
SIGMA 7 (2011), 081, 8 pages arXiv:1104.4034
https://doi.org/10.3842/SIGMA.2011.081
On a Recently Introduced FifthOrder BiHamiltonian 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 fifthorder biHamiltonian 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 thirdorder 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 fifthorder and thirdorder compatible Hamiltonian operators. Using the socalled trivial compositions of
lowerorder Hamiltonian operators, we give nonlocal generalizations of some higherorder Hamiltonian operators.
Key words:
biHamiltonian structure; Hamiltonian operators.
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References
 Gel'fand I.M., Dorfman I.Ya.,
Hamiltonian operators and algebraic structures related to them,
Funct. Anal. Appl. 13 (1979), 248262.
 Astashov A.M., Vinogradov A.M.,
On the structure of Hamiltonian operators in field theory,
J. Geom. Phys. 3 (1986), 263287.
 Mokhov O.I.,
Hamiltonian differential operators and contact geometry,
Funct. Anal. Appl. 21 (1987), 217223.
 Olver P.J., Darboux' theorem for Hamiltonian operators,
J. Differential Equations 71 (1988), 1033.
 Cooke D.B.,
Classification results and the Darboux theorem for loworder Hamiltonian operators,
J. Math. Phys. 32 (1991), 109119.
 de Sole A., Kac V.G., Wakimoto M.,
On classification of Poisson vertex algebras,
Transform. Groups 14 (2010), 883907,
arXiv:1004.5387.
 Mikhailov A.V., Shabat A.V., Sokolov V.V.,
The symmetry approach to classification of integrable equations,
in What is Integrability,
Editor V.E. Zakharov,
Springer Ser. Nonlinear Dynam., Springer, Berlin, 1991, 115184.
 Olver P.J., Nutku Y.,
Hamiltonian structures for systems of hyperbolic conservation laws,
J. Math. Phys. 29 (1988), 16101619.
 Fuchssteiner B., Fokas A.S.,
Symplectic structures, their Bäcklund transformations and hereditary symmetries,
Phys. D 4 (1981), 4766.
 Vodová J.,
The Darboux coordinates for a new family of Hamiltonian operators and linearization of associated evolution equations,
Nonlinearity 24 (2011), 25692574,
arXiv:1012.2365.
 Nutku Y.,
On a new class of completely integrable nonlinear wave equations. II. MultiHamiltonian structure,
J. Math. Phys. 28 (1987), 25792585.
 Ferapontov E.V., Pavlov M.V.,
Reciprocal transformations of Hamiltonian operator of hydrodynamic type: nonlocal Hamiltonian formalism for linearly degenerate systems,
J. Math. Phys. 44 (2003), 11501172,
nlin.SI/0212026.
 Turhan R.,
InfiniteHamiltonian structures of the Riemann equation,
Turk. J. Phys. 30 (2006), 445449,
nlin.SI/0512071.
 Sergyeyev A.,
Weakly nonlocal Hamiltonian structures: Lie derivative and compatibility,
SIGMA 3 (2007), 062, 14 pages,
mathph/0612048.
 Gürses M., Karasu A.,
Variable coefficient third order Kortewegde Vries type of equations,
J. Math. Phys. 36 (1995), 34853491,
solvint/9411004.
 Sakovich S.Yu.,
FujimotoWatanabe equations and differential substitutions,
J. Phys. A: Math. Gen. 24 (1991), L519L521.

