In this paper, we propose three new types of the integrable nonlocal combined nonlinear Schrödinger–Gerdjikov–Ivanov (NLS-GI) models. By the Riemann–Hilbert approach, we discuss the Cauchy problem of the reverse-space-time nonlocal combined NLS-GI model with step-like initial data: u(z, 0) = o(1) for z → −∞ and u(z, 0) = A + o(1) for z → +∞, where A is an arbitrary positive constant. First of all, we give an integrable nonlocal combined NLS-GI model and its Lax pair. Then, we consider the analytical and asymptotic behaviors, symmetries, and scattering matrix of the Jost solutions. Finally, we discuss the Cauchy problem for the nonlocal combined NLS-GI model with step-like initial data.
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