Asymptotic analysis of high order soliton for the Hirota equation

In this paper, we mainly analyze the long-time asymptotics of high-order soliton for the Hirota equation. Two different Riemann-Hilbert representations of Darboux matrix with high-order soliton are given to establish the relationships between inverse scattering method and Darboux transformation. The asymptotic analysis with single spectral parameter is derived through the formulas of determinant directly. Furthermore, the long-time asymptotics with $k$ spectral parameters is given by combining the iterated Darboux matrix and the result of high-order soliton with single spectral parameter, which discloses the structure of high-order soliton clearly and is possible to be utilized in the optic experiments.