
SIGMA 12 (2016), 091, 17 pages arXiv:1605.07770
https://doi.org/10.3842/SIGMA.2016.091
Recursion Operators and TriHamiltonian 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 twocomponent evolutionary form and obtain Lagrangian and Hamiltonian representations of this system. We study all point symmetries of the twocomponent 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 twocomponent 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 abovementioned 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 twocomponent form. Using P. Olver's theory of the functional multivectors, 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 triHamiltonian representation of the first heavenly equation. Our wellfounded conjecture applied here is that P. Olver's method works fine for nonlocal operators and our proof of the Jacobi identities and biHamiltonian 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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