In this paper, the two- and three-soliton propagation of Sharma–Tasso–Olver (STO) equation for inhomogeneous media (with a variable-coefficient) are obtained. It is shown, due to the inelastic collisions, they undergo space-fission (or fusions). These soliton waves show a fusion that their nonlinear interactions are not completely elastic (or inelastic). To study some of the propagation features of pulses and their collision dynamics, the numerical simulations of the solutions are used. It is found that fusion along quasi-periodic waves is exhibited. The graded or step index waveguide under nonlinear refractive components are shown to reduce to improve soliton intensity. Further, the properties for these solutions are shown with figures. It may be found that research on new structure of soliton control has been obtained to understand real physical problems. These results have been usefully applied for long-wave and high-power transmission in telecommunication system. Moreover, stability analysis for the governing equation is investigated via the aspect of linear stability.
