Hierarchical structure and N-soliton solutions of the generalized Gerdjikov–Ivanov equation via Riemann–Hilbert problem

This paper addresses the generalized Gerdjikov–Ivanov (GI) equations through the application of the inverse scattering method. First, the hierarchy of equations is constructed from the space spectrum problem, from which the generalized GI equations can be derived. Then, the Riemann–Hilbert problem is established through spectral analysis, and the N-soliton solution of the GI equation is obtained by solving the Riemann–Hilbert problem. An asymptotic analysis of the proposed two-soliton solution reveals that soliton collisions are elastic. Furthermore, infinite conservation laws are derived using the Lax pair and Laurent expansion. The study explores the dynamical properties of single, double and triple-soliton solutions, supplemented by corresponding three-dimensional and density figures. These findings contribute to a deeper understanding of nonlinear phenomena in nature, including weakly nonlinear dispersive wave fields, quantum field theories, nonlinear optics, etc.