Recursion operators and bi-Hamiltonian structure of the general heavenly equation

Abstract

We discover two additional Lax pairs and three nonlocal recursion operators for symmetries of the general heavenly equation introduced by Doubrov and Ferapontov. Converting the equation to a two-component form, we obtain Lagrangian and Hamiltonian structures of the two-component general heavenly 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 discover that in the two-component form we have only a single nonlocal recursion operator. Composing the recursion operator with the first Hamiltonian operator we obtain second Hamiltonian operator. We check the Jacobi identities for the second Hamiltonian operator and compatibility of the two Hamiltonian structures using P. Olver's theory of functional multi-vectors. Our well-founded conjecture is that P. Olver's method works fine for nonlocal operators. We show that the general heavenly equation in the two-component form is a bi-Hamiltonian system integrable in the sense of Magri. We demonstrate how to obtain nonlocal Hamiltonian flows generated by local Hamiltonians by using formal adjoint recursion operator.

34 pages. Section 2 is substantially revised. Applicability of P. Olver's criterion to nonlocal Hamiltonian operators is discussed. Some terminology is modified. The list of bibliographical references is considerably extended

References in corpus (3)

• A Simple Construction of Recursion Operators for Multidimensional Dispersionless Integrable Systems
• Weakly Nonlocal Hamiltonian Structures: Lie Derivative and Compatibility
• Doubrov-Ferapontov general heavenly equation and the hyper-Kähler hierarchy

Cited by in corpus (4)

• Recursion Operators and Tri-Hamiltonian Structure of the First Heavenly Equation of Plebański
• The integrable heavenly type equations and their Lie-algebraic structure
• 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