A new loop algebra and its application to the multi-component S-mKdV hierarchy

An efficient, straightforward method for obtaining multi-component integrable hierarchy and the multi-component integrable couplings system is proposed in this paper. A new 4M dimensional loop algebra X∼ is constructed firstly, whose commutation operations defined by us are as simple and straightforward as that in the loop algebra A1∼. As an application example, a new isospectral problem is established, then the well-known multi-component Schrödinger hierarchy and the multi-component mKdV hierarchy are obtained. So we call it the multi-component S-mKdV hierarchy. 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 S-mKdV 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 in this paper.