Meta-Symplectic Geometry of 3rd Order Monge-Ampère Equations and their Characteristics

Abstract

This paper is a natural companion of [Alekseevsky D.V., Alonso Blanco R., Manno G., Pugliese F., Ann. Inst. Fourier (Grenoble) 62 (2012), 497-524, arXiv:1003.5177], generalising its perspectives and results to the context of third-order (2D) Monge-Ampère equations, by using the so-called ''meta-symplectic structure'' associated with the 8D prolongation M⁽¹⁾ of a 5D contact manifold M. We write down a geometric definition of a third-order Monge-Ampère equation in terms of a (class of) differential two-form on M⁽¹⁾. In particular, the equations corresponding to decomposable forms admit a simple description in terms of certain three-dimensional distributions, which are made from the characteristics of the original equations. We conclude the paper with a study of the intermediate integrals of these special Monge-Ampère equations, herewith called of Goursat type.