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

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 have not only derived the pure soliton solutions from the defocusing–defocusing coupled Hirota equations but also provided the multiple double-pole solutions for the first time.