<p>The generalized Schrödinger equation with the power-law of nonlinearity, which describes propagation pulses in optical fibers, is analyzed. The well-known unified auxiliary equation approach is used to study the solitary wave solutions of the proposed equation. The solutions of solitary waves and periodic waves at zero or unity modulus of ellipticity are obtained by employing Jacobi elliptic function solutions. The obtained novel solutions which come in a variety of forms, including kink, periodic, dark, bright, W-shaped, singular, and singular periodic are extracted by the hyperbolic, trigonometric, and exponential functions. In addition, the impact of the degree of nonlinearity on the periodic and solitary wave structures is examined. Furthermore, a comprehensive bifurcation analysis is performed to explore the structural transitions and qualitative behaviors of the solutions. Phase portraits are constructed to visualize the dynamical nature of the equilibrium points and to elucidate the global behavior of the system in the phase space.</p>