Coupled Kdv Hierarchy And Their Bi-hamilton Structures

By introducing a 4×4 matrix spectral problem with two potentials,we propose two classes of nonlinear evolution equations.A typical equation in the first class is the coupled KdV equation by Satsuma and Hirota.It is shown that the hierarchy possesses the generalized Hamilton form with the help of trace identity.Then,two other coupled KdV equations are obtained.Finally,we proved the nonlinear evolution equations are Liouville intergral.