Integrable couplings of the hierarchies of evolution equations

An efficient, straightforward method for obtaining integrable couplings of a hierarchy of evolution equations is proposed in this paper. First a transformation of a kind of Lax pairs is presented, a new loop algebra is constructed, which can be used to obtain the integrable couplings of the Levi hierarchy. Using this method can also obtain the integrable couplings of the well-known generalized AKNS hierarchy. As special cases, the integrable couplings of KdV equation and mKdV equation are obtained, respectively. This method can be used generally.