A combined Kaup–Newell type integrable hierarchy with four potentials and its bi-Hamiltonian formulation

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This paper aims to introduce a Kaup–Newell type matrix eigenvalue problem with four potentials, based on a specific matrix Lie algebra, and construct its associated Liouville integrable Hamiltonian hierarchy, through the zero curvature formulation. The Liouville integrability of the resulting hierarchy is shown by determining its recursion operator and bi-Hamiltonian formulation. An illustrative example of combined derivative nonlinear Schrödinger equations with two arbitrary constants is explicitly presented.