A combined Liouville integrable hierarchy associated with a fourth-order matrix spectral problem

This paper aims to discuss a fourth-order matrix spectral problem involving four potentials and to generate an associated Liouville integrable hierarchy via the zero curvature formulation. A bi-Hamiltonian formulation is furnished by applying the trace identity and a recursion operator is explicitly worked out, which exhibits the Liouville integrability of each model in the resulting hierarchy. Two specific examples, consisting of novel generalized combined nonlinear Schroedinger equations and modified Korteweg-de Vries equations, are given.