This study investigates a generalized Schrödinger equation that governs the propagation of optical pulses in nonlinear media in the presence of Kerr nonlinearity. Using Kudryashov’s generalized method, we reduce the model to a Duffing-type ordinary differential equation whose first integral yields an exact bright-soliton solution. We validate this solution in three complementary ways. The admissible parameter regime is stated explicitly, and the values used in every figure are shown to fall within this regime, ensuring that the plotted solutions are real and physically meaningful. The solution is then recovered via a stable boundary-value integration of the reduced equation, which agrees with the closed form to R² = 1.0000 and a root-mean-square error of order 10⁻¹¹. Finally, the dependence of the wave on the Kerr coefficient is characterised quantitatively: rank-correlation analysis reveals that the peak amplitude is a perfectly monotone but nonlinear function of the Kerr coefficient (Spearman ρ = −1, Kendall τ = −1, Pearson r = −0.94) obeying the exact law |A|_max ∝ ℵ^−1/2, while the characteristic width is proved to be independent of it. Cramér–von Mises and Anderson–Darling goodness-of-fit tests, together with a cumulative-distribution-function outlier screening, confirm that the numerical samples are statistically consistent with the theoretical model and free of artefacts. A phase-plane portrait, a propagation map and a two-parameter amplitude map further clarify the dynamics. All computations were reproduced in a license-free open-source environment to ensure portability and reproducibility.