Integration of the nonlinear Schr\"{o}dinger equation with a self-consistent source and nonzero boundary conditions

TL;DR

This paper investigates the integrability of the defocusing nonlinear Schrödinger equation with a self-consistent source and nonzero boundary conditions using inverse scattering methods, revealing how the scattering data evolve.

Contribution

It extends the inverse scattering method to analyze the nonlinear Schrödinger equation with a self-consistent source and nonzero boundary conditions, demonstrating its complete integrability.

Findings

Derived the evolution equations for scattering data

Established the complete integrability of the system

Analyzed the spectral problem for the Zakharov-Shabat system

Abstract

This paper is devoted to the study of the defocusing nonlinear Schr\"{o}dinger equation with a self-consistent source and nonzero boundary conditions by the method of the inverse scattering problem. In cases where the source consists of a combination of eigenfunctions of the corresponding spectral problem for the Zakharov-Shabat system, the complete integrability of the nonlinear Schr\"{o}dinger equation is investigated. Namely, the evolutions of the scattering data of the self-adjoint Zakharov-Shabat system, whose potential is a solution of the defocusing nonlinear Schr\"{o}dinger equation with a self-consistent source, are obtained.