Inverse scattering transform of the general coupled Hirota system with nonzero boundary conditions

TL;DR

This paper develops a comprehensive inverse scattering transform for the general coupled Hirota system with nonzero boundary conditions, enabling the analysis of complex soliton solutions including breathers and solitons relevant to nonlinear wave phenomena.

Contribution

It introduces a rigorous inverse scattering framework for the coupled Hirota system with nonzero boundaries, including a Riemann-Hilbert problem formulation and soliton solution analysis.

Findings

Derived explicit soliton solutions including Akhmediev breathers and Kuznetsov-Ma solitons.

Analyzed the properties and symmetries of the scattering data and eigenfunctions.

Graphically discussed the characteristics of solutions in the self-focusing case.

Abstract

The initial value problem for the general coupled Hirota system with nonzero boundary conditions at infinity is solved by reporting a rigorous theory of the inverse scattering transform. With the help of a suitable uniformization variable, both the inverse and the direct problems are analyzed which allows us to develop the inverse scattering transform on the complex $z$-plane. Firstly, analyticity of the scattering eigenfunctions and scattering data, properties of the discrete spectrum, symmetries, and asymptotics are discussed in detail. Moreover, the inverse problem is posed as a Riemann-Hilbert problem for the eigenfunctions, and the reconstruction formula of the potential in terms of eigenfunctions and scattering data is presented. Finally, the main characteristics of these obtained soliton solutions are graphically discussed in the $2\times2$ self-focusing case. This family of solutions contains novel Akhmediev breather and Kuznetsov-Ma soliton. These results would be of much importance in understanding and enriching breather wave phenomena arising in nonlinear and complex systems, especially in Bose-Einstein condensates.