A combined integrable hierarchy with four potentials and its recursion operator and bi-Hamiltonian structure

Based on a specific matrix Lie algebra, we propose a spectral matrix with four potentials and generate its associated Liouville integrable Hamiltonian hierarchy. The zero curvature formulation and the trace identity are the basic tools. The Liouville integrability of the resulting hierarchy is shown by determining its recursion operator and bi-Hamiltonian structure. Two illustrative examples of generalized combined nonlinear Schrödinger equations and modified Korteweg-de Vries equations are explicitly presented. The success lies in introducing a specific \(4\times 4\) $4\times 4$ spectral matrix which keads to an integrable hierarchy.