<p>The paper focuses on the nonlinear Sasa-Satsuma equation, which describes the propagation of ultra-short pulses in single-mode fibres and accounts for third-order dispersion, self-steepening, and stimulated scattering. In contrast to previous studies that employed complex travelling-wave transformations, this study employs more effective methods of analysis to derive more general and meaningful soliton solutions. The generalised projective Riccati equation and modified auxiliary equation methods are efficiently applied to construct travelling-wave solutions and links to several known analytical techniques, though they are less suited for multi-soliton or complex non-periodic structures. In contrast, utilized methods are highly flexible and yield a wide range of solution types but depend strongly on the chosen auxiliary equation. Using these methods, a wide spectrum of solution structures can be derived, including high-amplitude solitons, combined dark–bright and kink–bright waves, bright and bright-lump solutions, anti-kink profiles, U-shaped bright solitons, and singular waveforms, all expressed in terms of hyperbolic and trigonometric functions. Their propagation characteristics are illustrated through three-dimensional surfaces, density distributions, and contour plots generated in Maple, which effectively show how key parameters influence the evolution and dynamics of these solutions. A further sensitivity analysis of a new dynamics model of the soliton wave speed factor is conducted. Furthermore, bifurcation and chaotic analyses are conducted to illustrate the system’s complex dynamical behavior. In general, the mathematical strategies adopted show that they are efficient in deriving effective and credible travelling wave solutions to a large set of nonlinear evolution equations.</p>