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In this paper, we investigate the modulation instability and the dynamics of solitary waves in a higher-order nonlinear Schrödinger equation. We use a powerful and robust method known as the modified sub-equation method to secure the solitary waves and other numerous solutions to this nonlinear model. Several constraint conditions that guarantee the existence of these solutions are highlighted. Utilizing a linearizing technique, we establish the modulation instability gain. The influence of the higher nonlinear component on the modulation instability is also discussed. Furthermore, we perform the numerical simulations of this model by using the split-step Fourier method. Finally, several solutions are shown in two and three dimensions to better explain the behavior of the considered model. The two-dimensional plots depict Peregrine soliton under the influence of the nonlinear dispersion term. These graphs are useful for understanding the dynamic properties of the obtained results. • New optical solitons for the HNSE were constructed. • The obtained solutions were found using the modified sub-equation technique. • The model's modulation instability is studied. • The physical meaning for the solutions is graphically investigated. [ABSTRACT FROM AUTHOR]