Four-component combined integrable equations possessing bi-Hamiltonian formulations

This paper aims to generate a Liouville integrable Hamiltonian hierarchy by introducing a specific matrix eigenvalue problem with four components. The adopted approach is the zero curvature formulation. A bi-Hamiltonian formulation is furnished through applying the trace identity, which shows the Liouville integrability of the resulting hierarchy. Two examples of generalized combined nonlinear Schrödinger equations and modified Korteweg–de Vries equations are presented.