Darboux-Backlund transformations, dressing & impurities in multi-component NLS

TL;DR

This paper develops Darboux-Backlund transformations for the vector non-linear Schrödinger model with space-like discontinuities, providing explicit matrix and integral representations for the transformations related to discrete and continuous spectra.

Contribution

It introduces explicit Darboux-Backlund transformations for vector NLS models with impurities, covering both discrete and continuous spectra.

Findings

Derived explicit matrix and integral forms of Darboux transformations.

Analyzed the effect of space-like discontinuities on the NLS evolution.

Provided a framework for incorporating impurities into integrable models.

Abstract

We consider the discrete and continuous vector non-linear Schrodinger (NLS) model. We focus on the case where space-like local discontinuities are present, and we are primarily interested in the time evolution on the defect point. This in turn yields the time part of a typical Darboux-Backlund transformation. Within this spirit we then explicitly work out the generic Backlund transformation and the dressing associated to both discrete and continuous spectrum, i.e. the Darboux transformation is expressed in the matrix and integral representation respectively.