We obtain new computational soliton solutions characterized by topological, rational, exponential, trigonometric, and hyperbolic functions for the modified Zakharov–Kuznetsov equation (mZKE). Utilizing two successful procedures, the extended rational sinh-cosh and Kudryashov expansion methods are applied to derive diverse dynamical wave structures of soliton solutions within the context of evolutionary dynamical structures of solitary wave solutions. It is permissible to choose parameters that show a solution. To improve understanding of the physical phenomena related to these dynamical models in mathematical physics, the empirical demonstration of the physical behavior of these solutions is provided. The technique of symmetry analysis is utilized to examine the governing equation. The implementation of a novel conservation theorem results in the formation of a comprehensive system of one-dimensional subalgebras. The study encompasses the Lie-Bäcklund symmetry generators. Additionally, the methodology for establishing conservation laws for nonlinear partial differential equations is clarified through the presentation of an innovative conservation theorem associated with Lie-Bäcklund symmetries. This strategy is used to come up with the conservation laws that apply to the governing equation.