Weakly Nonlocal Hamiltonian Structures: Lie Derivative and Compatibility

Abstract

We show that under certain technical assumptions any weakly nonlocal Hamiltonian structure compatible with a given nondegenerate weakly nonlocal symplectic structure $J$ can be written as the Lie derivative of $J^{-1}$ along a suitably chosen nonlocal vector field. Moreover, we present a new description for local Hamiltonian structures of arbitrary order compatible with a given nondegenerate local Hamiltonian structure of zero or first order, including Hamiltonian operators of the Dubrovin-Novikov type.

This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

References in corpus (5)

• On the local systems Hamiltonian in the weakly nonlocal Poisson brackets
• Compatible Poisson brackets of hydrodynamic type
• On recursion operators and nonlocal symmetries of evolution equations
• Weakly-nonlocal Symplectic Structures, Whitham method, and weakly-nonlocal Symplectic Structures of Hydrodynamic Type
• Compatible Dubrovin-Novikov Hamiltonian operators, Lie derivative and integrable systems of hydrodynamic type

Cited by in corpus (4)

• Recursion Operators and Tri-Hamiltonian Structure of the First Heavenly Equation of Plebański
• Recursion operators and bi-Hamiltonian structure of the general heavenly equation
• On a Recently Introduced Fifth-Order Bi-Hamiltonian Equation and Trivially Related Hamiltonian Operators
• Integrability of S-deformable surfaces: conservation laws, Hamiltonian structures and more