Darboux transformation and exact solutions of the variable-coefficient nonlocal Gerdjikov–Ivanov equation

We for the first time study the integrable nonlocal nonlinear Gerdjikov–Ivanov (GI) equation with variable coefficients. The variable-coefficient nonlocal GI equation is constructed using a Lax pair. On this basis, the Darboux transformation is studied. Exact solutions of the variable-coefficient nonlocal GI equation are then obtained by constructing the $2n$-fold Darboux transformation of the equation. The results show that the solution of the GI equation with variable coefficients is more general than that of its constant-coefficient form. By taking special values for the coefficient function, we can obtain specific exact solutions, such as a kink solution, a periodic solution, a breather solution, a two-soliton interaction solution, etc. The exact solutions are represented visually with the help of images.