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

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.

Some text overlap with our paper arXiv:1510.03666 is caused by our use here of basically the same method for discovering the Hamiltonian and bi-Hamiltonian structures of the equation, but the equation considered here and the results are totally different from arXiv:1510.03666

References in corpus (5)

• A Simple Construction of Recursion Operators for Multidimensional Dispersionless Integrable Systems
• Recursion operators and bi-Hamiltonian structure of the general heavenly equation
• Recursion Operators for Multidimensional Integrable PDEs
• Weakly Nonlocal Hamiltonian Structures: Lie Derivative and Compatibility
• Jacobi Structures of Evolutionary Partial Differential Equations

Cited by in corpus (6)

• A Simple Construction of Recursion Operators for Multidimensional Dispersionless Integrable Systems
• Evolutionary Hirota Type (2+1)-Dimensional Equations: Lax Pairs, Recursion Operators and Bi-Hamiltonian Structures
• Dispersionless Multi-Dimensional Integrable Systems and Related Conformal Structure Generating Equations of Mathematical Physics
• Integrability of dispersionless Hirota type equations in 4D and the symplectic Monge-Ampere property
• Lax pairs, recursion operators and bi-Hamiltonian representations of (3+1)-dimensional Hirota type equations
• Recursion operators and bi-Hamiltonian representations of cubic evolutionary (2+1)-dimensional systems