Chaotic behaviors, stability, and solitary wave propagations of M-fractional LWE equation in magneto-electro-elastic circular rod

This work studies the chaotic behaviors and solitary wave propagations for the M-fractional longitudinal wave equation (M-fLWE). Here, we explain some assertions of the M-fractional derivative. Initially, we employ bifurcation theory to examine the chaotic behaviors that arise from the incorporation of diverse perturbation terms. We depict the phase portraits using three-dimensional (3D) and two-dimensional (2D) representations, Poincaré diagrams, and time-series plots. Furthermore, we utilize an enhanced modified F-expansion method to examine ion acoustic waves in the fLWE. The derived solutions manifest as trigonometric, exponential, and hyperbolic functions. In the numerical discussion, we present novel phenomena not observed in previous studies. For particular values of the free parameters, we discern luminous and obscure bell-shaped waves, periodic waves, periodic bell-shaped rogue waves, periodic rogue waves featuring singular solitons, periodic rogue waves, and interactions between periodic rogue waves and kink-shaped formations. Additionally, we juxtapose our results with the current literature to emphasize unique attributes in 2D, 3D, and density-based representations. This research provides significant insights into the intricate behaviors and varied waveforms of the governing model via a thorough investigation. This study enhances the comprehension of real-world physical phenomena through the examination of waveform attributes, bifurcation analysis, chaotic dynamics, and solitary waves.