Space-time variable-order fractional analysis of nonlinear longitudinal wave propagation in magneto-electro-elastic materials.

This work presents a space-time variable-order (V-O) fractional framework for the analytical investigation of nonlinear longitudinal wave propagation in magneto-electro-elastic (MEE) materials. The model contains Caputo-type V-O derivatives, which account for spatial and temporal memory effects and nonlocal interactions inherent to coupled mechanical, electrical, and magnetic fields. On this basis, exact traveling-wave solutions of the resulting nonlinear fractional wave equation were derived using an exponential expansion methodology, in the form of periodic, kink-type, and solitary waves. The effect of the V-O parameters on the wave amplitude, dispersion characteristics, and stability behavior is systematically analyzed. Stability and bifurcation analyses show parameter-dependent transition of both stable and unstable regimes, whereas insertion of random perturbations illustrates the development of chaotic dynamics. The proposed V-O model offers increased modeling flexibility and parameter-dependent stability regimes in the reduced system compared to constant-order (C-O) fractional formulations. This enhancement provides increased analytical flexibility at the theoretical level for coupled field interactions in MEE media. These results suggest that the V-O fractional modeling represents a valid tool for the analysis of complex nonlinear electromechanical phenomena and could lead to a better understanding of the wave dynamics at a theoretical level in multifunctional material systems.