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

JOURNAL OF GEOMETRY AND PHYSICS

卷:116

Article

Sheftel, M. B.¹  Yazici, D.²  Malykh, A. A.³ 

[1] Bogazici Univ, Dept Phys, TR-34342 Istanbul, Turkey

[2] Yildiz Tech Univ, Dept Phys, TR-34220 Istanbul, Turkey

[3] Russian State Hydrometeorol Univ, Dept Numer Modelling, Malookhtinsky Pr 98, St Petersburg 195196, Russia

关键词: General heavenly equation;    Lax pair;    Recursion operator;    Symplectic form;    Hamiltonian operator;    Jacobi identities;   

DOI  :  10.1016/j.geomphys.2017.01.026

来源: Elsevier

【 摘 要 】

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. (C) 2017 Elsevier B.V. All rights reserved.

【 授权许可】

Free