Nuclei discovered new practical insights via optimized soliton-like pulse analysis in a space fractional-time beta-derivatives equations.

Nerve signal conduction, and particularly in myelinated nerve fibers, is a highly dynamic phenomenon that is affected by various biological and physical factors. The propagation of such moving electric signals may seemingly help elucidate the mechanisms underlying normal and abnormal functioning. This work aims to derive the exact physical wave solutions of the nonlinear partial differential equations with fractional beta-derivatives for the cases of transmission of nerve impulses in coupled nerves. To this end, the research uses a polynomial expansion approach to convert the problems of modeling nerve impulses into a second order elliptic nonlinear ordinary differential equation containing fractional beta-derivatives. Such transformation permits the study of solitary waves and their perturbation responses in the case of nerve fibers. The other direction of this study is applying the fixed-point theory to analyze the system dynamics and obtaining the Jacobian matrix to peruse the stability. Modulation instability regions are visualized, and nerve impulse waveforms are shown in three and two dimensions. The investigation depicts how impulse transmission amplitude and velocity are influenced by changing nerve fiber diameter and varying order physiological parameters. Soliton-like kink, anti-kink, and rogue wave solutions are revealed to explain nerve impulse propagation thoroughly. The analysis provides significant regions of equilibrium and modulational instability showing that the behavior of the nerve fibers is more dynamic than appreciated by most authors. Additionally, the authors suggest a refined mathematical formulation of the nerve impulse conduction with particular emphasis on the effect of fractional beta-derivatives on the transmission of waves. The obtained solutions and the graphs support their usefulness in various medical and biological industries, specifically the research on myelinated nerve fibers. The findings provide additional insights into the processes of nerve conduction which may be useful in the treatment of various diseases of the nervous system.