  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.
