Construction of diverse water wave structures for coupled nonlinear fractional Drinfel'd-Sokolov-Wilson model with Beta derivative and its modulus instability

Muhammad Shakeel; Salman A AlQahtani; Muhammad Junaid U Rehman; Grzegorz Kudra; Jan Awrejcewicz; Abdulaziz M Alawwad; Abdullilah A Alotaibi; Mejdl Safran

doi:10.1038/s41598-023-44428-5

Construction of diverse water wave structures for coupled nonlinear fractional Drinfel'd-Sokolov-Wilson model with Beta derivative and its modulus instability

Sci Rep. 2023 Oct 16;13(1):17528. doi: 10.1038/s41598-023-44428-5.

Authors

Muhammad Shakeel^{1

2}, Salman A AlQahtani³, Muhammad Junaid U Rehman⁴, Grzegorz Kudra⁴, Jan Awrejcewicz⁴, Abdulaziz M Alawwad⁵, Abdullilah A Alotaibi⁵, Mejdl Safran⁵

Affiliations

¹ Department of Mathematics, Quaid-I-Azam University, Islamabad, Pakistan. mshakeel@math.qau.edu.pk.
² School of Mathematics and Statistics, Central South University, Changsha, 410083, China. mshakeel@math.qau.edu.pk.
³ New Emerging Technologies and 5G Network and Beyond Research Chair, Department of Computer Engineering, College of Computer and Information Sciences, King Saud University, Riyadh, Saudi Arabia. salmanq@ksu.edu.sa.
⁴ Department of Automation, Biomechanics, and Mechatronics, Lodz University of Technology, 1/15 Stefanowski St. (Building A22), Lodz, 90-924, Poland.
⁵ Department of Computer Science, College of Computer and Information Sciences, King Saud University, Riyadh, Saudi Arabia.

Abstract

This paper aims to analyze the coupled nonlinear fractional Drinfel'd-Sokolov-Wilson (FDSW) model with beta derivative. The nonlinear FDSW equation plays an important role in describing dispersive water wave structures in mathematical physics and engineering, which is used to describe nonlinear surface gravity waves propagating over horizontal sea bed. We have applied the travelling wave transformation that converts the FDSW model to nonlinear ordinary differential equations. After that, we applied the generalized rational exponential function method (GERFM). Diverse types of soliton solution structures in the form of singular bright, periodic, dark, bell-shaped and trigonometric functions are attained via the proposed method. By selecting a suitable parametric value, the 3D, 2D and contour plots for some solutions are also displayed to visualize their nature in a better way. The modulation instability for the model is also discussed. The results show that the presented method is simple and powerful to get a novel soliton solution for nonlinear PDEs.