This study investigates typical solitons of the beta space–time fractional doubly dispersive equation to explore wave propagation in nonlinear elastic materials, introscopy, long-distance energy transformation, plasma, and acoustic waves using a reliable analytical approach named the \(({G}{\prime}/G, 1/G)\) -expansion method. We obtain diverse exact solitary wave solutions, including circular, hyperbolic, and rational functions. The physical framework of the obtained solitons is illustrated with three- and two-dimensional geometric figures along with contour plots, revealing various soliton structures such as bell-shaped, spatio-temporal anti-kink, quasi-periodic, periodic, W-shaped, lump, and singular solitons. The two-dimensional graphs emphasize the importance of fractional-order differentiation in mathematical modeling. The model’s stability is analyzed through a linear stability technique combined with perturbed solutions. We examine the nonlinear planar dynamical system through bifurcation theory, which refers to qualitative changes in the behavior of the system for small numerical changes in parametric values. The phase-plane analysis provides insights into the system’s long-term behavior, equilibrium points, and stability. We establish several intrinsic and generic soliton solutions of the model under consideration, presented in a comparison table. This research contributes significant findings to understanding acoustic waves, plasma waves, seismic waves, and earthquake analysis.
