Bifurcation analysis and the modulation instability in a nonlinear silica optical fibers

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.