On Some Novel Soliton Structures for the Beta-Time Fractional Benjamin–Ono Dynamical Equation in Fluids
Abstract
1. Introduction
2. The -Time Fractional Derivative
3. The Governing Model Presentation
4. Methodology
4.1. The Extended ShGEE Technique
4.2. Improved -Expansion Technique
5. Mathematical Analysis
5.1. Applications of the Extended ShGEE Technique
5.2. Soliton Solutions via the Improved -Expansion Technique
6. The Stability Analysis (SA)
7. The Modulation Instability Analysis
8. Physical Description
9. Results and Discussion
10. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
- Razzaq, W.; Zafar, A.; Raheel, M. Searching the new exact wave solutions to the beta-fractional Paraxial nonlinear Schrödinger model via three different approaches. Int. J. Mod. Phys. B 2024, 38, 2450132. [Google Scholar] [CrossRef]
- Qawaqneh, H.; Zafar, A.; Raheel, M.; Zaagan, A.A.; Zahran, E.H.; Cevikel, A.; Bekir, A. New soliton solutions of M-fractional Westervelt model in ultrasound imaging via two analytical techniques. Opt. Quantum Electron. 2024, 56, 737. [Google Scholar] [CrossRef]
- Khater, M.M.A. Novel constructed dark, bright and rogue waves of three models of the well-known nonlinear Schrödinger equation. Int. J. Mod. Phys. B 2024, 38, 2450023. [Google Scholar] [CrossRef]
- Khater, M.M.A. Exploring the rich solution landscape of the generalized Kawahara equation: Insights from analytical techniques. Eur. Phys. J. Plus 2024, 139, 184. [Google Scholar] [CrossRef]
- Walait, A.; Ashraf, H.; Chou, D.; Rehman, H.U. Stagnant Rings and Uniform Film Analysis of Phan-Thien Tanner Fluid Film Flow on a Vertically Upward Moving Tube. Phys. Fluids 2024, 36, 083613. [Google Scholar] [CrossRef]
- Khater, M.M.A. Comment on the paper of El-Ganaini et al. [Chaos, Solitons and Fractals 140 (2020) 110218]. Chaos Solitons Fractals 2024, 182, 114729. [Google Scholar] [CrossRef]
- Bezgabadi, A.S.; Bolorizadeh, M.A. Analytic combined bright-dark, bright and dark solitons solutions of generalized nonlinear Schrödinger equation using extended sinh-Gordon equation expansion method. Results Phys. 2021, 30, 104852. [Google Scholar] [CrossRef]
- Kumar, D.; Manafian, J.; Hawlader, F.; Ranjbaran, A. New closed form soliton and other solutions of the Kundu–Eckhaus equation via the extended sinh-Gordon equation expansion method. Optik 2018, 160, 159–167. [Google Scholar] [CrossRef]
- Ilhan, O.A.; Manafian, J. Analytical treatment in optical metamaterials with anti-cubic law of nonlinearity by improved exp (-Ω (η))-expansion method and extended sinh-Gordon equation expansion method. Rev. Mex. Física 2019, 65, 658–677. [Google Scholar] [CrossRef]
- Qawaqneh, H.; Hakami, K.H.; Altalbe, A.; Bayram, M. The Discovery of Truncated M-Fractional Exact Solitons and a Qualitative Analysis of the Generalized Bretherton Model. Mathematics 2024, 12, 2772. [Google Scholar] [CrossRef]
- Zayed EM, E.; Al-Joudi, S. Applications of an Improved (G′/G)-expansion Method to Nonlinear PDEs in Mathematical Physics. In AIP Conference Proceedings; American Institute of Physics: College Park, MD, USA, 2009; Volume 1168, pp. 371–376. [Google Scholar]
- Sahoo, S.; Ray, S.S.; Abdou, M.A. New exact solutions for time-fractional Kaup–Kupershmidt equation using improved (G′/G)-expansion and extended (G′/G)-expansion methods. Alex. Eng. J. 2020, 59, 3105–3110. [Google Scholar] [CrossRef]
- Zayed, E.M.E. A further improved (g′/g)-expansion method and the extended tanh-method for finding exact solutions of nonlinear pdes. Wseas Trans. Math. 2011, 10, 56–64. [Google Scholar]
- Qawaqneh, H.; Alrashedi, Y. Mathematical and Physical Analysis of Fractional Estevez–Mansfield–Clarkson Equation. Fractal Fract. 2024, 8, 467. [Google Scholar] [CrossRef]
- Alhefthi, R.K.; Tariq, K.U.; Wazwaz, A.M.; Mehboob, F. On the nonlinear wave structures and stability analysis for the new generalized stochastic fractional potential-kdv model in dispersive medium. Opt. Quantum Electron. 2024, 56, 662. [Google Scholar] [CrossRef]
- Abdeljawad, T. On conformable fractional calculus. J. Comput. Appl. Math. 2015, 279, 57–66. [Google Scholar] [CrossRef]
- Tariq, K.U.; Khater, M.M.; Ilyas, M.; Rezazadeh, H.; Inc, M. Soliton structures for a generalized unstable space-time fractional nonlinear Schrödinger model in mathematical physics. Int. J. Mod. Phys. B 2024, 38, 2450174. [Google Scholar] [CrossRef]
- Wu, G.-Z.; Yu, L.J.; Wang, Y.Y. Fractional optical solitons of the spacetime fractional nonlinear Schrödinger equation. Optik 2020, 207, 164405. [Google Scholar] [CrossRef]
- Almeida, R. A caputo fractional derivative of a function with respect to another function. Commun. Nonlinear Sci. Numer. Simul. 2017, 44, 460–481. [Google Scholar] [CrossRef]
- Jumarie, G. Modified riemann-liouville derivative and fractional taylor series of nondifferentiable functions further results. Comput. Math. Appl. 2006, 51, 1367–1376. [Google Scholar] [CrossRef]
- Gurefe, Y. The generalized kudryashov method for the nonlinear fractional partial differential equations with the beta-derivative. Rev. Mex. Fis. 2020, 66, 771–781. [Google Scholar] [CrossRef]
- Atangana, A.; Baleanu, D.; Alsaedi, A. Analysis of time-fractional Hunter–Saxton equation: A model of neumatic liquid crystal. Open Phys. 2016, 14, 145–149. [Google Scholar] [CrossRef]
- Uddin, M.F.; Hafez, M.G.; Hammouch, Z.; Baleanu, D. Periodic and rogue waves for Heisenberg models of ferromagnetic spin chains with fractional beta derivative evolution and obliqueness. Waves Random Complex Media 2020, 31, 2135–2149. [Google Scholar] [CrossRef]
- Ghanbari, B.; Gómez-Aguilar, J.F. The generalized exponential rational function method for Radhakrishnan–Kundu–Lakshmanan equation with Beta time derivative. Rev. Mex. Física 2019, 65, 503–518. [Google Scholar] [CrossRef]
- Khater, M.M.; Muhammad, S.; Al-Ghamdi, A.; Higazy, M. Abundant wave structures of the fractional Benjamin-Ono equation through two computational techniques. J. Ocean. Eng. Sci. 2022, in press.
- Tasbozan, O. New analytical solutions for time fractional Benjamin-Ono equation arising internal waves in deep water. China Ocean. Eng. 2019, 33, 593–600. [Google Scholar] [CrossRef]
- Alam, M.N.; Akbar, M.A. Traveling wave solutions of nonlinear evolution equations via the new generalized (G′/G)-expansion method. Univers. J. Comput. Math. 2013, 1, 129–136. [Google Scholar] [CrossRef]
- Islam, R.; Alam, M.N.; Hossain, A.K.M.K.S.; Roshid, H.O.; Akbar, M.A. Traveling wave solutions of nonlinear evolution equations via exp(−Φ(η))-expansion method. Glob. J. Sci. Front. Res. 2013, 13, 63–71. [Google Scholar]
- Aderyani, S.R.; Saadati, R.; Vahidi, J. Multiple exp-function method to solve the nonlinear space–time fractional partial differential symmetric regularized long wave (SRLW) equation and the (1+1)-dimensional Benjamin–Ono equation. Int. J. Mod. Phys. B 2023, 37, 2350213. [Google Scholar] [CrossRef]
- Karaman, B. The use of improved-F expansion method for the time-fractional Benjamin–Ono equation. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 2021, 115, 128. [Google Scholar] [CrossRef]
- Alizadeh, F.; Hashemi, M.S.; Haji Badali, A. Lie symmetries, exact solutions, and conservation laws of the nonlinear time-fractional Benjamin-Ono equation. Comput. Methods Differ. Equ. 2022, 10, 608–616. [Google Scholar]
- Sagar, B.; Ray, S.S. A localized meshfree technique for solving fractional Benjamin–Ono equation describing long internal waves in deep stratified fluids. Commun. Nonlinear Sci. Numer. Simul. 2023, 123, 107287. [Google Scholar] [CrossRef]
- Babajanov, B.; Abdikarimov, F. Soliton and periodic wave solutions of the nonlinear loaded (3+1)-dimensional version of the benjamin-ono equation by functional variable method. J. Nonlinear Model. Anal. 2023, 5, 782–789. [Google Scholar]
- Yang, H.; Sun, J.; Fu, C. Time-fractional Benjamin-Ono equation for algebraic gravity solitary waves in baroclinic atmosphere and exact multi-soliton solution as well as interaction. Commun. Nonlinear Sci. Numer. Simul. 2019, 71, 187–201. [Google Scholar] [CrossRef]
- Yang, X.L.; Tang, J.S. Travelling wave solutions for Konopelchenko-Dubrovsky equation using an extended sinh-Gordon equation expansion method. Commun. Theor. Phys. 2008, 50, 10471051. [Google Scholar]
- Tariq, K.U.; Bekir, A.; Nisar, S. The dynamical structures of the Sharma–Tasso–Olver model in doubly dispersive medium. Chaos Solitons Fractals 2023, 177, 114290. [Google Scholar] [CrossRef]
- Tariq, K.U.; Wazwaz, A.M.; Javed, R. Construction of different wave structures, stability analysis and modulation instability of the coupled nonlinear Drinfel’d-Sokolov-Wilson model. Chaos Solitons Fractals 2023, 166, 112903. [Google Scholar] [CrossRef]
- Zulfiqar, H.; Aashiq, A.; Tariq, K.U.; Ahmad, H.; Almohsen, B.; Aslam, M.; Rehman, H.U. On the solitonic wave structures and stability analysis of the stochastic nonlinear Schrödinger equation with the impact of multiplicative noise. Optik 2023, 289, 171250. [Google Scholar] [CrossRef]
- ur Rehman, S.; Ahmad, J. Modulation instability analysis and optical solitons in birefringent fibers to RKL equation without four wave mixing. Alex. Eng. J. 2021, 60, 1339–1354. [Google Scholar] [CrossRef]
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Alomair, M.A.; Tariq, K.U. On Some Novel Soliton Structures for the Beta-Time Fractional Benjamin–Ono Dynamical Equation in Fluids. Fractal Fract. 2025, 9, 185. https://doi.org/10.3390/fractalfract9030185
Alomair MA, Tariq KU. On Some Novel Soliton Structures for the Beta-Time Fractional Benjamin–Ono Dynamical Equation in Fluids. Fractal and Fractional. 2025; 9(3):185. https://doi.org/10.3390/fractalfract9030185
Chicago/Turabian StyleAlomair, Mohammed Ahmed, and Kalim U. Tariq. 2025. "On Some Novel Soliton Structures for the Beta-Time Fractional Benjamin–Ono Dynamical Equation in Fluids" Fractal and Fractional 9, no. 3: 185. https://doi.org/10.3390/fractalfract9030185
APA StyleAlomair, M. A., & Tariq, K. U. (2025). On Some Novel Soliton Structures for the Beta-Time Fractional Benjamin–Ono Dynamical Equation in Fluids. Fractal and Fractional, 9(3), 185. https://doi.org/10.3390/fractalfract9030185