Analytical analysis of the nonlinear fractional order Pochhammer-Chree equation with power-law nonlinearity in elastic materials

This study investigates the nonlinear fractional-order Pochhammer-Chree equation, featuring a power-law nonlinearity of order τ, a model that describes how nonlinear longitudinal waves travel in elastic materials with memory of prior deformations. Understanding nonlinear wave propagation in elastic materials with memory effects is important for accurately modeling complex physical phenomena in nonlinear elasticity, geophysics, and material science. To incorporate these memory effects, conformable fractional derivative is utilized that allowing us to examine the fractional-order spatial and temporal changes. Kumar-Malik method is used to extract analytical solutions like bright soliton, dark soliton, kink soliton and solitary wave forms, that demonstrates the dynamics of the model. To better understand these solutions behaviours, visual analysis is considered. In this analysis mathematica 14.2 is used to construct the two-dimensional, three-dimensional, and contour plots, which illustrate the influence of the fractional parameter χ on the waveforms. The shape, width and amplitude of the solitons also vary with the change in the value of the parameter, χ. These diagrams are able to make it clear how the system would respond to various physical situations. These findings indicate the efficiency of the Kumar-Malik method which guarantees accurate solutions of the fractional Pochhammer-Chree equation having power laws nonlinearity. The work enhances the theoretical knowledge on the nonlinear waves of the fractional-order systems and brings forth new uses in the field of mathematical physics and nonlinear elasticity.