We propose a unified analytical–numerical framework for exploring optical soliton evolution in a cubic (c)-quintic (q) nonlinear medium with dispersion (d), incorporating self-steepening (SS, \({s}_{p}\) ), and self-frequency shift (SFS, \({s}_{f}\) ). Combining ansatz-based analytical approach with the split-step Fourier and finite element method, the proposed model studies soliton stability, captures collisions, and profile deformation across varying parameters. The analytical solution is numerically validated by employing it as the initial condition for the numerical simulations. Then the influences of the physical parameters are first identified using numerical simulations, followed by an examination of the stability behavior under various parameter regimes. Finally, stable propagation is obtained for parameters \(d=0.2, c=0.2, q=-0.5, {s}_{p}=0.02, and {s}_{f}=0.02\) . The proposed model is investigated for soliton collision dynamics using the split-step Fourier method and a (2 + 2) soliton molecule is observed at \(d=0.1, c=0.1, q=-0.1, {s}_{p}=0.01, and {s}_{f}=0.01\) , whereas, \(d=0.05, c=0.02, q=-0.1, {s}_{p}=0.001, {s}_{f}=0.005\) give rise to the (2 + 1 + 2) soliton molecule from collision. Inelastic pumping is observed for \(d=0.05, c=0.08, q=-0.1, {s}_{p}=0.01, {s}_{f}=0.025\) . Modulational instability analysis of continuous-wave background reveals that SS governs phase modulation, whereas SFS induces instability. The proposed framework effectively reduces mathematical and computational complexity, offering new insight into ultrashort pulse control in nonlinear optical media. These findings have direct implications for tuneable laser design and wavelength-division multiplexing (WDM) technologies.
