Symmetries, Reductions and Exact Solutions of Nonstationary Monge–Ampère Type Equations

A family of strongly nonlinear nonstationary equations of mathematical physics with three independent variables is investigated, which contain an arbitrary degree of the first derivative with respect to time and a quadratic combination of second derivatives with respect to spatial variables of the Monge–Ampère type. Individual PDEs of this family are encountered, for example, in electron magnetohydrodynamics and differential geometry. The symmetries of the considered parabolic Monge–Ampère equations are investigated by group analysis methods. Formulas are obtained that make it possible to construct multiparameter families of solutions based on simpler solutions. Two-dimensional and one-dimensional symmetry and non-symmetry reductions are considered, which lead to the original equation to simpler partial differential equations with two independent variables or ordinary differential equations or systems of such equations. Self-similar and other invariant solutions are described. A number of new exact solutions are constructed by methods of generalized and functional separation of variables, many of which are expressed in elementary functions or in quadratures. To obtain exact solutions, the principle of the structural analogy of solutions was also used, as well as various combinations of all the above-mentioned methods. In addition, some solutions are constructed by auxiliary intermediate-point or contact transformations. The obtained exact solutions can be used as test problems intended to check the adequacy and assess the accuracy of numerical and approximate analytical methods for solving problems described by highly nonlinear equations of mathematical physics.