The Schäfer-Wayne equation (SWE), a crucial model for ultrashort pulse propagation in nonlinear silicon optical fibers, is investigated using the $F$-expansion method and enhanced modified extended tanh expansion method (EMETEM). We derive diverse solitary wave solutions, including dark, bright, periodic, multi-peak periodic, and breather-like periodic solutions, visualized through $2D$ and $3D$ graphics. Novel contributions include comprehensive bifurcation analysis via planar dynamical systems revealing phase portrait classifications, modulation instability analysis for solution stability evaluation, and sensitivity analysis assessing parameter dependence and initial condition effects. The diverse solitary wave solutions represent a new advancement in understanding SWE dynamics. The study demonstrates the methods' robustness in examining nonlinear wave dynamics with applications in optics, engineering, and telecommunications.

In this paper, we investigated higher-order smooth positon and breather-positon solutions of the Kuralay equation. Starting from the associated Lax pair, we constructed an explicit $N$-fold Darboux transformation (DT) in determinant form. By introducing a spectral parameter degeneration procedure combined with higher-order Taylor expansion, we derived smooth higher-order positon solutions from multi-soliton solutions. Furthermore, under nonvanishing boundary conditions, breather-positon solutions were obtained. The dynamical properties of these solutions were analyzed, revealing elastic interaction behavior and nontrivial phase shifts. The results provided a unified framework for constructing degenerate localized wave structures and extended existing studies on the Kuralay equation.