Inverse scattering transform for the focusing two-component Hirota equation with nonzero boundary conditions

TL;DR

This paper develops an inverse scattering transform method for the focusing two-component Hirota equation with nonzero boundary conditions, enabling the explicit construction of soliton solutions and analysis of their interactions.

Contribution

It systematically characterizes the spectral properties, formulates the Riemann-Hilbert problem, and derives explicit soliton solutions for the two-component Hirota equation with nonzero boundary conditions.

Findings

Characterization of three types of discrete eigenvalues.

Derivation of trace and reconstruction formulas.

Construction and interaction analysis of various solitons.

Abstract

Applying the inverse scattering transform to study a focusing two-component Hirota equation with nonzero boundary conditions at infinity. Through the spectral problem and the adjoint spectral problem, the analyticity properties and symmetry relations of the eigenfunctions and scattering coefficients are systematically investigated. Based on these properties, the discrete spectrum is comprehensively characterized, producing three types of discrete eigenvalues. The asymptotic behavior of the eigenfunctions and scattering coefficients is explored in detail. Furthermore, the inverse scattering problem is formulated through a suitable matrix Riemann-Hilbert problem, and the trace formula along with the reconstruction formula for the potential function are derived. The reflectionless condition facilitates the successful construction of N-soliton solution. Finally, various types of solitons are obtained by considering different types of discrete eigenvalues and their combinations, and the interactions of these solitons are demonstrated.