Reduction and Exact Solutions of the Ideal Magnetohydrodynamic Equations

TL;DR

This paper employs symmetry reduction to derive invariant solutions of the (3+1)-dimensional ideal magnetohydrodynamic equations, providing new exact solutions with potential physical significance.

Contribution

It introduces a systematic approach to find exact solutions of MHD equations using symmetry classification and reduction techniques.

Findings

Several explicit invariant solutions are obtained.

Some solutions have meaningful physical interpretations.

The method simplifies complex PDEs to ODEs for easier analysis.

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 ($1\leq r\leq 4$) 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.