The multi-component TA hierarchy and its multi-component integrable couplings system with six arbitrary functions

A new simple 3M dimensional loop algebra X˜ is produced, whose commutation operation defined by us as simple and straightforward as that in the loop algebra A˜1. 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 by making use of the Tu scheme the well-known multi-component TA hierarchy is obtained. Finally, an expanding loop algebra F˜M of the loop algebra X˜ is presented, based on the F˜M, the multi-component integrable couplings system of the multi-component TA hierarchy with six arbitrary functions are 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 in this paper.