GROUP ANALYSES, REDUCTIONS AND EXACT SOLUTIONS OF MONGE–AMPERE EQUATION OF MAGNETIC HYDRODYNAMICS

We study the Monge–Amp`ere equation with three independent variables which occurs in electron magnetohydrodynamics. A group analysis of this strongly nonlinear partial derivative equation is carried out. An eleven-parameter transformation preserving the form of the equation is found. A formula is obtained that makes it possible to construct multiparametric families of solutions based on simpler solutions. Two-dimensional reductions leading to simpler partial differential equations with two independent variables. One-dimensional reductions are described, which make it possible to obtain self-similar and other invariant solutions that satisfy ordinary differential equations. Exact solutions with additive, multiplicative and generalized separation of variables are constructed, many of which admit representation in elementary functions. The obtained results and exact solutions can be used to evaluate the accuracy and analyze the adequacy of numerical methods for solving initial boundary value problems described by strongly nonlinear partial differential equations.