A few new higher-dimensional Lie algebras and two types of coupling integrable couplings of the AKNS hierarchy and the KN hierarchy

Four higher-dimensional Lie algebras are introduced. With the help of their different loop algebras and the block matrices of Lax pairs for the zero curvature representations of two given integrable couplings, the two types of coupling integrable couplings of the AKNS hierarchy and the KN hierarchy are worked out, respectively, which fill up the consequences obtained by Ma and Gao (2009) [9]. The coupling integrable couplings of the AKNS hierarchy obtained in the paper again reduce to the coupling integrable couplings of the nonlinear Schrِdinger equation and the modified KdV (mKdV) equation, which are different from the resulting results given by Ma and Gao.