New integrable couplings and Hamiltonian structure of the KN hierarchy and the DLW hierarchy

Two new loop algebras F∼ and G∼ are constructed, which are devoted to establishing the resulting isospectral problems. By taking use of the compatibility of Lax pairs, the two corresponding zero curvature equations are presented from which the integrable couplings of the KN hierarchy and the dispersive long wave hierarchy (briefly called DLW hierarchy). As far as we can see, the above results are new. Again via employing the quadratic identity, the Hamiltonian structures of the two well-known integrable systems are obtained, respectively, and they are Liouville integrable.