Some exact solutions of the ideal MHD equations through symmetry reduction method

We use the symmetry reduction method based on Lie group theory to obtain some exact solutions, the so-called invariant solutions, of the ideal magnetohydrodynamic equations in (3 + 1) dimensions. In particular, these equations are invariant under a Galilean-similitude Lie algebra for which the classification by conjugacy classes of r-dimensional subalgebras (1 r 4) was already known.We restrict our study to the three-dimensional Galilean-similitude subalgebras that give us systems composed of ordinary differential equations. Here, some examples of these solutions are presented with a brief physical interpretation. © 2007 Elsevier Inc. All rights reserved.