New Exact Solutions of the Drinfeld-Sokolov System by the Generalized Unified Method

Year 2023, Issue: 44, 10 - 19, 30.09.2023

Tuğba Aydemir

https://doi.org/10.53570/jnt.1294322

Cited By: 1

Abstract

In this study, we apply the generalized unified method (GUM), an enhanced version of the unified method, to find novel exact solutions of the Drinfeld-Sokolov System (DSS) that models the dispersive water waves in fluid dynamics. Moreover, 3D and 2D graphs of some of the obtained exact solutions are plotted to present how various characteristic forms they have. The results show that the presented method simplifies the computation process on the computer in a highly reliable and straightforward manner while providing the solutions in more general forms. In addition, the GUM has great potential to apply to a wide range of problems, including nonlinear partial differential equations (NPDEs) and fractional partial differential equations (FPDEs) for finding exact solutions.

Keywords

Drinfeld-Sokolov system, unified method, generalized unified method, exact solution, traveling wave solutions

References

• V. G. Drinfeld, V. V. Sokolov, Lie Algebras and Equations of Korteweg-De Vries Type, Journal of Soviet Mathematics 30 (1985) 1975--2036.
• R. Hirota, B. Grammaticos, A. Ramani, Soliton Structure of the Drinfeld–Sokolov–Wilson Equation, Journal of Mathematical Physics 27 (1986) 1499--1505.
• R. Morris, A. H. Kara, Double Reductions/Analysis of the Drinfeld–Sokolov–Wilson Equation, Applied Mathematics and Computation 219 (2013) 6473--6483.
• A. M. Wazwaz, Exact and Explicit Traveling Wave Solutions for the Nonlinear Drinfeld–Sokolov System, Communications in Nonlinear Science and Numerical Simulation 11 (2006) 311--325.
• R. Arora, A. Kumar, Solution of the Coupled Drinfeld’s-Sokolov-Wilson System by Homotopy Analysis Method, Advanced Science Engineering and Medicine 5 (10) (2010) 1105--1111.
• Y. He, Y. Long, S. Li, Exact Solutions of the Drinfel’d-Sokolov-Wilson Equation Using the F-Expansion Method Combined with Exp-Function Method, International Mathematical Forum 5 (65) (2010) 3231--3242.
• C. A. G\'{o}mez, Generalized Drinfeld-Sokolov-Wilson Equation: Exact Solutions, Advanced Studies in Theoretical Physics 11 (4) (2017) 585--591.
• F. Düşünceli, Solutions for the Drinfeld-Sokolov Equation Using an IBSEFM Method, Mus Alparslan University Journal of Science 6 (1) (2018) 505--510.
• W. M. Zhang, Solitary Solutions and Singular Periodic Solutions of the Drinfeld-Sokolov-Wilson Equation by Variational Approach, Applied Mathematical Sciences 5 (38) (2011) 1887--1894.
• B. Günay, C. K. Kuo, W. X. Ma, An Application of the Exponential Rational Function Method to Exact Solutions to the Drinfeld–Sokolov System, Results in Physics 29 (2021) Article ID 104733 9 pages.
• B. J. Salim, O. A. Jasim, Z. Y. Ali, Numerical Solution of Drinfeld-Sokolov-Wilson System by Using Modified Adomian Decomposition Method, Indonesian Journal of Electrical Engineering and Computer Science 23 (1) (2021) 590--599.
• M. N. Alam, E. Bonyah, M. Fayz-Al-Asad, Reliable Analysis for the Drinfeld-Sokolov-Wilson Equation in Mathematical Physics, Palestine Journal of Mathematics 11 (1) (2022) 397--407.
• H. M. Jaradat, S. Al-Shar’a, J. A. Q. Khan, M. Alquran, K. Al-Khaled, Analytical Solution of Time-Fractional Drinfeld-Sokolov-Wilson System Using Residual Power Series Method, IAENG International Journal of Applied Mathematics 46 (1) (2016) 64--70.
• A. Bhatter, A. Mathur, D. Kumar, J. Singh, A New Analysis of Fractional Drinfeld–Sokolov–Wilson Model with Exponential Memory, Physica A: Statistical Mechanics and Its Applications 537 (1) (2020) Article ID 122578 13 pages.
• W. Gao, P. Veeresha, D. G. Prakasha, H. M. Başkonuş, G. Yel, A Powerful Approach for Fractional Drinfeld-Sokolov-Wilson Equation with Mittag-Leffler Law, Alexandria Engineering Journal 58 (2019) 1301--1311.
• O. Taşbozan, M. Şenol, A. Kurt, O. Özkan, New Solutions of Fractional Drinfeld-Sokolov-Wilson System in Shallow Water Waves, Ocean Engineering 161 (2018) 62--68.
• K. J. Wang, G. D. Wang, He's Variational Method for the Time-Space Fractional Nonlinear Drinfeld–Sokolov–Wilson System, Mathematical Methods in the Applied Sciences 46 (7) (2021) 1--9.
• S. Noor, A. S. Alshehry, H. M. Dutt, R. Nazir, A. Khan, R. Shah, Investigating the Dynamics of Time-Fractional Drinfeld–Sokolov–Wilson System through Analytical Solutions, Symmetry 15 (3) (2023) 703 13 pages.
• T. Aydemir, Application of the Generalized Unified Method to Solve (2+1)-Dimensional Kundu–Mukherjee–Naskar Equation, Optical and Quantum Electronics 55 (2023) Article Number 534 17 pages.
• Ö. F. Gözükızıl, Ş. Akçağıl, T. Aydemir, Unification of All Hyperbolic Tangent Function Methods, Open Physics 14 (2016) 524--541.
• S. Akçağıl, T. Aydemir, A New Application of the Unified Method, New Trends in Mathematical Sciences 6 (1) (2018) 166--180.
• M. S. Ullah, H. Roshid, M. S. Ali, A. Biswas, M. Ekici, S. Khan, L. Moraru, A. K. Alzahrani, M. R. Belic, Optical Soliton Polarization With Lakshmanan–Porsezian–Daniel Model by Unified Approach, Results in Physics 22 (2021) Article ID 103958 5 pages.
• M. Bilal, S. Rehman, J. Ahmad, Analysis in Fiber Bragg Gratings with Kerr Law Nonlinearity for Diverse Optical Soliton Solutions by Reliable Analytical Techniques, Modern Physics Letters B 36 (23) (2022) Article ID 2250122 14 pages.
• M. Bilal, S. Rehman, J. Ahmad, Dynamical Nonlinear Wave Structures of the Predator-Prey Model Using Conformable Derivative and Its Stability Analysis, Pramana Journal of Physics 96 (2022) Article ID 149 18 pages.
• S. Foyjonnesa, N. H. M. Shahen, M. M. Rahman, Dispersive Solitary Wave Structures with MI Analysis to the Unidirectional DGH Equation via the Unified Method, Partial Differential Equations in Applied Mathematics 6 (2022) Article ID 100444 11 pages.
• S. M. Y. Arafat, S. M. R. Islam, M. H. Bashar, Influence of the Free Parameters and Obtained Wave Solutions from CBS Equation, International Journal of Applied and Computational Mathematics 8 (2022) Article ID 99 17 pages.
• S. M. R. Islam, M. H. Bashar, S. M. Y. Arafat, H. Wang, M. M. Roshid, Effect of the Free Parameters on the Biswas-Arshed Model with a Unified Technique, Chinese Journal of Physics 77 (2022) 2501--2519.
• S. M. R. Islam, D. Kumar, E. F. Donfack, M. Inc, Impact of Nonlinearity and Wave Dispersion Parameters on the Soliton Pulses of the (2+1)-Dimensional Kundu-Mukherjee-Naskar Equation, Revista Mexicana de Fisica 68 (2022) 1--14.
• S. Uddin, S. Karim, F. S. Alshammari, H. O. Roshid, N. F. M. Noor, F. Hoque, M. Nadeem, A. Akgül, Bifurcation Analysis of Travelling Waves and Multi-Rogue Wave Solutions for a Nonlinear Pseudo-Parabolic Model of Visco-Elastic Kelvin-Voigt Fluid, Mathematical Problems in Engineering 2022 (2022) Article ID 8227124 16 pages.
• D. C. Nandi, M. S. Ullah, M. Z. Ali, Application of the Unified Method to Solve the Ion Sound and Langmuir Waves Model, Heliyon 8 (10) (2022) Article ID e10924 8 pages.
• M. S. Ullah, A. Abdeljabbar, H. Roshid, M. Z. Ali, Application of the Unified Method to Solve the Biswas–Arshed Model, Results in Physics 42 (2022) Article ID 105946 6 pages.
• D. Kumar, M. D. M. Hasan, G. C. Paul, D. Debnath, N. Mondal, O. Faruk, Revisiting the Spatiotemporal Dynamics of a Diffusive Predator-Prey System: An Analytical Approach, Results in Physics 44 (2023) Article ID106122 18 pages.
• A. Akbulut, D. Kumar, Conservation Laws and Optical Solutions of the Complex Modified Korteweg-De Vries Equation, Journal of Ocean Engineering and Science (in press).
• A. Akbulut, S. M. R. Islam, S. M. Y. Arafat, F. Tascan, A Novel Scheme for SMCH Equation with Two Different Approaches, Computational Methods for Differential Equations 11 (2) (2023) 263--280.
• S. M. Y. Arafat, K. Khan, S. M. R. Islam, M. M. Rahman, Parametric Effects on Paraxial Nonlinear Schrödinger Equation in Kerr Media, Chinese Journal of Physics 83 (2023) 361--378.
• M. Bilal, J. Ahmad, Investigation of Diverse Genres Exact Soliton Solutions to the Nonlinear Dynamical Model via Three Mathematical Methods, Journal of Ocean Engineering and Science (in press).