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arXiv · 1403.3521

Meta-Symplectic Geometry of $3^{\rm rd}$ Order Monge-Ampère Equations and their Characteristics

Gianni Manno·Giovanni Moreno
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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.

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Gianni Manno, Giovanni Moreno. 2016-05-10. Meta-Symplectic Geometry of $3^{\rm rd}$ Order Monge-Ampère Equations and their Characteristics. https://doi.org/10.3842/sigma.2016.032

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