Meta-Symplectic Geometry of $3^{\mathrm{rd}}$ 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.

References in corpus (4)

• Characteristics, Bicharacteristics, and Geometric Singularities of Solutions of PDEs
• Contact geometry of multidimensional Monge-Ampère equations: characteristics, intermediate integrals and solutions
• Normal forms for parabolic Monge-Ampere equations
• Remarks on non-maximal integral elements of the Cartan plane in jet spaces

Cited by in corpus (2)

• Completely exceptional $2^{\text{nd}}$ order PDEs via conformal geometry and BGG resolution
• Remarks on the Lagrangian representation of bi-Hamiltonian equations