Bifurcation analysis and solitary wave solution of fractional longitudinal wave equation in magneto-electro-elastic (MEE) circular rod

•M−Fractional Derivative in Longitudinal Wave Equation: The study investigates the longitudinal wave equation (LWE) using the M−fractional derivative to model wave propagation in materials with mechanical, electrical, and magnetic field interactions.•Bifurcation Analysis and Soliton Formation: Bifurcation analysis is applied to identify critical points and phase portraits, revealing transitions to new behaviors such as stability shifts and chaos, and the formation of static solitons through saddle-node bifurcation.•Modified Simple Equation (MSE) Technique: The study utilizes the MSE technique to find solitary wave solutions, which are expressed in hyperbolic, trigonometric, and exponential forms depending on free parameter relationships.•Complex Phenomena and Solution Comparison: The numerical solutions reveal complex wave phenomena, including dark and bright bell waves, kink periodic lump waves, and interactions of periodic and lump waves. The results are compared with existing literature, highlighting the efficacy of the methods for producing distinct soliton solutions. In this study, we investigate the longitudinal wave equation (LWE) with the M−fractional derivative, which describes the propagation of longitudinal waves along a rod while incorporating interactions between mechanical, electrical, and magnetic fields within the material. Initially, we apply bifurcation analysis to examine the critical points or phase portraits where the system transitions to new behaviors, such as stability shifts or the emergence of chaos, and observe the mechanism of static soliton formation through a saddle-node bifurcation. Subsequently, we utilize a modified simple equation (MSE) technique to find solitary wave solutions. Depending on the relationships between free parameters, the solutions are expressed as hyperbolic, trigonometric, and exponential functions. The numerical form of the obtained solutions reveals complex phenomena, including dark and bright bell waves, kink periodic lump waves, kink periodic waves, periodic lump waves, interaction waves between kink and periodic waves, linked lump waves, and interactions of periodic and lump waves. Additionally, we compare our results with previously published work, demonstrating that the discussed methods are valuable tools for providing distinct, accurate soliton solutions relevant to nonlinear science and technology applications.