Group Analysis, Reductions, and Exact Solutions of the Monge–Ampère Equation in Magnetic Hydrodynamics

We study the Monge–Ampère equation with three independent variables, whichoccurs in electron magnetohydrodynamics. A group analysis of this strongly nonlinear partialdifferential equation is carried out. An eleven-parameter transformation preserving the form of theequation is found. A formula is obtained that permits one to construct multiparameter families ofsolutions based on simpler solutions. Two-dimensional reductions leading to simpler partialdifferential equations with two independent variables are considered. One-dimensional reductionsare described that permit one to obtain self-similar and other invariant solutions that satisfyordinary differential equations. Exact solutions with additive, multiplicative, and generalizedseparation of variables are constructed, many of which admit representation in elementaryfunctions. The obtained results and exact solutions can be used to evaluate the accuracy andanalyze the adequacy of numerical methods for solving initial–boundary value problems describedby strongly nonlinear partial differential equations.