The multi-component Tu hierarchy of soliton equations and its multi-component integrable couplings system

A new simple loop algebra \widetildeG_M is constructed, which is devoted to the establishing of an isospectral problem. By making use of the Tu scheme, the multi-component Tu hierarchy is obtained. Furthermore, an expanding loop algebra \widetildeF_M of the loop algebra \widetildeG_M is presented. Based on the \widetildeF_M, the multi-component integrable coupling system of the multi-component Tu hierarchy has been worked out. The method can be applied to other nonlinear evolution equation hierarchies.