Weakly Nonlocal Hamiltonian Structures: Lie Derivative and Compatibility

TL;DR

This paper demonstrates that compatible weakly nonlocal Hamiltonian structures can be expressed as Lie derivatives of inverse symplectic structures, providing a new perspective on their characterization and compatibility.

Contribution

It introduces a novel description of local Hamiltonian structures of arbitrary order compatible with given structures, including Dubrovin-Novikov type operators.

Findings

Compatible structures can be written as Lie derivatives of inverse symplectic forms.

New description for local Hamiltonian structures of any order.

Includes Hamiltonian operators of Dubrovin-Novikov type.

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.