Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 081, 8 pages      arXiv:1104.4034      https://doi.org/10.3842/SIGMA.2011.081

On a Recently Introduced Fifth-Order Bi-Hamiltonian Equation and Trivially Related Hamiltonian Operators

Daryoush Talati and Refik Turhan
Department of Engineering Physics, Ankara University 06100 Tandogan Ankara, Turkey

Received April 25, 2011, in final form August 18, 2011; Published online August 20, 2011

Abstract
We show that a recently introduced fifth-order bi-Hamiltonian equation with a differentially constrained arbitrary function by A. de Sole, V.G. Kac and M. Wakimoto is not a new one but a higher symmetry of a third-order equation. We give an exhaustive list of cases of the arbitrary function in this equation, in each of which the associated equation is inequivalent to the equations in the remaining cases. The equations in each of the cases are linked to equations known in the literature by invertible transformations. It is shown that the new Hamiltonian operator of order seven, using which the introduced equation is obtained, is trivially related to a known pair of fifth-order and third-order compatible Hamiltonian operators. Using the so-called trivial compositions of lower-order Hamiltonian operators, we give nonlocal generalizations of some higher-order Hamiltonian operators.

Key words: bi-Hamiltonian structure; Hamiltonian operators.

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