.3 ¡Á 3lax Matrix Of Integrable Soliton Equations Solution

In the present paper, integrable decomposition is made for the soliton equations generated from the spectral problem with 3×3 matrix coefficients. The main purpose, is to study the Boussinesq equation, the three-wave interaction equation, the Manakov equation and the derivative Manakov equation (or the coupled Kaup-Newell equation). The main result is to decompose their finite-parameter solutions into solutions of compatible Hamiitonian systems of ordinary differential equations. With the Lax matrix as a basic tool, the conserved integrals are obtained with the proof of their involutivity.The Lax matrices, obtained through the nonlinearization technique, are all of 3×3 type, which are far more complicated than the 2×2 ones. A method of "generating function flow" is developed to get the basic condition of involutivity, which provides a convenient and effective approach to study the integrabilityin Liouville sense of the associated Hatniltonian systems for the difficult 3×3 spectral problem.A Neumarm system is obtained for the third order differential operator related to the Boussinesq equation, which is a generalization of the well-known KdV-Neumann system for the second order ones. This gives an answer to an open problem proposed by H. Flaschka in 1983. Moreover, it is proved rigorously that a solution of the Boussinesq equation is generated from the compatible solution of two conserved Hamiltonian flows of this Neumann system.