An integrable coupling hierarchy of Dirac integrable hierarchy, its Liouville integrability and Darboux transformation

An integrable coupling hierarchy of Dirac integrable hierarchy is presented by means of zero curvature representation. A Hamiltonian operator involving two parameters is introduced, and it is used to derive a pair of Hamiltonian operators. A bi-Hamiltonian structure of the obtained integrable coupling hierarchy is constructed with the aid of Magri pattern of bi-Hamiltonian formulation. Moreover, we prove the Liouville integrability of the obtained integrable coupling hierarchy and establish a Darboux transformation of the integrable coupling. As an application, an exact solution of the integrable coupling of Dirac equation is given.