The solitary wave phenomena of the fractional Calogero-Bogoyavlenskii-Schiff equation

Hussain Gissy; Abdullah Ali H. Ahmadini; Ali H. Hakami; Hussain Gissy; Abdullah Ali H. Ahmadini; Ali H. Hakami

doi:10.3934/math.2025020

AIMS Mathematics

2025, Volume 10, Issue 1: 420-437. doi: 10.3934/math.2025020

Previous Article Next Article

Research article Special Issues

The solitary wave phenomena of the fractional Calogero-Bogoyavlenskii-Schiff equation

Department of Mathematics, Faculty of Science, Jazan University, P.O. Box 2097, Jazan 45142, Saudi Arabia

Received: 22 November 2024 Revised: 21 December 2024 Accepted: 02 January 2025 Published: 09 January 2025
MSC : 34G20, 35A20, 35A22, 35R11

The Riemann waves in two spatial dimensions are described by the fractional Calogero-Bogoyavlenskii-Schiff equation, which has been used to explain numerous physical phenomena including magneto-sound waves in plasmas, tsunamis, and flows in rivers and internal oceans. This work concerned itself with obtaining new analytic soliton solutions for the fractional Calogero-Bogoyavlenskii-Schiff model based on the fractional conformable. By solving the model equation with the Riccati-Bernoulli sub-ODE technique in association with the Bäcklund transformation, the solution was found in terms of trigonometric, hyperbolic, and rational functions. To analyze the detailed features of the wave structures as well as the pattern of dynamics of these solutions, 3D and contour diagrams were plotted by using Wolfram Mathematica. A great advantage of these types of visualizations is that they demonstrate amplitude, shape, and propagation characteristics of the selected soliton solutions. The results reveal that the proposed approach is accurate, universal, and fast for the investigation of the different aspects of the Riemann problem and the related phenomena concerning the propagation of waves.
Citation: Hussain Gissy, Abdullah Ali H. Ahmadini, Ali H. Hakami. The solitary wave phenomena of the fractional Calogero-Bogoyavlenskii-Schiff equation[J]. AIMS Mathematics, 2025, 10(1): 420-437. doi: 10.3934/math.2025020

Related Papers:

Abstract

The Riemann waves in two spatial dimensions are described by the fractional Calogero-Bogoyavlenskii-Schiff equation, which has been used to explain numerous physical phenomena including magneto-sound waves in plasmas, tsunamis, and flows in rivers and internal oceans. This work concerned itself with obtaining new analytic soliton solutions for the fractional Calogero-Bogoyavlenskii-Schiff model based on the fractional conformable. By solving the model equation with the Riccati-Bernoulli sub-ODE technique in association with the Bäcklund transformation, the solution was found in terms of trigonometric, hyperbolic, and rational functions. To analyze the detailed features of the wave structures as well as the pattern of dynamics of these solutions, 3D and contour diagrams were plotted by using Wolfram Mathematica. A great advantage of these types of visualizations is that they demonstrate amplitude, shape, and propagation characteristics of the selected soliton solutions. The results reveal that the proposed approach is accurate, universal, and fast for the investigation of the different aspects of the Riemann problem and the related phenomena concerning the propagation of waves.

References

[1]	W. Gao, H. Rezazadeh, Z. Pinar, H. M. 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. https://doi.org/10.1007/s11082-019-2162-8 doi: 10.1007/s11082-019-2162-8
[2]	C. Zhu, M. Al-Dossari, S. Rezapour, B. Gunay, On the exact soliton solutions and different wave structures to the $(2+1)$-dimensional Chaffee-Infante equation, Results Phys., 57 (2024), 107431. https://doi.org/10.1016/j.rinp.2024.107431 doi: 10.1016/j.rinp.2024.107431
[3]	M. Ghasemi, High order approximations using spline-based differential quadrature method: implementation to the multi-dimensional PDEs, Appl. Math. Model., 46 (2017), 63–80. https://doi.org/10.1016/j.apm.2017.01.052 doi: 10.1016/j.apm.2017.01.052
[4]	N. Perrone, R. Kao, A general finite difference method for arbitrary meshes, Comput. Struct., 5 (1975), 45–57. https://doi.org/10.1016/0045-7949(75)90018-8 doi: 10.1016/0045-7949(75)90018-8
[5]	S. Mahmood, R. Shah, H. Khan, 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
[6]	M. A. Abdou, A. A. Soliman, New applications of variational iteration method, Phys. D, 211 (2005), 1–8. https://doi.org/10.1016/j.physd.2005.08.002 doi: 10.1016/j.physd.2005.08.002
[7]	E. Yusufoglu, 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. https://doi.org/10.1080/00207160601138756 doi: 10.1080/00207160601138756
[8]	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.
[9]	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
[10]	I. Ahmad, H. Seno, An epidemic dynamics model with limited isolation capacity, Theory Biosci., 142 (2023), 259–273. https://doi.org/10.1007/s12064-023-00399-9 doi: 10.1007/s12064-023-00399-9
[11]	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
[12]	A. A. Alderremy, N. Iqbal, S. Aly, K. Nonlaopon, Fractional series solution construction for nonlinear fractional reaction-diffusion Brusselator model utilizing Laplace residual power series, Symmetry, 14 (2022), 1944. https://doi.org/10.3390/sym14091944 doi: 10.3390/sym14091944
[13]	H. Yasmin, A. S. Alshehry, A. H. Ganie, A. M. Mahnashi, Perturbed Gerdjikov-Ivanov 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
[14]	M. M. Al-Sawalha, A. Khan, O. Y. Ababneh, T. Botmart, Fractional view analysis of Kersten-Krasil'shchik 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
[15]	E. M. Elsayed, R. Shah, K. Nonlaopon, The analysis of the fractional-order Navier-Stokes equations by a novel approach, J. Funct. Spaces, 2022 (2022), 8979447. https://doi.org/10.1155/2022/8979447 doi: 10.1155/2022/8979447
[16]	M. Naeem, H. Rezazadeh, A. A. Khammash, R. Shah, S. Zaland, Analysis of the fuzzy fractional-order solitary wave solutions for the KdV equation in the sense of Caputo-Fabrizio derivative, J. Math., 2022 (2022), 3688916. https://doi.org/10.1155/2022/3688916 doi: 10.1155/2022/3688916
[17]	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/10.1007/s10441-024-09478-w doi: 10.1007/s10441-024-09478-w
[18]	K. Hosseini, M. Mirzazadeh, J. F. Gomez-Aguilar, Soliton solutions of the Sasa-Satsuma equation in the monomode optical fibers including the beta-derivatives, Optik, 224 (2020), 165425. https://doi.org/10.1016/j.ijleo.2020.165425 doi: 10.1016/j.ijleo.2020.165425
[19]	R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65–70. https://doi.org/10.1016/j.cam.2014.01.002 doi: 10.1016/j.cam.2014.01.002
[20]	K. A. Abro, A. Atangana, J. F. Gomez-Aguilar, Role of bi-order Atangana-Aguilar fractional differentiation on Drude model: an analytic study for distinct sources, Opt. Quantum Electron., 53 (2021), 177. https://doi.org/10.1007/s11082-021-02804-3 doi: 10.1007/s11082-021-02804-3
[21]	J. F. Gomez-Aguilar, D. Baleanu, Schrödinger equation involving fractional operators with non-singular kernel, J. Electromagn. Waves Appl., 31 (2017), 752–761. https://doi.org/10.1080/09205071.2017.1312556 doi: 10.1080/09205071.2017.1312556
[22]	B. Ghanbari, M. S. Osman, D. Baleanu, Generalized exponential rational function method for extended Zakharov-Kuzetsov equation with conformable derivative, Mod. Phys. Lett. A, 34 (2019), 1950155. https://doi.org/10.1142/S0217732319501554 doi: 10.1142/S0217732319501554
[23]	A. Akbulut, F. Tascan, E. Ozel, Trivial conservation laws and solitary wave solution of the fifth-order Lax equation, Partial Differ. Equ. Appl. Math., 4 (2021), 100101. https://doi.org/10.1016/j.padiff.2021.100101 doi: 10.1016/j.padiff.2021.100101
[24]	H. Khatri, M. S. Gautam, A. Malik, Localized and complex soliton solutions to the integrable $(4+ 1)$-dimensional Fokas equation, SN Appl. Sci., 1 (2019), 1070. https://doi.org/10.1007/s42452-019-1094-z doi: 10.1007/s42452-019-1094-z
[25]	M. N. Alam, O. A. Ilhan, M. S. Uddin, M. A. Rahim, Regarding on the results for the fractional Clannish Random Walker's parabolic equation and the nonlinear fractional Cahn-Allen equation, Adv. Math. Phys., 2022 (2022), 5635514. https://doi.org/10.1155/2022/5635514 doi: 10.1155/2022/5635514
[26]	M. N. Alam, S. Islam, O. A. Ilhan, H. Bulut, Some new results of nonlinear model arising in incompressible visco-elastic Kelvin-Voigt fluid, Math. Methods Appl. Sci., 45 (2022), 10347–10362. https://doi.org/10.1002/mma.8372 doi: 10.1002/mma.8372
[27]	U. Younas, A. R. Seadawy, M. Younis, S. T. R. Rizvi, Optical solitons and closed-form solutions to the $(3+1)$-dimensional resonant Schrödinger dynamical wave equation, Int. J. Mod. Phys. B, 34 (2020), 2050291. https://doi.org/10.1142/S0217979220502914 doi: 10.1142/S0217979220502914
[28]	M. Arshad, A. Seadawy, D. Lu, J. Wang, Travelling wave solutions of generalized coupled Zakharov-Kuznetsov and dispersive long wave equations, Results Phys., 6 (2016), 1136–1145. https://doi.org/10.1016/j.rinp.2016.11.043 doi: 10.1016/j.rinp.2016.11.043
[29]	S. Kumar, K. S. Nisar, M. Niwas, On the dynamics of exact solutions to a $(3+1)$-dimensional YTSF equation emerging in shallow sea waves: Lie symmetry analysis and generalized Kudryashov method, Results Phys., 48 (2023), 106432. https://doi.org/10.1016/j.rinp.2023.106432 doi: 10.1016/j.rinp.2023.106432
[30]	A. Irshad, N. Ahmed, A. Nazir, U. Khan, S. T. Mohyud-Din, Novel exact double periodic soliton solutions to strain wave equation in micro structured solids, Phys. A, 550 (2020), 124077. https://doi.org/10.1016/j.physa.2019.124077 doi: 10.1016/j.physa.2019.124077
[31]	S. Bibi, S. T. Mohyud-Din, U. Khan, N. Ahmed, Khater method for nonlinear Sharma-Tasso-Olever (STO) equation of fractional order, Results Phys., 7 (2017), 4440–4450. https://doi.org/10.1016/j.rinp.2017.11.008 doi: 10.1016/j.rinp.2017.11.008
[32]	M. N. Alam, An analytical method for finding exact solutions of a nonlinear partial differential equation arising in electrical engineering, Open J. Math. Sci., 7 (2023), 10–18. https://doi.org/10.30538/oms2023.0195 doi: 10.30538/oms2023.0195
[33]	M. Alqhtani, K. M. Saad, R. Shah, W. M. 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. https://doi.org/10.1007/s11082-023-05407-2 doi: 10.1007/s11082-023-05407-2
[34]	M. Alqhtani, K. M. Saad, R. Shah, W. Weera, W. M. Hamanah, Analysis of the fractional-order local Poisson equation in fractal porous media, Symmetry, 14 (2022), 1323. https://doi.org/10.3390/sym14071323 doi: 10.3390/sym14071323
[35]	S. A. El-Tantawy, R. T. Matoog, R. Shah, A. W. Alrowaily, S. M. E. Ismaeel, On the shock wave approximation to fractional generalized Burger-Fisher equations using the residual power series transform method, Phys. Fluids, 36 (2024), 023105. https://doi.org/10.1063/5.0187127 doi: 10.1063/5.0187127
[36]	H. Yasmin, A. S. Alshehry, A. H. 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. https://doi.org/10.1038/s41598-024-52211-3 doi: 10.1038/s41598-024-52211-3
[37]	S. Noor, W. Albalawi, R. Shah, M. M. Al-Sawalha, S. M. Ismaeel, S. A. 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. https://doi.org/10.3389/fphy.2024.1374481 doi: 10.3389/fphy.2024.1374481
[38]	H. Tian, M. Zhao, J. Liu, Q. Wang, X. Yu, Z. Wang, Dynamic analysis and sliding mode synchronization control of chaotic systems with conditional symmetric fractional-order memristors, Fractal Fract., 8 (2024), 307. https://doi.org/10.3390/fractalfract8060307 doi: 10.3390/fractalfract8060307
[39]	L. Liu, S. Zhang, L. Zhang, G. Pan, J. Yu, Multi-UUV maneuvering counter-game for dynamic target scenario based on fractional-order recurrent neural network, IEEE Trans. Cybernetics, 53 (2023), 4015–4028. https://doi.org/10.1109/TCYB.2022.3225106 doi: 10.1109/TCYB.2022.3225106
[40]	M. Li, T. Wang, F. Chu, Q. Han, Z. Qin, M. J. Zuo, Scaling-basis chirplet transform, IEEE Trans. Ind. Electron., 68 (2021), 8777–8788. https://doi.org/10.1109/TIE.2020.3013537 doi: 10.1109/TIE.2020.3013537
[41]	A. Iftikhar, A. Ghafoor, T. Zubair, S. Firdous, S. T. Mohyud-Din, Solutions of $(2+1)$-dimensional generalized KdV, Sin-Gordon, and Landau-Ginzburg-Higgs equations, Sci. Res. Essays, 8 (2013), 1349–1359.
[42]	J. H. He, X. H. Wu, Exp-function method for nonlinear wave equations, Chaos Soliton. Fract., 30 (2006), 700–708. https://doi.org/10.1016/j.chaos.2006.03.020 doi: 10.1016/j.chaos.2006.03.020
[43]	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
[44]	M. M. Al-Sawalha, H. Yasmin, R. Shah, A. H. Ganie, K. Moaddy, Unraveling the dynamics of singular stochastic solitons in stochastic fractional Kuramoto-Sivashinsky equation, Fractal Fract., 7 (2023), 753. https://doi.org/10.3390/fractalfract7100753 doi: 10.3390/fractalfract7100753
[45]	K. J. Wang, F. Shi, Multi-soliton solutions and soliton molecules of the $(2+1)$-dimensional Boiti-Leon-Manna-Pempinelli equation for incompressible fluid, Europhys. Lett., 145 (2024), 42001. https://doi.org/10.1209/0295-5075/ad219d doi: 10.1209/0295-5075/ad219d
[46]	P. F. Han, Y. Zhang, Investigation of shallow water waves near the coast or in lake environments via the KdV-Calogero-Bogoyavlenskii-Schiff equation, Chaos Soliton. Fract., 184 (2024), 115008. https://doi.org/10.1016/j.chaos.2024.115008 doi: 10.1016/j.chaos.2024.115008
[47]	X. F. Yang, Z. C. Deng, Y. Wei, A Riccati-Bernoulli sub-ODE method for nonlinear partial differential equations and its application, Adv. Differ. Equ., 2015 (2015), 117. https://doi.org/10.1186/s13662-015-0452-4 doi: 10.1186/s13662-015-0452-4
[48]	P. F. Han, T. Bao, Bilinear auto-Bäcklund transformations and higher-order breather solutions for the $(3+1)$-dimensional generalized KdV-type equation, Nonlinear Dyn., 110 (2022), 1709–1721. https://doi.org/10.1007/s11071-022-07658-2 doi: 10.1007/s11071-022-07658-2
[49]	P. F. Han, T. Bao, Dynamical behavior of multiwave interaction solutions for the $(3+1)$-dimensional Kadomtsev-Petviashvili-Bogoyavlensky-Konopelchenko equation, Nonlinear Dyn., 111 (2023), 4753–4768. https://doi.org/10.1007/s11071-022-08097-9 doi: 10.1007/s11071-022-08097-9
[50]	D. Lu, Q. Shi, New Jacobi elliptic functions solutions for the combined KdV-mKdV equation, Int. J. Nonlinear Sci., 10 (2010), 320–325.
[51]	S. M. Mabrouk, Traveling wave solutions of the extended Calogero-Bogoyavlenskii-Schiff equation, Int. J. Eng. Res. Technol., 2019.
[52]	D. Baldwin, W. Hereman, A symbolic algorithm for computing recursion operators of nonlinear partial differential equations, Int. J. Comput. Math., 87 (2010), 1094–1119. https://doi.org/10.1080/00207160903111592 doi: 10.1080/00207160903111592
[53]	H. Rezazadeh, A. Korkmaz, M. Eslami, J. Vahidi, R. Asghari, Traveling wave solution of conformable fractional generalized reaction Duffing model by generalized projective Riccati equation method, Opt. Quant. Electron., 50 (2018), 150. https://doi.org/10.1007/s11082-018-1416-1 doi: 10.1007/s11082-018-1416-1
[54]	L. M. B. Alam, X. Jiang, A. A. Mamun, Exact and explicit traveling wave solution to the time-fractional phi-four and $(2+1)$-dimensional CBS equations using the modified extended tanh-function method in mathematical physics, Partial Differ. Equ. Appl. Math., 2021 (2021), 100039. https://doi.org/10.1016/j.padiff.2021.100039 doi: 10.1016/j.padiff.2021.100039

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)