The multi-component TD hierarchy and its multi-component integrable coupling system with five arbitrary functions

A new simple loop algebra G∼M is constructed, which is devoted to establishing an isospectral problem. By making use of Tu scheme, the multi-component TD hierarchy is obtained. Furthermore, an expanding loop algebra F∼M of the loop algebra G∼M is presented. Based on F∼M, the multi-component integrable couplings system with five arbitrary functions of the multi-component TD hierarchy is worked out. The method can be applied to other nonlinear evolution equations hierarchies.