Lie symmetry analysis for similarity reduction and exact solutions of modified KdV–Zakharov–Kuznetsov equation

In this paper, the Lie group analysis is used to carry out the similarity reduction and exact solutions of the ( 3 + 1 ) -dimensional modified KdV–Zakharov–Kuznetsov equation. This research deals with the similarity solutions of mKdV–ZK equation. The mKdV–ZK equation has been reduced into a new partial differential equation with less number of independent variables, and again using Lie group symmetry method, the new partial differential equation is reduced into an ordinary differential equation. We have obtained the infinitesimal generators, commutator table of Lie algebra, symmetry group, and similarity reduction for the mKdV–ZK equation. In addition to that, solitary wave solutions of the mKdV–ZK equation are derived from the reduction equation. Thus, we obtain some new exact explicit solutions of the ( 3 + 1 ) -dimensional mKdV–ZK equation which describes the dynamics of nonlinear waves in plasmas.