Review of exact solutions and reductions of Monge–Ampère type equations

We present a review of publications devoted to exact solutions, transformations, symmetries, reductions, and applications of strongly nonlinear stationary and nonstationary (parabolic) equations of the Monge–Ampère type. We study the strongly nonlinear nonstationary mathematical physics equations with three independent variables that contain a quadratic combination of second spatial derivatives of the Monge–Ampère type and an arbitrary degree of the first temporal derivative or an arbitrary function depending on this derivative. We study the symmetries of these equations using group analysis methods. We derive formulas that enable the construction of multiparameter families of solutions, based on simpler solutions. We consider two-dimensional and one-dimensional symmetry and nonsymmetry reductions, which transform the original equations into simpler partial differential equations with two independent variables, or to ordinary differential equations and systems of such equations. Self-similar and other invariant solutions are described. Using generalized and functional separation of variables methods, we constructed several new exact solutions, many of which are expressed in elementary functions or in quadratures. Some solutions are obtained using auxiliary intermediate-point or contact transformations. These exact solutions can be used as test problems to verify the adequacy of and evaluate the accuracy of numerical and approximate analytical methods for solving problems described by strongly nonlinear mathematical physics equations.