<p>This paper investigates the exact solutions of the nonlinear Schrödinger equation in a saturable medium. By analyzing the dynamical behavior of the corresponding traveling wave system, we derive explicit parametric representations for a wide range of wave solutions-including solitary waves, kink and anti-kink waves, pseudo-peakons, pseudo-periodic peakons, smooth periodic waves, and compactons-under various parameter conditions. The key novelty of this work lies in the systematic identification and classification of non-smooth wave solutions, particularly compactons and pseudo-peakons, which arise from system singularities. These results provide a comprehensive solution framework for understanding waveform evolution in saturable nonlinear media.</p>