Reduction and Exact Solutions of the Ideal Magnetohydrodynamic Equations

Abstract

In this paper we use the symmetry reduction method to obtain invariant solutions of the ideal magnetohydrodynamic equations in (3+1) dimensions. These equations are invariant under a Galilean-similitude Lie algebra for which the classification by conjugacy classes of r-dimensional subalgebras ($1\le r\le 4$) was already known. So we restrict our study to the three-dimensional Galilean-similitude subalgebras that give systems composed of ordinary differential equations. We present here several examples of these solutions. Some of these exact solutions show interesting physical interpretations.

Work-in-progress of some exact solutions of the ideal MHD solutions. We use an analytical method to obtain several particular solutions of this quasilinear and hyperbolic system of PDEs