Lax pair formulation in the simultaneous presence of boundaries and defects

TL;DR

This paper develops a systematic formulation of Lax pairs for integrable systems with both boundaries and defects, exemplified by the vector NLS model, revealing complex boundary-defect interactions.

Contribution

It introduces a unified approach to derive Lax pairs, sewing conditions, and equations of motion for systems with simultaneous boundaries and defects, extending previous isolated analyses.

Findings

Derived explicit Lax pairs for systems with boundaries and defects.

Identified two distinct boundary condition types based on algebraic structures.

Applied the framework to the vector NLS model demonstrating non-trivial behavior.

Abstract

Inspired by recent results on the effect of integrable boundary conditions on the bulk behavior of an integrable system, and in particular on the behavior of an existing defect we systematically formulate the Lax pairs in the simultaneous presence of integrable boundaries and defects. The respective sewing conditions as well as the relevant equations of motion on the defect point are accordingly extracted. We consider a specific prototype i.e. the vector non-linear Schr\"{o}dinger (NLS) model to exemplify our construction. This model displays a highly non-trivial behavior and allows the existence of two distinct types of boundary conditions based on the reflection algebra or the twisted Yangian.