Integration of the negative order Nonlinear Schr¨odinger Equation with self-consistent source

This paper focuses on the integrability properties of the negative-order nonlinear Schro¨dinger equation with a source. The source consists of the combination of the eigenfunctions of the corresponding spectral problem for the Dirac system which has not spectral singularities. The connection between the negative-order nonlinear Schro¨dinger equation with a self-consistent source and the Dirac system of equations is crucial, as it allows the complex dynamics of the original nonlinear model to be interpreted through the spectral theory of the Dirac operator. Building on this relationship, the evolution equations for the scattering data of the Dirac operator are derived, which is the central part in the inverse scattering transform (IST) framework. Due to the IST procedure, the rapidly decaying potential of the Dirac operator can be reconstructed from the derived differential equations for the scattering data, and this potential corresponds precisely to the solution of the problem under consideration. To illustrate the practical value of the theoretical results, the paper presents a detailed example demonstrating each stage of the method, from the formulation of the scattering data to the final reconstruction of the potential. This example clarifies the overall procedure and highlights the effectiveness of the approach in concrete applications.