On Multidimensional Exact Solutions to an Evolution Equation with the Wave Operator and the Monge–Ampère Operator

An evolution equation with the wave operator and the Monge–Ampère operator is studied. A version of the reduction method based on additive and multiplicative separation of variables is proposed for constructing exact multidimensional solutions. Parametric families of exact solutions that are anisotropic with respect to the spatial variables are obtained; these solutions are expressed explicitly in terms of elementary and special functions and/or in terms of solutions to ordinary differential equations. The case is considered in which known solutions to the linear wave equation with one spatial variable are used to construct exact solutions to the multidimensional nonlinear equation under study. Several examples illustrating the obtained results are presented.