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On the integrability of symplectic Monge-Amp'ere equations | | B. Doubrov
; E.V. Ferapontov
; | | Date: |
18 Oct 2009 | | Abstract: | Let $u$ be a function of $n$ independent variables $x^1, ..., x^n$, and
$U=(u_{ij})$ the Hessian matrix of $u$. The symplectic Monge-Amp’ere equation
is defined as a linear relation among all possible minors of $U$. Particular
examples include the equation $det U=1$ governing improper affine spheres, the
so-called heavenly equation $u_{13}u_{24}-u_{23}u_{14}=1$ describing self-dual
Ricci-flat 4-manifolds, etc.
In this paper we classify integrable symplectic Monge-Amp’ere equations in
four dimensions (for $n=3$ the integrability of such equations is known to be
equivalent to their linearizability). This problem can be reformulated
geometrically as the classification of "maximally singular" hyperplane sections
of the Lagrangian Grassmannian. We formulate a conjecture that any integrable
equation of the form $F(u_{ij})=0$ in more than three dimensions is necessarily
of the symplectic Monge-Amp’ere type. | | Source: | arXiv, 0910.3407 | | Services: | Forum | Review | PDF | Favorites | |
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