The Coupling Of Restrained CKdV System

In this paper, based on the Lie-Poisson structure we take the cKdV hierarchy as an il-lustrative example to study the integrability of finite-dimensional coupling systems. First, we define two new Lie-Poisson structures associated with the semidirect product of Lie algebra sl(2,R), then with these Poisson structures the spectral problem of the coupled cKdV hierarchy in Lax equations are nonlinearized into two coupled finite-dimensional sys-tems. Finally the integrability of coupled finite-dimensional systems are proven by using Lax representations and generating functions.