Symmetry extensions and their physical reasons in the kinetic and hydrodynamic plasma models
Plasma theory is based on Maxwell equations with sources determined by different kinetic or hydrodynamic models. Simplifying assumptions are made to build up the models describing correctly concrete experiments. Due to such simplifications, some symmetry properties are lost, but some new appear and must be investigated. For partial differential equations of hydrodynamic plasma models this can be done by the usual Lie group theory. Symmetries of the integro differential equations of the kinetic plasma theory can be obtained from the symmetries of an infinite set of partial differential equations for the moments of distribution functions. Other direct and indirect methods were proposed. On the other hand, it was shown that there exist more general extended symmetry transformations which can also help us to solve nonlinear plasma theory problems. For the collisionless plasmas containing components with equal charge to mass ratio of particles, symmetries allow us to reduce the number of Vlasov equations by one. For the electron-positron plasma the combination of the continuous symmetries with the discrete PCT symmetry leads to the existence of exact solutions satisfying the reduced system of equations. In the hydrodynamical Hasegawa-Mima model for vortices in planet atmospheres and drift waves in plasmas, conditional symmetries appear. These symmetries produce exact solutions describing the nonlinear interaction between zonal flows and harmonic waves.