org/10.1016/j.chaos.2017.06.029 doi: 10.1016/j.chaos.2017.06.029
[21]	H. Yasmin, N. H. Aljahdaly, A. M. Saeed, Investigating symmetric soliton solutions for the fractional coupled konnoonno system using improved versions of a novel analytical technique, Mathematics, 11 (2023), 2686. https://doi.org/10.3390/math11122686 doi: 10.3390/math11122686
[22]	M. M. Al-Sawalha, A. Khan, O. Y. Ababneh, T. Botmart, Fractional view analysis of Kersten-Krasilshchik coupled KdV-mKdV systems with non-singular kernel derivatives, AIMS Math., 7 (2022), 18334–18359. https://doi.org/10.3934/math.20221010 doi: 10.3934/math.20221010
[23]	H. Yasmin, A. S. Alshehry, A. H. Ganie, A. M. Mahnashi, Perturbed GerdjikovIvanov equation: Soliton solutions via Backlund transformation, Optik, 298 (2024), 171576. https://doi.org/10.1016/j.ijleo.2023.171576 doi: 10.1016/j.ijleo.2023.171576
[24]	I. Ahmad, H. Seno, An epidemic dynamics model with limited isolation capacity, Theor. Biosci., 142 (2023), 259–273. https://doi.org/10.1007/s12064-023-00399-9 doi: 10.1007/s12064-023-00399-9
[25]	W. Hamali, J. Manafian, M. Lakestani, A. M. Mahnashi, A. Bekir, Optical solitons of M-fractional nonlinear Schrdingers complex hyperbolic model by generalized Kudryashov method, Opt. Quant. Electron., 56 (2024), 7. https://doi.org/10.1007/s11082-023-05602-1 doi: 10.1007/s11082-023-05602-1
[26]	M. A. E. Abdelrahman, M. A. Sohaly, Solitary waves for the modified Korteweg-de Vries equation in deterministic case and random case, J. Phys. Math., 8 (2017). https://doi.org/10.4172/2090–0902.1000214.
[27]	M. A. E. Abdelrahman, M. A. Sohaly, Solitary waves for the nonlinear Schrdinger problem with the probability distribution function in the stochastic input case, Eur. Phys. J. Plus., 132 (2017), 339. https://doi.org/10.1140/epjp/i2017-11607-5 doi: 10.1140/epjp/i2017-11607-5
[28]	X. F. Yang, Z. C. Deng, Y. Wei, A Riccati-Bernoulli sub-ODE method for nonlinear partial differential equations and its application, Adv. Diff. Equa., 1 (2015), 117–133. https://doi.org/10.1140/epjp/i2017-11607-5 doi: 10.1140/epjp/i2017-11607-5
[29]	S. Meng, F. Meng, F. Zhang, Q. Li, Y. Zhang, A. Zemouche, Observer design method for nonlinear generalized systems with nonlinear algebraic constraints with applications, Automatica, 162 (2024), 111512. https://doi.org/10.1016/j.automatica.2024.111512 doi: 10.1016/j.automatica.2024.111512
[30]	X. Cai, R. Tang, H. Zhou, Q. Li, S. Ma, D. Wang, et al., Dynamically controlling terahertz wavefronts with cascaded metasurfaces, Adv. Photonics, 3 (2021), 036003. https://doi.org/10.1117/1.AP.3.3.036003 doi: 10.1117/1.AP.3.3.036003
[31]	T. A. A. Ali, Z. Xiao, H. Jiang, B. Li, A class of digital integrators based on trigonometric quadrature rules, IEEE T. Ind. Electron., 71 (2024), 6128–6138. https://doi.org/10.1109/TIE.2023.3290247 doi: 10.1109/TIE.2023.3290247
[32]	C. Guo, J. Hu, Time base generator based practical predefined-time stabilization of high-order systems with unknown disturbance. IEEE Transactions on Circuits and Systems II: Express Briefs, (2023). https://doi.org/10.1109/TCSII.2023.3242856
[33]	Y. Kai, J. Ji, Z. Yin, Study of the generalization of regularized long-wave equation, Nonlinear Dynam., 107 (2022), 2745–2752. https://doi.org/10.1007/s11071-021-07115-6 doi: 10.1007/s11071-021-07115-6
[34]	Y. Kai, Z. Yin, Linear structure and soliton molecules of Sharma-Tasso-Olver-Burgers equation, Phys. Let. A, 452 (2022), 128430. https://doi.org/10.1016/j.physleta.2022.128430 doi: 10.1016/j.physleta.2022.128430
[35]	L. Debnath, Nonlinear partial differential equations for scientists and engineers, Boston: Birkhauser, (2005), 528–529. https://doi.org/10.1007/b138648
[36]	M. Z. Sarikaya, H. Budak, H. Usta, On generalized the conformable fractional calculus, TWMS J. Appl. Eng. Math., 9 (2019), 792–799.
[37]	M. A. Ramadan, M. S. Al-Luhaibi, Application of Sumudu decomposition method for solving linear and nonlinear Klein-Gordon equations, Int. J. Soft Comput. Eng., 3 (2016), 138–140.
[38]	M. Hussain, M. Khan, A variational iterative method for solving the linear and nonlinear Klein-Gordon equations, Appl. Math. Sci., 4 (2010), 1931–1940.
[39]	H. Hosseinzadeh, H. Jafari, M. Roohani, Application of laplace decomposition method for solving Klein-Gordon equation, World Appl. Sci. J., 8 (2010), 809–813.
[40]	S. Kulkarni, K. Takale, Application of Adomian decomposition method for solving linear and nonlinear Klein-Gordon equations, Int. J. Eng. Contemp. Math. Sci., 1 (2015), 21–27.
[41]	A. K. Adio, Natural decomposition method for solving the linear and nonlinear Klein Gordon equations, Int. J. Res. Appl., 4 (2016), 59–72.
[42]	D. Kumar, J. Singh, D. Baleanu, A hybrid computational approach for KleinGordon equations on Cantor sets, Nonlinear Dynam., 87 (2017), 511–517. https://doi.org/10.1007/s11071-016-3057-x doi: 10.1007/s11071-016-3057-x
[43]	B. Grebert, E. Paturel, KAM for the Klein Gordon equation on SdSd, Bollettino dellUnione Matematica Italiana, 9 (2016), 237–288. https://doi.org/10.1007/s40574-016-0072-2 doi: 10.1007/s40574-016-0072-2
[44]	D. A. Nugraha, A. Suparmi, C. Cari, B. N. Pratiwi, Asymptotic iteration method for solution of the Kratzer potential in D-dimensional Klein-Gordon equation, J. Phys. Conf. Ser., 820 (2017), 1–8. https://doi.org/10.1088/1742-6596/820/1/012014 doi: 10.1088/1742-6596/820/1/012014
[45]	Y. Luo, X. Li, C. Gu, Fourth-order compact and energy conservative scheme for solving nonlinear Klein-Gordon equation, Numer. Meth. Part. D. E., (2017), 1283–1304. https://doi.org/10.1002/num.22143
[46]	D. Lu, Q. Shi, New Jacobi elliptic functions solutions for the combined KdV-mKdV equation, Int. J. Nonlinear Sci., 10, 320–325.
[47]	Y. Zhang, Solving STO and KD equations with modified RiemannLiouville derivative using improved ($G/G'$)-expansion function method, Int. J. Appl. Math. 45 (2015), 16–22.

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