Methods and Applications of Analysis

Volume 17 (2010)

Number 1

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

Pages: 123 – 136

DOI: http://dx.doi.org/10.4310/MAA.2010.v17.n1.a5

Author

Chun-Te Lee

Abstract

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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