We present a comprehensive analytical and qualitative study of the Sharma–Tasso–Olver (STO) equation. Exact solutions are obtained using the new auxiliary equation method and compared with the \(({G'}/{G})\) expansion and Tan–Cot function methods. We considered STO due to its advances in fluid dynamics, nonlinear optics, and plasma theory, as it models dispersive interactions and wave propagation that affect wave profiles and stability. It displays solitary, periodic, and singular wave characteristics that give rise to nonlinear phenomena. This provides an analytical and geometrical framework that focuses on both theoretical and potential physical significance of nonlinear wave dynamics. Bifurcation and Hamiltonian analyses of the reduced dynamical system, analysing equilibrium structures, energy surfaces, and phase portraits for varying bifurcation parameter \(\epsilon \) . This dual framework reveals complex transitions such as homoclinic orbits and nonlinear oscillations.
