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Keywords = extended sinh-Gordon equation expansion scheme

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15 pages, 654 KiB  
Article
On Some Novel Soliton Structures for the Beta-Time Fractional Benjamin–Ono Dynamical Equation in Fluids
by Mohammed Ahmed Alomair and Kalim U. Tariq
Fractal Fract. 2025, 9(3), 185; https://doi.org/10.3390/fractalfract9030185 - 17 Mar 2025
Cited by 2 | Viewed by 291
Abstract
This paper consists of an exploration of the wave structures of the Benjamin–Ono equation along with a β-time fractional derivative. The model concerned is utilized to demonstrate internal waves of deep-stratified fluids. Bright, rational, periodic, and many more kinds of solutions for [...] Read more.
This paper consists of an exploration of the wave structures of the Benjamin–Ono equation along with a β-time fractional derivative. The model concerned is utilized to demonstrate internal waves of deep-stratified fluids. Bright, rational, periodic, and many more kinds of solutions for waves are achieved by utilizing the extended sinh-Gordon equation expansion (EShGEE) technique and the improved G′/G-expansion scheme. An influence of fractional-order derivatives was also explored which gives the non-existing results. The Mathematica tool is utilized to gain and verify the results. The results are represented by 3-D, 2-D, and contour graphs. A stability analysis is utilized to confirm that results are precise as well as exact. Modulation instability (MI) is also performed for the steady-state solutions to the concerned model. Full article
(This article belongs to the Special Issue Mathematical and Numerical Analysis of Fractional Evolution Equations and Applications)
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29 pages, 1528 KiB  
Article
Exact Solutions of M-Fractional Kuralay Equation via Three Analytical Schemes
by Asim Zafar, Muhammad Raheel, Mohamed R. Ali, Zhaidary Myrzakulova, Ahmet Bekir and Ratbay Myrzakulov
Symmetry 2023, 15(10), 1862; https://doi.org/10.3390/sym15101862 - 4 Oct 2023
Cited by 21 | Viewed by 1885
Abstract
This article concerns new analytical wave solutions of the Kuralay-II equations (K-IIAE and K-IIBE) with exploration of a new definition of the derivative. This model is used in various fields, like nonlinear optics, ferromagnetic materials and optical fibers. For this purpose, the [...] Read more.
This article concerns new analytical wave solutions of the Kuralay-II equations (K-IIAE and K-IIBE) with exploration of a new definition of the derivative. This model is used in various fields, like nonlinear optics, ferromagnetic materials and optical fibers. For this purpose, the expa function, the extended sinh-Gordon equation expansion scheme, and the generalized Kudryashov schemes were utilized. The resulting solutions are dark, bright, dark-bright, periodic, singular and other kinds of solitons. These results are obtained and also verified by the Mathematica tool. Some of the solutions are explained with 2-D, 3-D and contour plots using the Mathematica tool. The solutions obtained succede the present solutions in the literature. For the first time, the effect of the fractional derivative on the solutions is also shown graphically for this model. The analytical wave solutions are highly desirable as they offer insights into the underlying physics or mathematics of a system and provide a framework for further analysis. The results obtained can also be fruitful for the development of models in the future. The schemes used in this research are effective, easy to apply, and reliably handle other fractional non-linear partial differential equations. Full article
(This article belongs to the Special Issue Analytic/Numeric Solutions of Schrödinger-Type Equations: Applications of Lie Symmetry and Other Methods)
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16 pages, 2427 KiB  
Article
New Three Wave and Periodic Solutions for the Nonlinear (2+1)-Dimensional Burgers Equations
by Waseem Razzaq, Asim Zafar, Abdulaziz Khalid Alsharidi and Mohammed Ahmed Alomair
Symmetry 2023, 15(8), 1573; https://doi.org/10.3390/sym15081573 - 12 Aug 2023
Cited by 3 | Viewed by 1368
Abstract
This research paper is about the new three wave, periodic wave and other analytical wave solutions of (2+1)-Dimensional Burgers equations by utilizing Hirota bilinear and extended sinh-Gordon equation expansion (EShGEE) schemes. Achieved solutions are verified and demonstrated by different plots with the use [...] Read more.
This research paper is about the new three wave, periodic wave and other analytical wave solutions of (2+1)-Dimensional Burgers equations by utilizing Hirota bilinear and extended sinh-Gordon equation expansion (EShGEE) schemes. Achieved solutions are verified and demonstrated by different plots with the use of Mathematica 11.01 software. Some of the achieved solutions are also described graphically by two-dimensional, three-dimensional and contour plots. The gained solutions are helpful for the future study of concerned models. Finally, these two schemes are simple, fruitful and reliable to handle the nonlinear PDEs. Full article
(This article belongs to the Special Issue Analytic/Numeric Solutions of Schrödinger-Type Equations: Applications of Lie Symmetry and Other Methods)
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