Solitary wave solutions and sensitivity analysis to the space-time β-fractional Pochhammer–Chree equation in elastic medium

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 \(\beta\) -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.