New solitons and bifurcation dynamics in the fractional Huxley model: Numerical and analytical approaches

This research explores the higher-order nonlinear fractional Huxley equation formulated with the $\beta$, and $M$-truncated fractional derivatives to account for memory and hereditary effects present in nonlinear diffusion and excitation wave dynamics. The fractional formulation expands on the standard Huxley model by incorporating nonlocal temporal and spatial correlations, providing a more realistic description of finite-amplitude wave propagation and spectral energy transfer in complex media. Two advanced analytical techniques are used to derive exact solutions: the enhanced modified extended tanh expansion method (EMETEM) and the improved $F$-expansion technique. Compared to classic perturbation and variational techniques, these methods offer greater algebraic freedom and faster convergence, resulting in a diverse family of closed-form traveling-wave solutions expressed in trigonometric, hyperbolic, exponential, and rational forms. The analytical results are further validated, and the spatiotemporal evolution of the resulting wave structures is investigated using a finite-difference numerical scheme. The accuracy and robustness of the suggested framework are confirmed by the numerical findings, which show good agreement with the analytical results. Phase-plane and bifurcation analyses demonstrate transitions between periodic, quasi-periodic, and chaotic regimes by revealing both stable and unstable spiral formations. The findings show that fractional derivatives significantly improve the dynamical characteristics of the Huxley system by allowing for finer control of diffusion, dispersion, and localized energy concentration, advancing our understanding of nonlinear fractional wave behavior in excitable and dispersive media.