Direct and inverse scattering problems for the first-order discrete system associated with the derivative NLS system

TL;DR

This paper analyzes the direct and inverse scattering problems for a discrete system linked to the semi-discrete derivative NLS system, introducing new methods and explicit solutions for the inverse problem.

Contribution

It introduces a novel approach to the inverse scattering problem using a discrete Marchenko system and describes the bound-state data with matrix triplets.

Findings

Bound-state data described by matrix triplets.

A discrete Marchenko system method is developed.

Explicit solutions in the reflectionless case are provided.

Abstract

The direct and inverse scattering problems are analyzed for a first-order discrete system associated with the semi-discrete version of the derivative NLS system. The Jost solutions, the scattering coefficients, the bound-state dependency and norming constants are investigated and related to the corresponding quantities for two particular discrete linear systems associated with the semi-discrete version of the NLS system. The bound-state data set with any multiplicities is described in an elegant manner in terms of a pair of constant matrix triplets. Several methods are presented to the solve the inverse problem. One of these methods involves a discrete Marchenko system using as input the scattering data set consisting of the scattering coefficients and the bound-state information, and this method is presented in a way generalizable to other first-order systems both in the discrete and continuous cases for which a Marchenko system is not yet available. Finally, using the time-evolved scattering data set, the inverse scattering transform is applied on the corresponding semi-discrete derivative NLS system, and in the reflectionless case certain explicit solution formulas are presented in closed form expressed in terms of the two matrix triplets.