Self-dual gravity is completely integrable

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.

LaTeX2e source file, 18 pages. Title has been changed and two more authors are added: Y. Nutku and J. Kalayci. A new, simpler version of the Lagrangian is used in sections 2 and 3. Cosmetic changes are made in Introduction, section 6 and Conclusion

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• Twistor theory of hyper-K{ä}hler metrics with hidden symmetries
• Multi-Hamiltonian structure of Plebanski's second heavenly equation
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• Partner symmetries and non-invariant solutions of four-dimensional heavenly equations
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Cited by in corpus (4)

• A Simple Construction of Recursion Operators for Multidimensional Dispersionless Integrable Systems
• Lax pairs, recursion operators and bi-Hamiltonian representations of (3+1)-dimensional Hirota type equations
• Nonlocal symmetry of CMA generates ASD Ricci-flat metric with no Killing vectors
• On Integrating the Left-Flat Vacuum Einstein Equations