Liouville integrable defects: the non-linear Schrodinger paradigm

TL;DR

This paper develops a systematic algebraic framework for Liouville integrable defects in the non-linear Schrödinger model, constructing local integrals of motion and sewing conditions that preserve integrability.

Contribution

It introduces a Poisson algebraic approach to integrable defects and derives compatible sewing conditions for the non-linear Schrödinger model.

Findings

Constructed local integrals of motion for the defect model

Derived sewing conditions compatible with integrability hierarchy

Confirmed results match continuum limit of previous discrete models

Abstract

A systematic approach to Liouville integrable defects is proposed, based on an underlying Poisson algebraic structure. The non-linear Schrodinger model in the presence of a single particle-like defect is investigated through this algebraic approach. Local integrals of motions are constructed as well as the time components of the corresponding Lax pairs. Continuity conditions imposed upon the time components of the Lax pair to all orders give rise to sewing conditions, which turn out to be compatible with the hierarchy of charges in involution. Coincidence of our results with the continuum limit of the discrete expressions obtained in earlier works further confirms our approach.