Inverse scattering transform for the nonlocal Gerdjikov–Ivanov equation with simple and double poles

We systematically investigate the nonlocal Gerdjikov-Ivanov (nGI) equation with non-vanishing boundary conditions by means of the inverse scattering transform method. We define eigenfunctions and scattering matrix, then analyze their analytical, symmetric and asymptotic properties. With the help of the inverse scattering transform, an appropriate Riemann-Hilbert problem is constructed. The nGI equation displays drastically different symmetry properties compared to its local counterpart, which leads to a disparate discrete spectral distribution. We then deduce the general expressions of N -simple and N -double poles solitons of the nGI equation under the reflectionless potential. What’s more, novel dynamical behaviors of these solutions are not only exhibited graphically with 3D and projection profiles, wave propagation with the x -axis, but also analyzed detailedly. These solutions play a crucial role in revealing the abundant dynamics of solitons and advancing our comprehension of nonlocal nonlinear phenomena.