Shahen Nur Hasan Mahmud, Al Amin Md, Rahman M M
Department of Mathematics, Bangladesh University of Engineering and Technology, Dhaka, 1000, Bangladesh.
Department of Arts and Sciences, Bangladesh Army University of Science and Technology, Saidpur, 5310, Bangladesh.
Sci Rep. 2024 Aug 14;14(1):18894. doi: 10.1038/s41598-024-64348-2.
This article delves into the dynamic constructions of distinctive traveling wave solutions for wave circulation in shallow water mechanics, specifically addressing the time-fractional couple Drinfel'd-Sokolov-Wilson (DSW) equation. Introducing the previously untapped -expansion method, we successfully generate a diverse set of analytic solutions expressed in hyperbolic, trigonometric, and rational functions, each with permitted parameters. Visualization through three-dimensional (3D) as well two-dimensional (2D) plots, including contour plots, reveals inherent wave phenomena in the DSW equation. These newly obtained wave solutions serve as a catalyst for refining theories in applied science, offering a means to validate mathematical simulations for the proliferation of waves in shallow water as well as other nonlinear scenarios. Obtained wave solutions demonstrate the bright soliton, periodic, multiple soliton, and kink soliton shape. The simplicity and efficacy of the implemented methods are demonstrated, providing a valuable tool for approximating the considered equation. All figures are devoted to demonstrate the complete wave futures of the attained solutions to the studied equation with the collaboration of specific selections of the chosen parameters. Moreover, it may have summarized that the attained wave solutions and their physical phenomena might be useful to comprehend the various kind of wave propagation in mathematical physics and engineering.
本文深入探讨了浅水波动力学中波环流独特行波解的动态构造，特别针对时间分数阶耦合的Drinfel'd-Sokolov-Wilson（DSW）方程。引入此前未被利用的 -展开法，我们成功生成了一系列以双曲函数、三角函数和有理函数表示的解析解，每个解都有允许的参数。通过三维（3D）和二维（2D）图（包括等高线图）进行可视化，揭示了DSW方程中固有的波动现象。这些新获得的波解成为完善应用科学理论的催化剂，为验证浅水波以及其他非线性场景中波的传播的数学模拟提供了一种手段。所获得的波解呈现出亮孤子、周期、多孤子和扭结孤子形状。展示了所采用方法的简单性和有效性，为近似所考虑的方程提供了一个有价值的工具。所有图形都致力于通过所选参数的特定选择来展示所研究方程的所得解的完整波动特征。此外，可以总结出，所获得的波解及其物理现象可能有助于理解数学物理和工程中的各种波传播。
