The classical nonlinear Schr\"odinger model with a new integrable boundary

TL;DR

This paper introduces a new integrable boundary condition for the classical nonlinear Schrödinger model by dressing a boundary with a defect, thoroughly analyzing its integrability and conserved charges.

Contribution

It derives and proves the integrability of a novel boundary condition in the classical nonlinear Schrödinger model using a boundary dressing method and modified Poisson brackets.

Findings

Derived the boundary ${\ m K}$ matrix ensuring integrability.

Proved integrability via classical $r$-matrix approach.

Confirmed the integrability of the defect used in the boundary.

Abstract

A new integrable boundary for the classical nonlinear Schr\"odinger model is derived by dressing a boundary with a defect. A complete investigation of the integrability of the new boundary is carried out in the sense that the boundary ${\cal K}$ matrix is derived and the integrability is proved via the classical $r$-matrix. The issue of conserved charges is also discussed. The key point in proving the integrability of the new boundary is the use of suitable modified Poisson brackets. Finally, concerning the kind of defect used in the present context, this investigation offers the opportunity to prove - beyond any doubts - their integrability.