Inverse scattering transform for the defocusing-defocusing coupled Hirota equations with non-zero boundary conditions: double-pole solutions

TL;DR

This paper develops an inverse scattering transform method for the defocusing-defocusing coupled Hirota equations with non-zero boundary conditions, deriving explicit double-pole soliton solutions and analyzing their spectral properties.

Contribution

It introduces a comprehensive inverse scattering framework for these equations, including the derivation of double-pole solutions and the analysis of spectral symmetries.

Findings

Explicit double-pole soliton solutions are obtained.

The spectral properties and symmetries of the scattering data are characterized.

A Riemann-Hilbert problem formulation is established for the inverse scattering transform.

Abstract

The inverse scattering transform for the defocusing-defocusing coupled Hirota equations with non-zero boundary conditions at infinity is thoroughly discussed. We delve into the analytical properties of the Jost eigenfunctions and scrutinize the characteristics of the scattering coefficients. To enhance our investigation of the fundamental eigenfunctions, we have derived additional auxiliary eigenfunctions with the help of the adjoint problem. Two symmetry conditions are studied to constrain the behavior of the eigenfunctions and scattering coefficients. Utilizing these symmetries, we precisely delineate the discrete spectrum and establish the associated symmetries of the scattering data. By framing the inverse problem within the context of the Riemann-Hilbert problem, we develop suitable jump conditions to express the eigenfunctions. Consequently, we deduce the pure soliton solutions from the defocusing-defocusing coupled Hirota equations, and the double-poles solutions are provided explicitly for the first time in this work.