SIGMA 3 (2007), 034, 6 pages      nlin.SI/0702055      https://doi.org/10.3842/SIGMA.2007.034
Contribution to the Proceedings of the Coimbra Workshop on Geometric Aspects of Integrable Systems

By Magri's Theorem, Self-Dual Gravity is Completely Integrable

Yavuz Nutku
Feza Gürsey Institute, P.O.Box 6, Çengelköy, Istanbul, 81220 Turkey

Received September 08, 2006, in final form February 08, 2007; Published online February 27, 2007

Abstract
By Magri's theorem the bi-Hamiltonian structure of Plebanski's second heavenly equation proves that (anti)-self-dual gravity is a completely integrable system in four dimensions.

Key words: self-dual gravity; Plebanski equation; Magri's theorem.

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References

1. Neyzi F., Nutku Y., Sheftel M.B., Multi-Hamiltonian structure of Plebanski's second heavenly equation, J. Phys. A: Math. Gen. 38 (2005), 8473-8485, nlin.SI/0505030.
2. Magri F., A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978), 1156-1162.
3. Magri F., A geometrical approach to the nonlinear solvable equations, in Nonlinear Evolution Equations and Dynamical Systems, Editors M. Boiti, F. Pempinelli and G. Soliani, Lecture Notes in Phys., Vol. 120, Springer, New York, 1980, 233-263.
4. Plebanski J.F., Some solutions of complex Einstein equations, J. Math. Phys. 16 (1975), 2395-2402.
5. Nutku Y., Halil M., Colliding impulsive gravitational waves, Phys. Rev. Lett. 39 (1977), 1379-1382.
6. Nambu Y., Generalized Hamiltonian dynamics, Phys. Rev. D 7 (1973), 2405-2412.
7. Nutku Y., On a new class of completely integrable nonlinear wave equations. II. Multi-Hamiltonian structure, J. Math. Phys. 28 (1987), 2579-2585.
8. Olver P.J., Nutku Y., Hamiltonian structures for systems of hyperbolic conservation laws, J. Math. Phys. 29 (1988), 1610-1619.
9. Calabi E., The space of Kähler metrics, in Proceedings of the International Congress of Mathematicians (1954, Amsterdam), Vol. 2, North-Holland, Amsterdam, 1956, 206-207.
10. Nutku Y., Hamiltonian structure of real Monge-Ampère equations, J. Phys. A: Math. Gen. 29 (1996), 3257-3280, solv-int/9812023.
11. Dirac P.A.M., Lectures on quantum mechanics, Belfer Graduate School of Science Monographs Series 2, New York, 1964.
12. Malykh A.A., Nutku Y., Sheftel M.B., Partner symmetries of the complex Monge-Ampère equation yield hyper-Kahler metrics without continuous symmetries, J. Phys. A: Math. Gen. 36 (2003), 10023-10037, math-ph/0403020.
13. Santini P.M., Fokas A.S., Recursion operators and bi-Hamiltonian structures in multidimensions. I, Comm. Math. Phys. 115 (1988), 375-419.