Liouville integrable defects: the non-linear Schrödinger paradigm

Abstract

A systematic approach to Liouville integrable defects is proposed, based on an underlying Poisson algebraic structure. The non-linear Schrödinger 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.

The classical nonlinear Schrödinger model with a new integrable boundary

“…The main difference with respect to the type I defect is that the type II requires the presence of extra degrees of freedom at the defect location. In the case of the NLS model, see also [10] for different types of defects.…”

“…In fact, it is worth reminding that a complete proof of the defect integrability via the classical r-matrix was missing so far, despite the fact that strong evidence has been collected to support this statement. On this issue, see [4,8,9,[11][12][13] and also [14][15][16].…”

The sine-Gordon model with integrable defects revisited

“…In the presence of an integrable defect the monodromy matrix of the field theory is modified (see also [1,6,13]), and takes the generic form…”

“…We recently proposed [1] a fully algebraic picture for a description of a Liouville integrable defect. It was successfully exemplified in the case of the continuous non-linear Schrödinger model (NLS), inducing us to now extend this procedure to the situation of the sine Gordon model.…”

“…It was successfully exemplified in the case of the continuous non-linear Schrödinger model (NLS), inducing us to now extend this procedure to the situation of the sine Gordon model. It is worth noting that the investigation of integrable defects has been a quite challenging problem, and there is a wealth of relevant articles in recent years at both classical and quantum level [1]- [23,24].…”