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Infinite-Dimensional Lie Algebra With a New Poisson Bracket 

Zhang Baoshan, Lu Dongqiang, Dai Shiqiang
(Shanghai Institute of Applied Mathematics & Mechanics, Shanghai University, Shanghai, China, 200072)

Abstract: For the  Hamiltonian formulation of the Korteweg-de Vires equation (KdV equation), C.S. Gardner defined a Poisson bracket. In this paper a brand-new bracket is defined. It is easily verified that new bracket possesses three properties of the Poisson bracket, bilinearity, skew symmetry, Jacobi identity. The new Poisson bracket has a close connection with C.S. Gardner's definition. In the framework of the new Poisson bracket, all the first integrals of the KdV equation constitute an infinite-dimensional Lie algebra. Then the necessary and sufficient conditions for identifying the first integrals are obtained. Finally, the method for finding first integrals of  KdV equation is investigated.
Keywords: KdV equation, Hamiltonian formulation, first integral, Poisson bracket, infinite-dimensional, Lie algebra..

The project supported by the National Natural Science Foundation of China.
Received 27,December 1996, revised 24 November 1997.

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