Hamiltonian Tri-Integrable Couplings of the AKNS Hierarchy

We propose a systematic approach for generating Hamiltonian tri-integrable couplings of soliton hierarchies. The resulting approach is based on semi-direct sums of matrix Lie algebras consisting of 4×4 block matrix Lie algebras. We apply the approach to the AKNS soliton hierarchy, and Hamiltonian structures of the obtained tri-integrable couplings are constructed by the variational identity.