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

Daryoush Talati ; Refik Turhan

Symmetry, integrability and geometry: methods and applications, Tome 7 (2011)

Résumé

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.

Détail

Keywords: bi-Hamiltonian structure; Hamiltonian operators.

Bibliographie

[1] Gel'fand I.M., Dorfman I.Ya., “Hamiltonian operators and algebraic structures related to them”, Funct. Anal. Appl., 13 (1979), 248–262 | DOI | MR

[2] Astashov A.M., Vinogradov A.M., “On the structure of Hamiltonian operators in field theory”, J. Geom. Phys., 3 (1986), 263–287 | DOI | MR | Zbl

[3] Mokhov O.I., “Hamiltonian differential operators and contact geometry”, Funct. Anal. Appl., 21 (1987), 217–223 | DOI | MR | Zbl

[4] Olver P.J., “Darboux' theorem for Hamiltonian operators”, J. Differential Equations, 71 (1988), 10–33 | DOI | MR | Zbl

[5] Cooke D.B., “Classification results and the Darboux theorem for low-order Hamiltonian operators”, J. Math. Phys., 32 (1991), 109–119 | DOI | MR | Zbl

[6] de Sole A., Kac V.G., Wakimoto M., “On classification of Poisson vertex algebras”, Transform. Groups, 14 (2010), 883–907 ; arXiv: 1004.5387 | DOI | MR

[7] Mikhailov A.V., Shabat A.V., Sokolov V.V., “The symmetry approach to classification of integrable equations”, What is Integrability, Springer Ser. Nonlinear Dynam., ed. V.E. Zakharov, Springer, Berlin, 1991, 115–184 | MR | Zbl

[8] Olver P.J., Nutku Y., “Hamiltonian structures for systems of hyperbolic conservation laws”, J. Math. Phys., 29 (1988), 1610–1619 | DOI | MR | Zbl

[9] Fuchssteiner B., Fokas A.S., “Symplectic structures, their Bäcklund transformations and hereditary symmetries”, Phys. D, 4 (1981), 47–66 | DOI | MR | Zbl

[10] Vodová J., “The Darboux coordinates for a new family of Hamiltonian operators and linearization of associated evolution equations”, Nonlinearity, 24 (2011), 2569–2574 ; arXiv: 1012.2365 | DOI | Zbl

[11] Nutku Y., “On a new class of completely integrable nonlinear wave equations. II. Multi-Hamiltonian structure”, J. Math. Phys., 28 (1987), 2579–2585 | DOI | MR | Zbl

[12] 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), 1150–1172 ; arXiv: nlin.SI/0212026 | DOI | MR | Zbl

[13] Turhan R., “Infinite-Hamiltonian structures of the Riemann equation”, Turk. J. Phys., 30 (2006), 445–449; arXiv: nlin.SI/0512071

[14] Sergyeyev A., “Weakly nonlocal Hamiltonian structures: Lie derivative and compatibility”, SIGMA, 3 (2007), 062, 14 pp. ; arXiv: math-ph/0612048 | DOI | MR | Zbl

[15] Gürses M., Karasu A., “Variable coefficient third order Korteweg–de Vries type of equations”, J. Math. Phys., 36 (1995), 3485–3491 ; arXiv: solv-int/9411004 | DOI | MR

[16] Sakovich S.Yu., “Fujimoto–Watanabe equations and differential substitutions”, J. Phys. A: Math. Gen., 24 (1991), L519–L521 | DOI | MR | Zbl