On a Systematic Approach to Defects in Classical Integrable Field Theories

Abstract

We present an inverse scattering approach to defects in classical integrable field theories. Integrability is proved systematically by constructing the generating function of the infinite set of modified integrals of motion. The contribution of the defect to all orders is explicitely identified in terms of a defect matrix. The underlying geometric picture is that those defects correspond to Bäcklund transformations localized at a given point. A classification of defect matrices as well as the corresponding defect conditions is performed. The method is applied to a collection of well-known integrable models and previous results are recovered (and extended) directly as special cases. Finally, a brief discussion of the classical r-matrix approach in this context shows the relation to inhomogeneous lattice models and the need to resort to lattice regularizations of integrable field theories with defects.

The sine-Gordon model with integrable defects revisited

“…Similarities with corresponding equations emerging in [14,23,24] from the inverse scattering point of view are apparent as expected, given that one essentially solves the same fundamental equations (2.1) (W ij → Γ ij ). Of course Liouville integrability is guaranteed within the present approach by construction (at least formally), whereas in the methodology of [14,23,24] only the conservation of the charges for a singled-out time-evolution is shown through the zero curvature condition i.e.…”

“…It is instructive to point out that comparison of the extracted charges (3.24), (3.25), and the latter equations (3.41) with similar results obtained for instance in [13,14] reveal manifest discrepancies. We shall further comment on this matter in the discussion section.…”

“…Of course Liouville integrability is guaranteed within the present approach by construction (at least formally), whereas in the methodology of [14,23,24] only the conservation of the charges for a singled-out time-evolution is shown through the zero curvature condition i.e. explicit use of the equations of motion.…”

The classical nonlinear Schrödinger model with a new integrable boundary

“…Actually, making use of this fact a slightly more general defect contribution was found in [14]. In that case another real parameter β is added and the defect potential D is modified as follows…”

“…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].…”