<p>The simultaneous analysis of stability at equilibrium points and soliton dynamics is crucial to nonlinear systems and to initiatives involving various challenging physical phenomena. In this study, we investigate the Manakov equation, an essential model in nonlinear optical science, plasma physics, and Bose-Einstein condensates. Initially, the unified method is used to identify soliton solutions to this nonlinear problem. Then, a traveling wave transformation is used to change this model into a first-order dynamical structure. We also illustrate the stability of equilibrium locations utilizing the planar dynamics approach. We demonstrate different types of solitons, such as breather waves, periodic waves, local breathers, periodic breathers, and kink waves, by using 2D, 3D, contour, and polar plots. Additionally, an intriguing theorem is included to describe the dynamics of equilibrium points. The results improve our understanding of how to shape soliton waveforms and the stability of equilibrium points. They also offer innovative strategies for controlling nonlinear wave processes.</p>