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SIGMA 3 (2007), 062, 14 pages math-ph/0612048
https://doi.org/10.3842/SIGMA.2007.062
Contribution to the Vadim Kuznetsov Memorial Issue
Weakly Nonlocal Hamiltonian Structures: Lie Derivative and Compatibility
Artur Sergyeyev
Mathematical Institute, Silesian University in Opava, Na
Rybnícku 1, 746 01 Opava, Czech Republic
Received December 15, 2006, in final form April 23, 2007; Published online April 26, 2007
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.
Key words:
weakly nonlocal Hamiltonian structure; symplectic structure; Lie derivative.
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