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