Similarity Reductions and Analytic Solutions for a (3+1)-Dimensional Modified Korteweg-De Vries-Zakharov-Kuznetsov Equation Relevant to the Pulsars’ Magnetospheres, Planetary Aurorae, Earth’s Magnetosphere and Solar Wind

Recently, nonlinear research endeavors focused on the cosmic electron-positron plasmas have yielded productive outcomes. Of astrophysical relevance to such contexts as the pulsars’ magnetospheres, auroral magnetoplasmas, Earth’s magnetosphere, and solar wind, this paper examines a (3+1)-dimensional modified Korteweg-de Vries-Zakharov-Kuznetsov equation. The investigation centers on a magnetized electron-positron plasma comprising the equal proportions of the hot and cool components for each species, where the hot components obey a Boltzmann distribution and the cool components are characterized by certain fluid equations. Employing symbolic computation, we derive two sets of the (3+1)-dimensional similarity reductions, mapping that (3+1)-dimensional equation onto two known ordinary differential equations, each corresponding to one set. Additionally, several (3+1)-dimensional analytic solutions for that (3+1)-dimensional equation are obtained through symbolic computation. Leveraging those (3+1)-dimensional findings, nonlinear wave structures can be explored in such contexts as the previously mentioned cosmic plasma. In turn, electrostatic oscillations within such a cosmic plasma might be examined in terms of their dependence on the plasma frequencies for the adiabatic species, phase velocity, parallel equilibrium streaming of the adiabatic species, thermal velocities for the adiabatic species, charges of the cool and hot species, adiabatic compression indices, common mass of an electron and a positron, Debye lengths for the isothermal species, Boltzmann constant, temperatures for the isothermal species, gyrofrequencies for the adiabatic species as well as magnitude of the ambient magnetic field. Future cosmic-plasma observations made by the spacecrafts or satellites may uncover certain phenomena predicted in this paper, and pertinent astrophysical insights might emerge.