Darboux transformation, positon solution, and breather solution of the third-order flow Gerdjikov–Ivanov equation

Abstract The third-order flow Gerdjikov–Ivanov (TOFGI) equation is studied, and the Darboux transformation (DT) is used to obtain the determinant expression of the solution of this equation. On this basis, the soliton solution, rational solution, positon solution, and breather solution of the TOFGI equation are obtained by taking zero seed solution and non-zero seed solution. The exact solutions and dynamic properties of the Gerdjikov–Ivanov (GI) equation and the TOFGI equation are compared in detail under the same conditions, and it is found that there are some differences in the velocities and trajectories of the solutions of the two equations.