This study investigates a widely recognized Hirota-Maccari model used in plasmas, fluid mechanics, and optics. Novel dynamics of this nonlinear structure are introduced through the computational software Maple 18. Three reliable and effective methods, such as the unified scheme, the G ' G '+G+A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{{G<^>{\prime } }}{{G<^>{\prime } + G + A}}$$\end{document} technique, and the modified rational sine-cosine approach, are applied to investigate the governing model. The stability analysis of the adopted model is also studied using the Hamiltonian approach. Our investigation yields various outcomes, such as a single dark-bright breather wave, multiple dark-bright breather waves, a local breather wave, a dark soliton, a kink wave, an anti-kink wave, a periodic wave, and an M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M$$\end{document}-shaped periodic wave. Finally, the acquired outcomes are presented in two-dimensional and three-dimensional plots for various parametric values. It is shown that the mentioned techniques are reliable, effective, and applicable mathematical algorithms for solving nonlinear models.
