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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