### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 032, 35 pages      arXiv:1403.3521      https://doi.org/10.3842/SIGMA.2016.032
Contribution to the Special Issue on Analytical Mechanics and Differential Geometry in honour of Sergio Benenti

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

Gianni Manno a and Giovanni Moreno b
a) Dipartimento di Scienze Matematiche ''G.L. Lagrange'', Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
b) Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warsaw, Poland

Received October 29, 2015, in final form March 16, 2016; Published online March 26, 2016

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.

Key words: Monge-Ampère equations; prolongations of contact manifolds; characteristics of PDEs; distributions on manifolds; third-order PDEs.

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