Optical solitons are solitary waves that propagate without changing shape due to a balance between dispersion and nonlinearity in the medium. Therefore, solitary optical waves are important solutions to nonlinear partial differential equations for modeling pulse propagation in optics. Our work derives new solitary wave solutions to the Kundu–Mukherjee–Naskar (KMN) equation, which governs complex nonlinear optical wave phenomena. Using innovative logarithmic transformation-based analytical techniques, various solution forms are obtained and expressed in closed form via elementary functions. The solutions are validated through direct substitution into the original KMN equation. Our new solutions provide fresh perspectives into the intricate soliton landscape described by this model. Since the KMN equation finds use in fiber optic communications, fluid dynamics, and other domains, these findings have broad implications. The methods showcase promising new pathways for unraveling soliton behaviors by fractional- and integer-order nonlinear models alike. Researchers can build upon these techniques to further advance understanding of the profound mathematical structures underlying real-world physical systems. •Derives new closed-form solitary wave solutions to the KMN equation using logarithmic transformations.•Validates solutions and provides fresh perspectives on solitons described by KMN model.•Showcases promising analytical techniques for nonlinear fractional- and integer-order models.•Advances understanding of mathematical structures in real-world physical systems.•Broad implications for fiber optics, fluid dynamics, and other KMN applications.