High-order soliton matrix for the third-order flow equation of the Gerdjikov-Ivanov hierarchy through the Riemann-Hilbert method

The Gerdjikov-Ivanov (GI) hierarchy is derived via recursion operator, in this paper, we mainly consider the third-order flow GI equation. In the framework of the Riemann-Hilbert method, through a standard dressing procedure, soliton matrices for simple zeros and elementary high-order zeros in the Riemann-Hilbert problem (RHP) for the third-order flow GI equation are constructed. Taking advantage of this result, some properties and asymptotic analysis of single soliton solutions and two-soliton solutions are discussed, and the simple elastic interaction of two-soliton is proved. Compared with soliton solution of the classical second-order flow, we found that the higher-order dispersion term affects the propagation velocity, propagation direction and amplitude of the soliton. Finally, by means of certain limit technique, the high-order soliton solution matrix for the third-order flow GI equation is derived.