Lie group analysis, solitons, self-adjointness and conservation laws of the modified Zakharov-Kuznetsov equation in an electron-positron-ion magnetoplasma

Electron-positron-ion plasmas are found in the primordial Universe, active galactic nuclei, surroundings of black holes and peripheries of neutron stars. We focus our attention on a modified Zakharov-Kuznetsov (mZK) equation which describes the ion acoustic drift solitary waves in an electron-positron-ion magnetoplasma. Lie symmetry generators and groups are presented by virtue of the Lie symmetry method. Optimal system of the one-dimensional subalgebras is presented, which is influenced via the ratio of the unperturbed ion density to electron density nio/neo, the ratio of the unperturbed positron density to electron density npo/neo, the ratio of the electron temperature to positron temperature Te/Tp and the normalized ion drift velocity vo*. Based on the optimal system, we construct the power-series, multi-soliton, breather-like and periodic-wave solutions. Two types of the elastic interactions, including the overtaking and head-on interactions between (among) two (three) solitons are discussed. We find that the amplitudes of the solitons and periodic waves are positively related to the electron Debye length λDe and negatively related to |ρi| with ρi as the ion Larmor radius. Besides, we find that the mZK equation is not only strictly self-adjoint but also nonlinearly self-adjoint. Condition for the nonlinear self-adjointness is related to nio/neo, npo/neo, Te/Tp and vo*. Based on the nonlinear self-adjointness of the mZK equation, conservation laws, which are related to nio/neo, npo/neo, Te/Tp, vo*,λDe, ρi and may be associated with the conservation of momentum and energy, are obtained.