Asymptotic analysis of high order soliton for the Hirota equation

TL;DR

This paper investigates the long-time behavior of high-order solitons in the Hirota equation, establishing connections between inverse scattering and Darboux transformations, and deriving asymptotics for multiple spectral parameters.

Contribution

It introduces two Riemann-Hilbert representations for high-order solitons and derives their long-time asymptotics with multiple spectral parameters, revealing the soliton structure.

Findings

Asymptotic formulas for high-order solitons with single spectral parameter.

Extension of asymptotic analysis to multiple spectral parameters.

Clarification of high-order soliton structure for potential optical applications.

Abstract

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.