The three-component defocusing nonlinear Schrodinger equation with nonzero boundary conditions

TL;DR

This paper develops a rigorous inverse scattering transform theory for a three-component defocusing nonlinear Schrödinger equation with non-zero boundary conditions, extending existing methods and introducing new symmetry analysis for soliton solutions.

Contribution

It combines advanced tensor, matrix decomposition, and Hodge star techniques to analyze the three-component NLS with non-zero boundaries, providing a comprehensive framework for soliton solutions.

Findings

Established a set of analytic eigenfunctions.

Characterized the discrete spectrum.

Derived exact dark-bright soliton solutions.

Abstract

We present a rigorous theory of the inverse scattering transform (IST) for the three-component defocusing nonlinear Schrodinger (NLS) equation with initial conditions approaching constant values with the same amplitude as $x\to\pm\infty$. The theory combines and extends to a problem with non-zero boundary conditions three fundamental ideas: (i) the tensor approach used by Beals, Deift and Tomei for $n$-th order scattering problems, (ii) the triangular decompositions of the scattering matrix used by Novikov, Manakov, Pitaevski and Zakharov for the $N$-wave interaction equations, and (iii) a generalization of the cross product via the Hodge star duality, which, to the best of our knowledge, is used in the context of the IST for the first time in this work. The combination of the first two ideas allows us to rigorously obtain a fundamental set of analytic eigenfunctions. The third idea allows us to establish the symmetries of the eigenfunctions and scattering data. The results are used to characterize the discrete spectrum and to obtain exact soliton solutions, which describe generalizations of the so-called dark-bright solitons of the two-component NLS equation.