ON MULTIDIMENSIONAL EXACT SOLUTIONS OF GENERALIZED MONGE–AMPE`RE EVOLUTION EQUATIONS

We consider evolutionary multidimensional generalized Monge–Amp`ere equations, the right parts of which, in addition to the determinant of the Hesse matrix, may depend on the Laplace operator and the gradient of the desired function. A variant of the reduction method for constructing exact multidimensional solutions of the generalized Monge–Amp`ere evolution equations using separation of variables is proposed. Multivariate exact solutions expressed explicitly through elementary functions and through solutions of ordinary differential equations are obtained. A number of examples of exact solutions, both radially symmetric and anisotropic in spatial variables, expressed through combinations of elementary functions are given.