Contact linearization of nondegenerate Monge-Ampère equations

The present paper is devoted to the problem of transforming the classical Monge-Ampère equations to the linear equations by change of variables. The class of Monge-Ampère equations is distinguished from the variety of second-order partial differential equations by the property that this class is closed under contact transformations. This fact was known already to Sophus Lie who studied the Monge-Ampère equations using methods of contact geometry. Therefore it is natural to consider the classification problems for the Monge-Ampère equations with respect to the pseudogroup of contact transformations. In the present paper we give the complete solution to the problem of linearization of regular elliptic and hyperbolic Monge-Ampère equations with respect to contact transformations. In order to solve this problem, we construct invariants of the Monge-Ampère equations and the Laplace differential forms, which involve the classical Laplace invariants as coefficients.