On the two nonzero boundary problems of the AB system with multiple poles

The Riemann–Hilbert method is used to study AB systems with two nonzero boundary conditions. In order to reconstruct the potentials, spectral analysis of Lax pair and the corresponding Riemann–Hilbert problem are discussed. For these two nonzero boundary problems, we consider the cases where the sectional analytic functions have double and triple poles, respectively. The derived scattering data has multiple zeros or pairs of multiple zeros, from which the relationship between the eigenfunctions at the corresponding poles can be obtained. By solving algebraic equations, we can derive the formulas of $N$-order solutions in different cases. As an application, we obtain different types of solitons and breathers in the reflectionless case and the dynamic analysis of these solutions is revealed.