Self-dual gravity is completely integrable

TL;DR

This paper demonstrates that the complex Monge-Ampere equation, expressed in a two-component form, possesses a multi-Hamiltonian structure, making it a completely integrable system in four real dimensions, with new recursion operators and an infinite hierarchy of flows.

Contribution

It introduces a multi-Hamiltonian framework for the complex Monge-Ampere equation, including new recursion operators and an infinite hierarchy of commuting flows.

Findings

Discovery of multi-Hamiltonian structure for CMA

Identification of two new real recursion operators

Construction of an infinite hierarchy of integrable flows

Abstract

We discover multi-Hamiltonian structure of complex Monge-Ampere equation (CMA) set in a real first-order two-component form. Therefore, by Magri's theorem this is a completely integrable system in four real dimensions. We start with Lagrangian and Hamiltonian densities and obtain a symplectic form and the Hamiltonian operator that determines the Dirac bracket. We have calculated all point symmetries of two-component CMA system and Hamiltonians of the symmetry flows. We have found two new real recursion operators for symmetries which commute with the operator of a symmetry condition on solutions of the CMA system. These operators form two Lax pairs for the two-component system. The recursion operators, being applied to the first Hamiltonian operator, generate infinitely many real Hamiltonian structures. We show how to construct an infinite hierarchy of higher commuting flows together with the corresponding infinite chain of their Hamiltonians.