Higher-order algebraic soliton solutions of the Gerdjikov–Ivanov equation: Asymptotic analysis and emergence of rogue waves

In this paper, we derive the determinant representation of higher-order algebraic soliton solutions of the Gerdjikov–Ivanov equation by using the Darboux transformation and some limit technique. Based on the asymptotic balance between different algebraic terms, we obtain the asymptotic expressions of algebraic soliton solutions with the order 2 ≤ N ≤ 4 . It turns out that all the asymptotic solitons have the same amplitudes, most of them are located in the parabolic curves and thus have the varying velocities with the rate O ( | t | − 1 2 ) (except that one pair of asymptotic solitons are located in the straight lines for the odd-order cases), and they exhibit the elastic interactions of the attractive type. In addition, we reveal that the transient rogue waves are generated in the soliton-interaction region and the peak value is exactly N times the amplitude of individual soliton.