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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. [ABSTRACT FROM AUTHOR]

In this paper, we construct the Riemann–Hilbert problem to the Lax pair of Chen–Lee–Liu (CLL) equation. As far as we know, many researchers have studied various equations with Riemann–Hilbert method before, but no one compared the N-soliton solutions calculated by different symmetries of potential matrix. Using different symmetries of potential matrix, we get two N-soliton solution formulae of the CLL equation. The interesting thing is that we find the equivalence of these two N-soliton solutions. [ABSTRACT FROM AUTHOR]