This paper focuses on the investigation of optical soliton solutions of the Biswas-Milovic equation with parabolic law nonlinearity, which models nonlinear pulse propagation in optical fibers. Numerous studies have focused on the Biswas-Milovic equation, which is one of the key equations for simulating soliton propagation through optical waveguides. Utilizing the new Kudryashov’s method and the \((\frac{1}{\psi (\zeta )},\frac{\psi ^{\prime }(\zeta )}{\psi (\zeta )})\) method, we derive bright, singular and kink soliton solutions. The derived optical soliton solutions provide a distinct analytical perspective on nonlinear phenomena and establish an independent foundation for further understanding and exploration of physical applications in different nonlinear wave behaviors. The study employs visually captivating three-dimensional, density, and two-dimensional plots that emphasis the distinctive characteristics of the solutions. A key novelty of this work lies in the integration of an improved Adomian decomposition method, which avoids linearization and discretization while providing excellent agreement with analytical results. The results are presented alongside the established error analysis. Our numerical findings demonstrate complete concordance with the suggested analytical optical solutions. In other words, our approach offers a significant level of reliability and accuracy. In addition, the research discusses the modulation instability of steady state waves in order to ascertain conditions under which small perturbations develop.
