On the Differential Operators of the Generalized Fifth-order Korteweg-de Vries Equation

In this paper, we present the differential operators of the generalized fifth-order KdV equation. We give formal proofs on the Hamiltonian property including the skew-adjoint property and Jacobi identity by the use of prolongation method. Our results show that there are five 3-order Hamiltonian operators, which can be used to construct the Hamiltonians, and no 5-order operators are shown to pass the Hamiltonian test, although there are infinite number of them, and are skew-adjoint.

Keywords

Hamiltonian system, nonlinear differential equation, nonlinear partial differential equation, fifth-order KdV equation, Ito equation, Sawada-Kotera equation, Caudrey-Dodd-Gibbon equation, Kaup-Kupershmidt equation, Lax equation, Jacobi identity, skew-adjoint operator, prolongation

2010 Mathematics Subject Classification

35G20, 35L05, 35Q53, 37K05, 37K10, 47J35

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Published 1 January 2010