The Riemann–Hilbert approach for the higher-order Gerdjikov–Ivanov equation, soliton interactions and position shift

In this paper, we are concerned with the Riemann–Hilbert approach for the higher-order Gerdjikov–Ivanov equation with nonzero boundary conditions. A uniformization variable will be introduced to make it forms a one-to-one mapping with the spectral parameter. The analyticity, symmetric and asymptotic behaviors of the eigenfunctions and scattering datas will be calculated in detail and the Riemann–Hilbert problem will be constructed to obtain the expression for the N-soliton solutions. In addition, the case of Jost solutions and scattering coefficients near the branch points will also be analyzed. On the other hand, when the boundary conditions turn to zero, the formula for the position shifts and phase shifts of solitons resulting from soliton interactions will be solved.