Generalized multi-component TC hierarchy and its multi-component integrable coupling system

A new W-dimensional Lie algebra X is constructed firstly. Then, the corresponding loop algebra X is produced, whose commutation operation defined by us is as simple and straightforward as that in the loop algebra A,. It follows that a general scheme for generating multi-component integrable hierarchy is proposed. By taking advantage of X, a new isospectral problem is established, and then well-known multi-component TC hierarchy is obtained. Finally, an expanding loop algebra F-M of the loop algebra X is presented. Based on the Pm, the multi-component integrable coupling system of the generalized multi-coinponent TC hierarchy has been worked out. The method in this paper can be applied to other nonlinear evolution equations hierarchies. It is easy to find that we can construct any finite-dimensional Lie algebra by this approach.