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

Abstract

We study the Monge–Ampère equation with three independent variables, which occurs in electron magnetohydrodynamics. A group analysis of this strongly nonlinear partial differential equation is carried out. An eleven-parameter transformation preserving the form of the equation is found. A formula is obtained that permits one to construct multiparameter families of solutions based on simpler solutions. Two-dimensional reductions leading to simpler partial differential equations with two independent variables are considered. One-dimensional reductions are described that permit one 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.