3 years ago

Integrability of dispersionless Hirota type equations in 4D and the symplectic Monge-Ampere property.

E.V. Ferapontov, V. Novikov, B. Kruglikov

We prove that integrability of a dispersionless Hirota type equation implies the symplectic Monge-Ampere property in any dimension $\geq 4$. In 4D this yields a complete classification of integrable dispersionless PDEs of Hirota type through a list of heavenly type equations arising in self-dual gravity. As a by-product of our approach we derive an involutive system of relations characterising symplectic Monge-Ampere equations in any dimension.

Moreover, we demonstrate that in 4D the requirement of integrability is equivalent to self-duality of the conformal structure defined by the characteristic variety of the equation on every solution, which is in turn equivalent to the existence of a dispersionless Lax pair. We also give a criterion of linerisability of a Hirota type equation via flatness of the corresponding conformal structure, and study symmetry properties of integrable equations.

Publisher URL: http://arxiv.org/abs/1707.08070

DOI: arXiv:1707.08070v2

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