新型Poisson括号意义下的无穷维Lie代数

INFINITE DIMENSIONAL LIE ALGEBRA WITH A NEW POISSON BRACKET 1)

本文首先针对ＫｄＶ方程的Ｈａｍｉｌｔｏｎ形式，建立一种比较容易验证的新型Ｐｏｉｓｓｏｎ括号和无穷维Ｌｉｅ代数．其次，研究ＫｄＶ方程的Ｈａｍｉｌｔｏｎ形式的第一积分与新型Ｐｏｉｓｏｎ括号的关系，得到判定第一积分的充分必要条件．最后，构造ＫｄＶ方程的第一积分．

For the Hamiltonian formulation of the Korteweg de Vires equation (KdVequation), C.S. Gardner defined a Poisson bracket. In this paper a brand new bracket is defined. It is easily verified that the 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.