Dynamics of Breather and Peakon Excitations in the Nonlinear Fractional Generalized Reaction Duffing and Fisher Models via Generalized Bernoulli Equation Method

This study examines precise analytical traveling wave solutions of two significant nonlinear space-time fractional partial differential equations, the fractional generalized reaction Duffing model and the fractional Fisher equation. The fractional derivatives are defined using Jumarie’s modified Riemann–Liouville operator, which facilitates the inclusion of memory and nonlocal effects in the governing equations. Utilizing an appropriate fractional wave transformation and the generalized Bernoulli equation method, several categories of closed-form solutions are obtained in terms of hyperbolic, exponential, and rational functions. The impact of the fractional-order parameter on wave dynamics is analyzed by graphical and parametric methods, in addition to the development of precise solutions. The findings indicate that alterations in the fractional order substantially influence wave shape, resulting in transitions between kink, breather, soliton, peakon, and multi-peakon structures. Specifically, peakon-type solutions arise from fractional nonlocality, demonstrating the fractional framework’s capacity to represent sharp and localized wave fronts that are lacking in classical integer-order models. To further validate the analytical findings, a numerical investigation is carried out using a FDM. The numerical solutions are compared with the corresponding exact solutions at selected grid points, and the absolute error is evaluated. The results demonstrate that the numerical approximations are in excellent agreement with the analytical solutions, with very small error magnitudes. The results indicate that fractional-order nonlinear models offer a more comprehensive characterization of wave propagation in complex media, with possible applications in nonlinear dynamical systems and biological processes. The suggested method provides a systematic and efficient framework for investigating analytical solutions and wave phenomena of nonlinear fractional evolution equations.