### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 091, 17 pages      arXiv:1605.07770      https://doi.org/10.3842/SIGMA.2016.091

### Recursion Operators and Tri-Hamiltonian Structure of the First Heavenly Equation of Plebański

Mikhail B. Sheftel a and Devrim Yazıcı b
a) Department of Physics, Boğaziçi University, Bebek, 34342 Istanbul, Turkey
b) Department of Physics, Yıldız Technical University, Esenler, 34220 Istanbul, Turkey

Received June 28, 2016, in final form September 10, 2016; Published online September 14, 2016

Abstract
We present first heavenly equation of Plebański in a two-component evolutionary form and obtain Lagrangian and Hamiltonian representations of this system. We study all point symmetries of the two-component system and, using the inverse Noether theorem in the Hamiltonian form, obtain all the integrals of motion corresponding to each variational (Noether) symmetry. We derive two linearly independent recursion operators for symmetries of this system related by a discrete symmetry of both the two-component system and its symmetry condition. Acting by these operators on the first Hamiltonian operator $J_0$ we obtain second and third Hamiltonian operators. However, we were not able to find Hamiltonian densities corresponding to the latter two operators. Therefore, we construct two recursion operators, which are either even or odd, respectively, under the above-mentioned discrete symmetry. Acting with them on $J_0$, we generate another two Hamiltonian operators $J_+$ and $J_-$ and find the corresponding Hamiltonian densities, thus obtaining second and third Hamiltonian representations for the first heavenly equation in a two-component form. Using P. Olver's theory of the functional multi-vectors, we check that the linear combination of $J_0$, $J_+$ and $J_-$ with arbitrary constant coefficients satisfies Jacobi identities. Since their skew symmetry is obvious, these three operators are compatible Hamiltonian operators and hence we obtain a tri-Hamiltonian representation of the first heavenly equation. Our well-founded conjecture applied here is that P. Olver's method works fine for nonlocal operators and our proof of the Jacobi identities and bi-Hamiltonian structures crucially depends on the validity of this conjecture.

Key words: first heavenly equation; Lax pair; recursion operator; Hamiltonian operator; Jacobi identities; variational symmetry.

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References

1. Antonowicz M., Fordy A.P., Coupled KdV equations with multi-Hamiltonian structures, Phys. D 28 (1987), 345-357.
2. Antonowicz M., Fordy A.P., Coupled Harry Dym equations with multi-Hamiltonian structures, J. Phys. A: Math. Gen. 21 (1988), L269-L275.
3. Antonowicz M., Fordy A.P., Factorisation of energy dependent Schrödinger operators: Miura maps and modified systems, Comm. Math. Phys. 124 (1989), 465-486.
4. De Sole A., Kac V.G., Nonlocal Hamiltonian structures and applications to the theory of integrable systems I, arXiv:1210.1688.
5. Dirac P.A.M., Lectures on quantum mechanics, Belfer Graduate School of Science Monographs Series, Vol. 2, Belfer Graduate School of Science, New York, 1967.
6. Doubrov B., Ferapontov E.V., On the integrability of symplectic Monge-Ampère equations, J. Geom. Phys. 60 (2010), 1604-1616, arXiv:0910.3407.
7. Ferapontov E.V., Differential geometry of nonlocal Hamiltonian operators of hydrodynamic type, Funct. Anal. Appl. 25 (1991), 195-204.
8. Ferapontov E.V., Nonlocal matrix Hamiltonian operators, differential geometry and applications, Theoret. Math. Phys. 91 (1992), 642-649.
9. Fuchssteiner B., Fokas A.S., Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D 4 (1981), 47-66.
10. Krasil'shchik I.S., Verbovetsky A.M., Vitolo R., A unified approach to computation of integrable structures, Acta Appl. Math. 120 (2012), 199-218, arXiv:1110.4560.
11. Kupershmidt B.A., Mathematics of dispersive water waves, Comm. Math. Phys. 99 (1985), 51-73.
12. Liu S.-Q., Zhang Y., Jacobi structures of evolutionary partial differential equations, arXiv:0910.2085.
13. Magri F., A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978), 1156-1162.
14. Magri F., A geometrical approach to the nonlinear solvable equations, in Nonlinear Evolution Equations and Dynamical Systems (Proc. Meeting, Univ. Lecce, Lecce, 1979), Lecture Notes in Phys., Vol. 120, Editors M. Boiti, F. Pempinelli, G. Soliani, Springer, Berlin-New York, 1980, 233-263.
15. Malykh A.A., Nutku Y., Sheftel M.B., Partner symmetries of the complex Monge-Ampère equation yield hyper-Kähler metrics without continuous symmetries, J. Phys. A: Math. Gen. 36 (2003), 10023-10037, math-ph/0305037.
16. Malykh A.A., Sheftel M.B., Recursions of symmetry orbits and reduction without reduction, SIGMA 7 (2011), 043, 13 pages, arXiv:1005.0153.
17. Mokhov O.I., Symplectic and Poisson structures on loop spaces of smooth manifolds, and integrable systems, Russ. Math. Surv. 53 (1998), 515-622.
18. Mokhov O.I., Nonlocal Hamiltonian operators of hydrodynamic type with flat metrics, integrable hierarchies, and associativity equations, Funct. Anal. Appl. 40 (2006), 11-23, math.DG/0406292.
19. 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.
20. Nutku Y., Sheftel M.B., Kalayci J., Yazıcı D., Self-dual gravity is completely integrable, J. Phys. A: Math. Theor. 41 (2008), 395206, 13 pages, arXiv:0802.2203.
21. Olver P.J., Applications of Lie groups to differential equations, Graduate Texts in Mathematics, Vol. 107, Springer-Verlag, New York, 1986.
22. Olver P.J., Nonlocal symmetries and ghosts, in New Trends in Integrability and Partial Solvability, NATO Sci. Ser. II Math. Phys. Chem., Vol. 132, Kluwer Acad. Publ., Dordrecht, 2004, 199-215.
23. Olver P.J., Poisson structures and integrability, Talk given at Institut des Hautes Etudes Scientifiques, France, 2010, available at http://www.math.umn.edu/~olver/t_/poisson.pdf.
24. Olver P.J., Private communication, 2016.
25. Olver P.J., Sanders J.A., Wang J.P., Ghost symmetries, J. Nonlinear Math. Phys. 9 (2002), suppl. 1, 164-172.
26. Plebański J.F., Some solutions of complex Einstein equations, J. Math. Phys. 16 (1975), 2395-2402.
27. Sergyeyev A., Weakly nonlocal Hamiltonian structures: Lie derivative and compatibility, SIGMA 3 (2007), 062, 14 pages, math-ph/0612048.
28. Sergyeyev A., Recursion operators for multidimensional integrable systems, arXiv:1501.01955.
29. Sheftel M.B., Recursions, in CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 3, New Trends in Theoretical Developments and Computational Methods, Editor N.H. Ibragimov, CRC Press, Boca Raton, FL, 1996, 91-137.
30. Sheftel M.B., Malykh A.A., On classification of second-order PDEs possessing partner symmetries, J. Phys. A: Math. Theor. 42 (2009), 395202, 20 pages, arXiv:0904.2909.
31. Sheftel M.B., Malykh A.A., Partner symmetries, group foliation and ASD Ricci-flat metrics without Killing vectors, SIGMA 9 (2013), 075, 21 pages, arXiv:1306.3195.
32. Sheftel M.B., Malykh A.A., Yazıcı D., Recursion operators and bi-Hamiltonian structure of the general heavenly equation, arXiv:1510.03666.
33. Sheftel M.B., Yazıcı D., Bi-Hamiltonian representation, symmetries and integrals of mixed heavenly and Husain systems, J. Nonlinear Math. Phys. 17 (2010), 453-484, arXiv:0904.3981.
34. Yazıcı D., Generalization of bi-Hamiltonian systems in $(3+1)$ dimension, possessing partner symmetries, J. Geom. Phys. 101 (2016), 11-18.