LAUSR

The Gerdjikov–Ivanov (GI) equation is one type of derivative nonlinear Schrödinger equation used widely in quantum field theory, nonlinear optics, weakly nonlinear dispersion water waves and other fields. In this paper, the coupled GI equation on a time–space scale is deduced from Lax pairs and the zero curvature equation on a time–space scale, which can be reduced to the classical and the semi-discrete GI equation by considering different time–space scales. Furthermore, the Darboux transformation (DT) of the GI equation on a time–space scale is constructed via a gauge transformation. Finally, N-soliton solutions of the GI equation are given through applying its DT, which are expressed by the Cayley exponential function. At the same time, one-solition solutions are obtained on three different time–space scales ( X = R, X = C and X = Kp ).