Analytical and numerical investigations, data-driven modeling via an analytical-machine learning approach, and...

This study investigates a nonlinear fractional soliton neuron model that arises in fluid mechanics, nonlinear dynamics, mathematical physics, engineering, neuroscience, plasma physics, and related scientific fields. An efficient mapping approach is employed after transforming the governing nonlinear partial differential equation into an ordinary differential equation through a suitable wave transformation. A machine learning approach is also applied for data-driven analysis of the obtained solution. These two proposed methods yield a variety of exact analytical solutions, including lump, local breather, periodic, anti-kink, multiple bright-dark breather, singular soliton, bright soliton, and kink waves. Their propagation characteristics are illustrated using three-dimensional, two-dimensional, contour, polar, and surface-of-revolution plots generated in Maple. Stability properties of the model and the obtained solutions are also examined. Furthermore, nonlinear dynamics are explored through MATLAB-based chaos diagnostics, including bifurcation diagrams, strange attractors, recurrence plots, fractal dimensions, phase portraits, basins of attraction, return maps, power spectra, time series, and multistability. The results demonstrate that variations in the amplitude and frequency of external forcing significantly influence the system's dynamical behavior.