The generalized non-linear Schrodinger model on the interval

TL;DR

This paper explores integrable boundary conditions in the generalized non-linear Schrödinger model, deriving conserved quantities, equations of motion, and Bethe Ansatz solutions for both soliton preserving and non-preserving cases.

Contribution

It introduces and analyzes boundary conditions in the classical and quantum generalized NLS model, providing explicit conserved quantities and Bethe Ansatz equations.

Findings

Conserved quantities are computed for both boundary conditions.

Explicit equations of motion are derived.

Bethe Ansatz equations are established for SNP boundary conditions.

Abstract

The generalized (1+1)-D non-linear Schrodinger (NLS) theory with particular integrable boundary conditions is considered. More precisely, two distinct types of boundary conditions, known as soliton preserving (SP) and soliton non-preserving (SNP), are implemented into the classical $gl_N$ NLS model. Based on this choice of boundaries the relevant conserved quantities are computed and the corresponding equations of motion are derived. A suitable quantum lattice version of the boundary generalized NLS model is also investigated. The first non-trivial local integral of motion is explicitly computed, and the spectrum and Bethe Ansatz equations are derived for the soliton non-preserving boundary conditions.