Kink soliton solution of integrable Kairat-X equation via two integration algorithms

Raed Qahiti; Naher Mohammed A. Alsafri; Hamad Zogan; Abdullah A. Faqihi; Raed Qahiti; Naher Mohammed A. Alsafri; Hamad Zogan; Abdullah A. Faqihi

doi:10.3934/math.20241456

AIMS Mathematics

2024, Volume 9, Issue 11: 30153-30173. doi: 10.3934/math.20241456

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Kink soliton solution of integrable Kairat-X equation via two integration algorithms

1.
Department of Mathematics, Faculty of Science, Jazan University, P.O. Box 2097, Jazan 45142, kingdom of Saudi Arabia
2.
Department of Mathematics, Faculty of Science, University of Jeddah, kingdom of Saudi Arabia
3.
Department of Computer Science, College of Computer Science and Information Technology, Jazan University, Jazan 45142, kingdom of Saudi Arabia
4.
Department of Industrial Engineering, College of Engineering and Computer Science, Jazan University, P.O. Box 706, Jazan 45142, kingdom of Saudi Arabia

Received: 30 August 2024 Revised: 17 October 2024 Accepted: 18 October 2024 Published: 23 October 2024
MSC : 34A25, 35C08, 35D99

In order to establish and assess the dynamics of kink solitons in the integrable Kairat-X equation, which explains the differential geometry of curves and equivalence aspects, the present investigation put forward two variants of a unique transformation-based analytical technique. These modifications were referred to as the generalized ($ r+\frac{G'}{G} $)-expansion method and the simple ($ \frac{G'}{G} $)-expansion approach. The proposed methods spilled over the aimed Kairat-X equation into a nonlinear ordinary differential equation by means of a variable transformation. Immediately following that, it was presumed that the resultant nonlinear ordinary differential equation had a closed form solution, which turned it into a system of algebraic equations. The resultant set of algebraic equations was solved to find new families of soliton solutions which took the forms of hyperbolic, rational and periodic functions. An assortment of contour, 2D and 3D graphs were used to visually show the dynamics of certain generated soliton solutions. This indicated that these soliton solutions likely took the structures of kink solitons prominently. Moreover, our proposed methods demonstrated their use by constructing a multiplicity of soliton solutions, offering significant understanding into the evolution of the focused model, and suggesting possible applications in dealing with related nonlinear phenomena.
- nonlinear partial differential equations,
- ($ \frac{G'}{G} $)-expansion methods,
- integrable model,
- closed-form solutions,
- soliton phenomena
Citation: Raed Qahiti, Naher Mohammed A. Alsafri, Hamad Zogan, Abdullah A. Faqihi. Kink soliton solution of integrable Kairat-X equation via two integration algorithms[J]. AIMS Mathematics, 2024, 9(11): 30153-30173. doi: 10.3934/math.20241456

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Abstract

In order to establish and assess the dynamics of kink solitons in the integrable Kairat-X equation, which explains the differential geometry of curves and equivalence aspects, the present investigation put forward two variants of a unique transformation-based analytical technique. These modifications were referred to as the generalized ($ r+\frac{G'}{G} $)-expansion method and the simple ($ \frac{G'}{G} $)-expansion approach. The proposed methods spilled over the aimed Kairat-X equation into a nonlinear ordinary differential equation by means of a variable transformation. Immediately following that, it was presumed that the resultant nonlinear ordinary differential equation had a closed form solution, which turned it into a system of algebraic equations. The resultant set of algebraic equations was solved to find new families of soliton solutions which took the forms of hyperbolic, rational and periodic functions. An assortment of contour, 2D and 3D graphs were used to visually show the dynamics of certain generated soliton solutions. This indicated that these soliton solutions likely took the structures of kink solitons prominently. Moreover, our proposed methods demonstrated their use by constructing a multiplicity of soliton solutions, offering significant understanding into the evolution of the focused model, and suggesting possible applications in dealing with related nonlinear phenomena.

References

[1]	Y. Swapna, Applications of partial differential equations in fluid physics, Commun. Appl. Nonlinear Anal., 31 (2024), 207–220. https://doi.org/10.52783/cana.v31.396 doi: 10.52783/cana.v31.396
[2]	A. Cheviakov, P. Zhao, Analytical properties of nonlinear partial differential equations: with applications to shallow water models, Vol. 10, Springer Cham, 2024. https://doi.org/10.1007/978-3-031-53074-6
[3]	A. H. Ganie, L. H. Sadek, M. M. Tharwat, M. A. Iqbal, M. M. Miah, M. M. Rasid, et al., New investigation of the analytical behaviors for some nonlinear PDEs in mathematical physics and modern engineering, Partial Differ. Equations Appl. Math., 9 (2024), 100608. https://doi.org/10.1016/j.padiff.2023.100608 doi: 10.1016/j.padiff.2023.100608
[4]	J. L. Kazdan, Applications of partial differential equations to problems in geometry, Graduate Texts in Mathematics, 1983.
[5]	H. Khan, R. Shah, J. F. Gómez-Aguilar, Shoaib, D. Baleanu, P. Kumam, Travelling waves solution for fractional-order biological population model, Math. Model. Nat. Phenom., 16 (2021), 32. https://doi.org/10.1051/mmnp/2021016 doi: 10.1051/mmnp/2021016
[6]	A. P. Bassom, P. A. Clarkson, A. C. Hicks, On the application of solutions of the fourth Painlev equation to various physically motivated nonlinear partial differential equations, Adv. Differ. Equations, 1 (1996), 175–198. https://doi.org/10.57262/ade/1366896236 doi: 10.57262/ade/1366896236
[7]	P. Albayrak, M. Ozisik, M. Bayram, A. Secer, S. E. Das, A. Biswas, et al., Pure-cubic optical solitons and stability analysis with Kerr law nonlinearity, Contemp. Math., 4 (2023), 530-548. https://doi.org/10.37256/cm.4320233308 doi: 10.37256/cm.4320233308
[8]	S. Altun, M. Ozisik, A. Secer, M. Bayram, Optical solitons for Biswas-Milovic equation using the new Kudryashov's scheme, Optik, 270 (2022), 170045. https://doi.org/10.1016/j.ijleo.2022.170045 doi: 10.1016/j.ijleo.2022.170045
[9]	E. M. Zayed, A. H. Arnous, A. Secer, M. Ozisik, M. Bayram, N. A. Shah, et al., Highly dispersive optical solitons in fiber Bragg gratings for stochastic Lakshmanan-Porsezian-Daniel equation with spatio-temporal dispersion and multiplicative white noise, Results Phys., 55 (2023), 107177. https://doi.org/10.1016/j.rinp.2023.107177 doi: 10.1016/j.rinp.2023.107177
[10]	M. S. Islam, K. Khan, M. A. Akbar, The generalized Kudrysov method to solve some coupled nonlinear evolution equations, Asian J. Math. Comput. Res., 3 (2015), 104–121.
[11]	R. Ali, E. Tag-eldin, A comparative analysis of generalized and extended ($\frac{G'}{G}$)-Expansion methods for travelling wave solutions of fractional Maccari's system with complex structure, Alexandria Eng. J., 79 (2023), 508–530. https://doi.org/10.1016/j.aej.2023.08.007 doi: 10.1016/j.aej.2023.08.007
[12]	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. https://doi.org/10.1007/s11082-022-03819-0 doi: 10.1007/s11082-022-03819-0
[13]	M. Dehghan, J. Manafian Heris, A. Saadatmandi, Application of the Exp-function method for solving a partial differential equation arising in biology and population genetics, Int. J. Numer. Methods Heat Fluid Flow, 21 (2011), 736–753. https://doi.org/10.1108/09615531111148482 doi: 10.1108/09615531111148482
[14]	A. Bekir, E. Aksoy, A. C. Cevikel, Exact solutions of nonlinear time fractional partial differential equations by sub-equation method, Math. Methods Appl. Sci., 38 (2015), 2779–2784. https://doi.org/10.1002/mma.3260 doi: 10.1002/mma.3260
[15]	M. Kamrujjaman, A. Ahmed, J. Alam, Travelling waves: interplay of low to high Reynolds number and Tan-Cot function method to solve Burgers equations, J. Appl. Math. Phys., 7 (2019), 861. https://doi.org/10.4236/jamp.2019.74058 doi: 10.4236/jamp.2019.74058
[16]	S. Noor, A. S. Alshehry, A. Khan, I. Khan, Analysis of soliton phenomena in $(2+ 1)$-dimensional Nizhnik-Novikov-Veselov model via a modified analytical technique, AIMS Math., 8 (2023), 28120–28142. https://doi.org/10.3934/math.20231439 doi: 10.3934/math.20231439
[17]	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. https://doi.org/10.1088/1402-4896/ad4784 doi: 10.1088/1402-4896/ad4784
[18]	M. M. Tariq, M. B. Riaz, M. Aziz-ur-Rehman, Investigation of space-time dynamics of Akbota equation using Sardar sub-equation and Khater methods: unveiling bifurcation and chaotic structure, Int. J. Theor. Phys., 63 (2024), 210. https://doi.org/10.1007/s10773-024-05733-5 doi: 10.1007/s10773-024-05733-5
[19]	X. Yang, Z. Wang, Z. Zhang, Decay mode ripple waves within the $(3+ 1)$-dimensional Kadomtsev-Petviashvili equation, Math. Methods Appl. Sci., 47 (2024), 10444-10461. https://doi.org/10.1002/mma.10132 doi: 10.1002/mma.10132
[20]	A. H. Ganie, M. M. AlBaidani, A. M. Wazwaz, W. X. Ma, U. Shamima, M. S. Ullah, Soliton dynamics and chaotic analysis of the Biswas-Arshed model, Opt. Quant. Electron., 56 (2024), 1379. https://doi.org/10.1007/s11082-024-07291-w doi: 10.1007/s11082-024-07291-w
[21]	M. Wang, X. Li, J. Zhang, The ($\frac{G'}{G}$)-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
[22]	E. H. M. Zahran, M. M. A. Khater, Modified extended tanh-function method and its applications to the Bogoyavlenskii equation, Appl. Math. Model., 40 (2016), 1769–1775. https://doi.org/10.1016/j.apm.2015.08.018 doi: 10.1016/j.apm.2015.08.018
[23]	Z. Myrzakulova, S. Manukure, R. Myrzakulov, G. Nugmanova, Integrability, geometry and wave solutions of some Kairat equations, arXiv, 2023. https://doi.org/10.48550/arXiv.2307.00027
[24]	M. Awadalla, A. Zafar, A. Taishiyeva, M. Raheel, R. Myrzakulov, A. Bekir, The analytical solutions to the M-fractional Kairat-Ⅱ and Kairat-X equations, Front. Phys., 11 (2023), 1335423.
[25]	S. Roy, S. Raut, R. Myrzakulov, Z. Umurzakhova, A Kairat-X equation and its integrability: shocks, lump-kink and kinky-breather, CC BY 4.0, 2023. https://doi.org/10.13140/RG.2.2.23245.20963 doi: 10.13140/RG.2.2.23245.20963
[26]	S. Ghazanfar, N. Ahmed, M. S. Iqbal, A. Akgül, M. Bayram, M. De la Sen, Imaging ultrasound propagation using the Westervelt equation by the generalized Kudryashov and modified Kudryashov methods, Appl. Sci., 12 (2022), 11813. https://doi.org/10.3390/app122211813 doi: 10.3390/app122211813
[27]	G. H. Tipu, W. A. Faridi, Z. Myrzakulova, R. Myrzakulov, S. A. AlQahtani, N. F. AlQahtani, et al., On optical soliton wave solutions of non-linear Kairat-X equation via new extended direct algebraic method, Opt. Quant. Electron., 56 (2024), 655. https://doi.org/10.1007/s11082-024-06369-9 doi: 10.1007/s11082-024-06369-9
[28]	M. Iqbal, D. Lu, A. R. Seadawy, F. A. H. Alomari, Z. Umurzakhova, R. Myrzakulov, Constructing the soliton wave structure to the nonlinear fractional Kairat-X dynamical equation under computational approach, Mod. Phys. Lett. B, 2024, 2450396. https://doi.org/10.1142/S0217984924503962 doi: 10.1142/S0217984924503962
[29]	S. Sirisubtawee, S. Koonprasert, S. Sungnul, New exact solutions of the conformable space-time Sharma-Tasso-Olver equation using two reliable methods, Symmetry, 12 (2020), 644. https://doi.org/10.3390/sym12040644 doi: 10.3390/sym12040644

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