Electron–positron–ion plasmas emerge in active galactic nuclei, the primordial Universe, peripheries of neutron stars, and surroundings of black holes. Thus, this article showcases the analytical examination of a multi-dimensional fifth-order generalized Zakharov–Kuznetsov model with dual power-law nonlinearities in an electron–positron–ion magnetoplasma. This interesting electron–positron–ion plasma model, with enough nonlinear mathematics and astrophysics/cosmology considerations, is observed to possess various copious real-world scenarios, especially in cosmic plasmas. In essence, a thorough investigation of the model is carried out with a view to see the application of its results in various science and engineering disciplines. Abundant soliton solutions to the models are to be generated, and various wave structures of interest are to be simulated numerically. In the wake of the robust Lie group theory, a comprehensive Lie group analysis of this equation with power-law nonlinearities is further performed. This consequently leads to the emergence of diverse invariants and solutions associated with the model. In addition, the equation is reduced to diverse ordinary differential equations using its point symmetries, and consequently, diverse closed-form solutions of interest are achieved for some particular cases of n. These outcomes are obtained in the form of complex and non-complex dark solitons, topological solitons, as well as various algebraic solutions with arbitrary functions. Moreover, by utilizing the power series method, one derives some series solutions of the understudy models for some specific cases of some of the consequential difficult nonlinear ordinary differential equations. A deep understanding of the found solutions is aided by simulating some of the solutions. Consequently, various soliton collisions ensued, thus giving rise to diverse structures of psychedelic bump waves, parabolic waves, concentric wave shapes with strata, as well as other wave forms of interest which are discussed. The real-world applications of the various achieved wave dynamics are presented in detail to bring the pertinence of the research results home. Thereafter, strict self-adjointness as well as formal Lagrangian formulation, leading to various conservation laws via Ibragimov’s theorem, are entrenched. Consequently, conservation of energy, momentum, and angular momentum is achieved, which has a very wide spectrum of pertinence and significance, especially in the fields of physics and mathematics.
