Some ideas and remarks are presented concerning a possible Lagrangian approach to the study of internal boundary conditions relating integrable fields at the junction of two domains. The main example given in the article concerns single real scalar fields in each domain and it is found that these may be free, of Liouville type, or of sinh-Gordon type.

A Lagrangian approach is proposed and developed to study defects within affine Toda field theories. In particular, a suitable Lax pair is constructed together with examples of conserved charges. It is found that only those models based on a (1) r data appear to allow defects preserving integrability. Surprisingly, despite the explicit breaking of Lorentz and translation invariance, modified forms of both energy and momentum are conserved. Some, but apparently not all, of the higher spin conserved charges are also preserved after the addition of contributions from the defect. This fact is illustrated by noting how defects may preserve a modified form of just one of the spin 2 or spin -2 charges but not both of them.

The classical sine-Gordon model permits integrable discontinuities, or jump-defects, where the conditions relating the fields on either side of a defect are Bäcklund transformations frozen at the defect location. The purpose of this article is to explore the extent to which this idea may be extended to the quantum sine-Gordon model and how the striking features of the classical model may translate to the quantum version. Assuming a positive defect parameter there are two types of defect. One type, carrying even charge, is stable, but the other type, carrying odd charge, is unstable and may be considered as a resonant bound state of a soliton and a stable defect. The scattering of solitons with defects is considered in detail, as is the scattering of breathers, and in all cases the jump-defect is purely transmitting. One surprising discovery concerns the lightest breather. Its transmission factor is independent of the bulk coupling -a property susceptible to a perturbative check, but not shared with any of the other breathers. It is argued that classical jump-defects can move and some comments are made concerning their quantum scattering matrix. a

An alternative Lagrangian definition of an integrable defect is provided and analyzed. The new approach is sufficiently broad to allow a description of defects within the Tzitzéica model, which was not possible in previous approaches, and may be generalizable. New, two-parameter, sine-Gordon defects are also described, which have characteristics resembling a pair of 'fused' defects of a previously considered type. The relationship between these defects and Bäcklund transformations is described and a Hamiltonian description of integrable defects is proposed.

Recent work on purely transmitting 'jump-defects' in the sine-Gordon model and other relativistic field theories is extended to non-relativistic models. In all the cases investigated the defect conditions are provided by 'frozen' Bäcklund transformations and it is also shown via a Lax pair argument how integrability will be preserved in the presence of this type of defect. Explicit examples of the scattering of solitons by defects are given, and bound states associated with 'jump-defects' in the nonlinear Schrödinger model are described. Although the nonlinear Schrödinger model provides the principal example, some results are also presented for the Korteweg de Vries and modified Korteweg de Vries equations.