Bi-Hamiltonian Structure and Liouville Integrablity for a  Gerdjikov--Ivanov Equation Hierarchy

Chin. Phys. Lett.

2001, Vol. 18

Issue (1): 1-3 DOI:

Original Articles

Bi-Hamiltonian Structure and Liouville Integrablity for a Gerdjikov--Ivanov Equation Hierarchy

FAN En-Gui

Institute of Mathematics, Fudan University, Shanghai 200433

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FAN En-Gui 2001 Chin. Phys. Lett. 18 1-3

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Abstract A new spectral problem is introduced and the associated hierarchy of nonlinear evolution equations is derived with the help of the zero curvature equation as well as recursive operators. It is found that the Gerdjikov--Ivanov (GI) equation, which is one of three important derivative nonlinear Schrödinger equations, exactly belongs to the hierarchy as a special reduction. A powerful tool of the trace identity is used to establish the bi-Hamiltonian structure for the whole GI hierarchy. Moreover, it is shown that GI hierarchy admits an infinite common set of conserved quantities which are in involution in pairs under Poisson's bracket. This indicates that the whole GI hierarchy is completely integrable in Liouville's sense.

Keywords: 03.40.Kf 02.30.Jr

Published: 01 January 2001

PACS:	03.40.Kf
	02.30.Jr	(Partial differential equations)

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https://cpl.iphy.ac.cn/ OR https://cpl.iphy.ac.cn/Y2001/V18/I1/01

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