Double Integrable Couplings and Their Constructing Method

To extend the study scopes of integrable couplings, the notion of double integrable couplings is proposed in the paper. The zero curvature equation appearing in the constructing method built in the paper consists of the elements of a new loop algebra which is obtained by using perturbation method. Therefore, the approach given in the paper has extensive applicablevalues, that is, it applies to investigate a lot of double integrable couplings of the known integrable hierarchies of evolution equations. As for explicit applications of the method proposed in the paper, the double integrable couplings of the AKNS hierarchy and the KN hierarchy are worked out, respectively.