In this research, the Fokas method is adopted to examine the coupled Gerdjikov–Ivanov equation within the half line interval \((-\infty ,0]\) . Meanwhile, the Riemann–Hilbert technique is engaged to work out the potential function associated with the equation. We initially partition the matrix into segments and identify the jump matrix linking each segment based on the positive feature of the segment. The jump matrix comes from the spectral matrix, the latter of which is decided by the initial value and the boundary value. The research indicates that these spectral functions display correlativity instead of being independently separated, and they abide by a global connection while being associated via a compatibility condition. Then, we explore the coupled Gerdjikov–Ivanov equation under the zero boundary condition at infinity. The initial value problem related to the equation is capable of being converted into a Riemann–Hilbert problem on the strength of the analytic and symmetric properties of the eigenfunctions. Ultimately, through the settlement of both the regular and non-regular Riemann–Hilbert problems, a general pattern of N-soliton solutions with respect to the coupled Gerdjikov–Ivanov equation is put forward.
