The travelling wave phenomena of the space-time fractional Whitham-Broer-Kaup equation

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

doi:10.3934/math.2025116

AIMS Mathematics

2025, Volume 10, Issue 2: 2492-2508. doi: 10.3934/math.2025116

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The travelling wave phenomena of the space-time fractional Whitham-Broer-Kaup equation

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

Received: 26 November 2024 Revised: 08 January 2025 Accepted: 17 January 2025 Published: 12 February 2025
MSC : 34G20, 35R11

In the present work, we consider the space-time fractional Whitham-Broer-Kaup (FWBK) equation and then find its analytical solutions under the framework of the Riccati-Bernoulli sub-ordinary differential equation method along with the Bäcklund transformation. The derived solutions are described in terms of the hyperbolic rational and trigonometric functions and resolve unique wave features. A unique feature of this work is the investigation of the impact of the fractional order on wave steepness in $ 2D $, $ 3D $, and contour plots. Also, it should be pointed that the found relationship shows that the change of the fractional order parameter really describes the traveling periodic waves, for example, Stokes waves, with potential importance for the analysis of wave propagation and stability in the nonlinear fractional structures. This work carries theoretical developments of fractional calculus and ideas for applications in fluid dynamics and wave mechanics.
- fractional partial differential equations,
- fractional Whitham-Broer-Kaup equation,
- Bäcklund transformation,
- Riccatti equation,
- travelling wave solutions
Citation: Hussain Gissy, Abdullah Ali H. Ahmadini, Ali H. Hakami. The travelling wave phenomena of the space-time fractional Whitham-Broer-Kaup equation[J]. AIMS Mathematics, 2025, 10(2): 2492-2508. doi: 10.3934/math.2025116

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Abstract

In the present work, we consider the space-time fractional Whitham-Broer-Kaup (FWBK) equation and then find its analytical solutions under the framework of the Riccati-Bernoulli sub-ordinary differential equation method along with the Bäcklund transformation. The derived solutions are described in terms of the hyperbolic rational and trigonometric functions and resolve unique wave features. A unique feature of this work is the investigation of the impact of the fractional order on wave steepness in $ 2D $, $ 3D $, and contour plots. Also, it should be pointed that the found relationship shows that the change of the fractional order parameter really describes the traveling periodic waves, for example, Stokes waves, with potential importance for the analysis of wave propagation and stability in the nonlinear fractional structures. This work carries theoretical developments of fractional calculus and ideas for applications in fluid dynamics and wave mechanics.

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. N. Rafiq, M. H. Rafiq, H. Alsaud, Chaotic response, multistability and new wave structures for the generalized coupled Whitham-Broer-Kaup-Boussinesq-Kupershmidt system with a novel methodology, Chaos Soliton. Fract., 190 (2025), 115755. https://doi.org/10.1016/j.chaos.2024.115755 doi: 10.1016/j.chaos.2024.115755
[4]	X. H. Zhao, Multi-solitons and integrability for a $(2+1)$-dimensional variable coefficients Date-Jimbo-Kashiwara-Miwa equation, Appl. Math. Lett., 149 (2024), 108895. https://doi.org/10.1016/j.aml.2023.108895 doi: 10.1016/j.aml.2023.108895
[5]	Z. Z. Lan, Semirational rogue waves of the three coupled higher-order nonlinear Schrödinger equations, Appl. Math. Lett., 147 (2024), 108845. https://doi.org/10.1016/j.aml.2023.108845 doi: 10.1016/j.aml.2023.108845
[6]	Z. Z. Lan, Multi-soliton solutions, breather-like and bound-state solitons for complex modified Korteweg-de Vries equation in optical fibers, Chinese Phys. B, 33 (2024), 060201. https://doi.org/10.1088/1674-1056/ad39d7 doi: 10.1088/1674-1056/ad39d7
[7]	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
[8]	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
[9]	M. N. Rafiq, H. Chen, Multiple interaction solutions, parameter analysis, chaotic phenomena and modulation instability for a $(3+1)$-dimensional Kudryashov-Sinelshchikov equation in ideal liquid with gas bubbles, Nonlinear Dyn., 2024. https://doi.org/10.1007/s11071-024-10164-2
[10]	S. Mahmood, R. Shah, H. Khan, M. Arif, Laplace Adomian Decomposition method for multidimensional time fractional model of Navier-Stokes equation, Symmetry, 11 (2019), 149. https://doi.org/10.3390/sym11020149 doi: 10.3390/sym11020149
[11]	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
[12]	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
[13]	E. Az-Zo'bi, L. Akinyemi, A. O. Alleddawi, Construction of optical solitons for conformable generalized model in nonlinear media, Mod. Phys. Lett. B, 35 (2021), 2150409. https://doi.org/10.1142/S0217984921504091 doi: 10.1142/S0217984921504091
[14]	E. A. Az-Zo'bi, Exact analytic solutions for nonlinear diffusion equations via generalized residual power series method, Int. J. Math. Comput. Sci., 14 (2019), 69–78.
[15]	C. K. Kuo, New solitary solutions of the Gardner equation and Whitham-Broer-Kaup equations by the modified simplest equation method, Optik, 147 (2017), 128–135. https://doi.org/10.1016/j.ijleo.2017.08.048 doi: 10.1016/j.ijleo.2017.08.048
[16]	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
[17]	M. Lei, H. Liao, S. Wang, H. Zhou, J. Zhu, H. Wan, et al., Electro-sorting creates heterogeneity: constructing a multifunctional Janus film with integrated compositional and microstructural gradients for guided bone regeneration, Adv. Sci., 11 (2024), 2307606. https://doi.org/10.1002/advs.202307606 doi: 10.1002/advs.202307606
[18]	R. Ali, M. M. Alam, S. Barak, Exploring chaotic behavior of optical solitons in complex structured conformable perturbed Radhakrishnan-Kundu-Lakshmanan model, Phys. Scr., 99 (2024), 095209. https://doi.org/10.1088/1402-4896/ad67b1 doi: 10.1088/1402-4896/ad67b1
[19]	A. Secer, S. Altun, A new operational matrix of fractional derivatives to solve systems of fractional differential equations via Legendre wavelets, Mathematics, 6 (2018), 238. https://doi.org/10.3390/math6110238 doi: 10.3390/math6110238
[20]	H. R. Ghehsareh, A. Majlesi, A. Zaghian, Lie symmetry analysis and conservation laws for time fractional coupled Whitham-Broer-Kaup equations, UPB Sci. Bull. A: Appl. Math. Phys., 80 (2018), 153–168.
[21]	S. M. El-Sayed, D. Kaya Exact and numerical traveling wave solutions of Whitham-Broer-Kaup equations, Appl. Math. Comput., 167 (2005), 1339–1349. https://doi.org/10.1016/j.amc.2004.08.012 doi: 10.1016/j.amc.2004.08.012
[22]	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
[23]	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
[24]	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
[25]	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
[26]	F. Meng, A. Pang, X. Dong, C. Han, X. Sha, $H_\infty$ optimal performance design of an unstable plant under bode integral constraint, Complexity, 2018 (2018), 4942906. https://doi.org/10.1155/2018/4942906 doi: 10.1155/2018/4942906
[27]	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
[28]	Y. Hu, Y. Sugiyama, Well-posedness of the initial-boundary value problem for 1D degenerate quasilinear wave equations, Adv. Differ. Equations, 30 (2025), 177–206. https://doi.org/10.57262/ade030-0304-177 doi: 10.57262/ade030-0304-177
[29]	L. Zheng, Y. Zhu, Y. Zhou, Meta-transfer learning-based method for multi-fault analysis and assessment in power system, Appl. Intell., 54 (2024), 12112–12127. https://doi.org/10.1007/s10489-024-05772-9 doi: 10.1007/s10489-024-05772-9
[30]	Y. Zhu, Y. Zhou, L. Yan, Z. Li, H. Xin, W. Wei, Scaling graph neural networks for large-scale power systems analysis: empirical laws for emergent abilities, IEEE Trans. Power Syst., 39 (2024), 7445–7448. https://doi.org/10.1109/TPWRS.2024.3437651 doi: 10.1109/TPWRS.2024.3437651
[31]	C. Yang, K. Meng, L. Yang, W. Guo, P. Xu, S. Zhou, Transfer learning-based crashworthiness prediction for the composite structure of a subway vehicle, Int. J. Mech. Sci., 248 (2023), 108244. https://doi.org/10.1016/j.ijmecsci.2023.108244 doi: 10.1016/j.ijmecsci.2023.108244
[32]	Z. Cheng, Y. Chen, H. Huang, Identification and validation of a novel prognostic signature based on ferroptosis-related genes in ovarian cancer, Vaccine, 11 (2023), 205. https://doi.org/10.3390/vaccines11020205 doi: 10.3390/vaccines11020205
[33]	Y. Yang, L. Zhang, C. Xiao, Z. Huang, F. Zhao, J. Yin, Highly efficient upconversion photodynamic performance of rare-earth-coupled dual-photosensitizers: ultrafast experiments and excited-state calculations, Nanophotonics, 13 (2024), 443–455. https://doi.org/10.1515/nanoph-2023-0772 doi: 10.1515/nanoph-2023-0772
[34]	B. Zhuang, Y. Yang, K. Huang, J. Yin, Plasmon-enhanced NIR-Ⅱ photoluminescence of rare-earth oxide nanoprobes for high-sensitivity optical temperature sensing, Opt. Lett., 49 (2024), 6489–6492. https://doi.org/10.1364/OL.540944 doi: 10.1364/OL.540944
[35]	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
[36]	Y. Gu, X. Zhang, Z. Huang, L. Peng, Y. Lai, N. Aminakbari, Soliton and lump and travelling wave solutions of the $(3+ 1)$ dimensional KPB like equation with analysis of chaotic behaviors, Sci. Rep., 14 (2024), 20966. https://doi.org/10.1038/s41598-024-71821-5 doi: 10.1038/s41598-024-71821-5
[37]	A. Yokuş, S. Duran, D. Kaya, An expansion method for generating travelling wave solutions for the $(2+ 1)$-dimensional Bogoyavlensky-Konopelchenko equation with variable coefficients, Chaos Soliton. Fract., 178 (2024), 114316. https://doi.org/10.1016/j.chaos.2023.114316 doi: 10.1016/j.chaos.2023.114316
[38]	R. Ali, E. Tag-eldin, A comparative analysis of generalized and extended ($\frac{G'}{G}$)-expansion methods for travelling wave solutions of fractional Maccari 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
[39]	J. A. Haider, S. Gul, J. U. Rahman, F. D. Zaman, Travelling wave solutions of the non-linear wave equations, Acta Mech. Autom., 17 (2023), 239–245. https://doi.org/10.2478/ama-2023-0027 doi: 10.2478/ama-2023-0027
[40]	K. Khan, M. A. Akbar, Study of explicit travelling wave solutions of nonlinear evolution equations, Partial Differ. Equ. Appl. Math., 7 (2023), 100475. https://doi.org/10.1016/j.padiff.2022.100475 doi: 10.1016/j.padiff.2022.100475
[41]	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), 1–6. https://doi.org/10.4172/2090-0902.1000214 doi: 10.4172/2090-0902.1000214
[42]	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
[43]	M. A. E. Abdelrahman, M. A. Sohaly, Solitary waves for the nonlinear Schrödinger 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
[44]	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
[45]	D. Lu, Q. Shi, New Jacobi elliptic functions solutions for the combined KdV-mKdV equation, Int. J. Nonlinear Sci., 10 (2010), 320–325.
[46]	H. G. Sun, Y. Zhang, D. Baleanu, W. Chen, Y. Q. Chen, A new collection of real world applications of fractional calculus in science and engineering, Commun. Nonlinear Sci. Numer. Simul., 64 (2018), 213–223. https://doi.org/10.1016/j.cnsns.2018.04.019 doi: 10.1016/j.cnsns.2018.04.019
[47]	G. B. Whitham, Variational methods and applications to water waves, Proc. R. Soc. Lond. A, 299 (1967), 6–25. https://doi.org/10.1098/rspa.1967.0119 doi: 10.1098/rspa.1967.0119
[48]	L. J. F. Broer, Approximate equations for long water waves, Appl. Sci. Res., 31 (1975), 377–395. https://doi.org/10.1007/BF00418048 doi: 10.1007/BF00418048
[49]	D. J. Kaup, A higher-order water-wave equation and the method for solving it, Prog. Theor. Phys., 54 (1975), 396–408. https://doi.org/10.1143/PTP.54.396 doi: 10.1143/PTP.54.396

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