Reduction groups and related integrable difference systems of NLS type

TL;DR

This paper extends the reduction group method to construct integrable difference systems related to nonlinear Schrödinger equations, covering all finite reduction groups and their associated Lax operators, transformations, and hierarchies.

Contribution

It introduces a comprehensive extension of the reduction group method to NLS-type systems, including all finite groups and their integrable difference equations.

Findings

Constructed Lax operators for all finite reduction groups.

Developed hierarchies of integrable differential-difference equations.

Derived scalar partial difference equations related to NLS type.

Abstract

We extend the reduction group method to the Lax-Darboux schemes associated with nonlinear Schr\"odinger type equations. We consider all possible finite reduction groups and construct corresponding Lax operators, Darboux transformations, hierarchies of integrable differential-difference equations, integrable partial difference systems and associated scalar partial difference equations.