Exact soliton solutions, bifurcation, sensitivity and stability analysis of the fractional longitudinal wave equation in magneto-electro-elastic circular rod

• Different wave structures for abundant solutions to the fractional longitudinal wave equation are investigated. • Performance was done using the strategy of the ( Φ ′ / Φ , 1 / Φ ) − expansion method. • Stability, bifurcation, and sensitivity analysis are discussed for this model. • Three-dimensional, two-dimensional, and contour plots are portrayed for the obtained solutions. This article examines the exact wave solutions, stability, bifurcation, and sensitivity analysis of the beta space-time fractional longitudinal wave equation in the magneto-electro-elastic circular rod. The governing model has wide-ranging applications in diverse fields of engineering, physical sciences, and technology like, aerodynamics, magneto-hydrodynamics, plasma physics, and others. We adopt a straightforward scheme named the ( Φ ′ / Φ , 1 / Φ ) -expansion method to scrutinize analytic solutions of the deliberated model. The present study offers several novel solitons for this equation, such as multi-soliton, periodic, kink, bell-shaped, W-shaped, breather, and singular solitons. These soliton solutions help to describe how energy and information propagate in magneto-electro-elastic circular rod, which are crucial for advanced applications in sensing, actuation, and energy conversion. Kink solitons represent topological waves or transition waves that connect two different equilibrium states of the system, bell-shaped soliton represents a concentrated energy packet moving through the medium without dispersion, breather solitons represent localized energy bursts that do not dissipate over time. Three-dimensional, two-dimensional, and contour plots are portrayed by selecting suitable values of the parameters to comprehend the physical feature of the obtained solutions. The Hopf and transcritical bifurcation have been investigated and phase-plane of the corresponding dynamical system are portrayed to study the bifurcation and equilibrium state of the model. Besides, the sensitivity analysis reveals the impact of free parameters involved in the focused equation.