Muhammad Jan, Younas Usman, Hussain Ejaz, Ali Qasim, Sediqmal Mirwais, Kedzia Krzysztof, Jan Ahmed Zubair
Department of Mathematics, Shanghai University, No. 99 Shangda Road, Shanghai, 200444, China.
Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore, 54590, Pakistan.
Sci Rep. 2024 Nov 17;14(1):28383. doi: 10.1038/s41598-024-79102-x.
Solitary wave solutions to the nonlinear evolution equations have recently attracted widespread interest in engineering and physical sciences. In this work, we investigate the fractional generalised nonlinear Pochhammer-Chree equation under the power-law of nonlinearity with order m. This equation is used to describe longitudinal deformation wave propagation in an elastic rod. In this study, we have secured a variety of exact solitary wave solutions by the assistance of the recently developed technique known as modified generalized exponential rational function method. Exact solutions of various categories, such as bright-dark, bright, mixed, singular, dark, complex, and combined solitons, are extracted. The applied approach is highly efficient and has a significant computational capability to efficiently tackle the solutions with a high degree of accuracy in nonlinear systems. To analyze the governing system, the equation under investigation is converted to an ordinary differential equation through the application of a suitable wave transformation with a β-derivative. In addition to illustrate the behavior of the solution at various parameter values, we generate 2D and 3D graphs that incorporate pertinent parameters. Moreover, the Galilean transformation is employed to investigate the sensitivity analysis. This research's results have the potential to enhance comprehension of the nonlinear dynamic characteristics displayed by the defined system and to verify the efficacy of the strategies that have been implemented. The results obtained are a substantial contribution to the comprehension of nonlinear science and nonlinear wave fields that are associated with higher dimensions.
非线性演化方程的孤立波解最近在工程和物理科学领域引起了广泛关注。在这项工作中，我们研究了在幂律非线性（阶数为m）下的分数阶广义非线性Pochhammer-Chree方程。该方程用于描述弹性杆中的纵向变形波传播。在本研究中，我们借助最近开发的称为修正广义指数有理函数方法的技术，获得了各种精确的孤立波解。提取了各类精确解，如亮-暗、亮、混合、奇异、暗、复和组合孤子。所应用的方法效率很高，具有显著的计算能力，能够在非线性系统中高效地以高精度求解。为了分析控制系统，通过应用带有β导数的合适波变换，将所研究的方程转化为常微分方程。除了说明解在各种参数值下的行为外，我们还生成了包含相关参数的二维和三维图形。此外，采用伽利略变换进行敏感性分析。本研究结果有可能增进对所定义系统显示的非线性动态特性的理解，并验证所实施策略的有效性。所获得的结果对理解与高维相关的非线性科学和非线性波场有重大贡献。
