Boundary Lax pairs for the Toda field theories

Abstract

Based on the recent formulation of a general scheme to construct boundary Lax pairs, we develop this systematic construction for the A (1) n affine Toda field theories (ATFT). We work out explicitly the first two models of the hierarchy, i.e. the sineGordon (A Toda theory is the first non-trivial example of the hierarchy that exhibits two distinct types of boundary conditions. We provide here novel expressions of boundary Lax pairs associated to both types of boundary conditions.

Integrable Defects and Bäcklund Transformations in Yang‐Baxter Models

“…In addition, one has to impose sewing conditions across the defect to avoid singular contributions from the zero curvature condition of the Lax connection. [ 26,27 ] In particular, at the location of the defect the zero curvature condition translates into the condition $$\begin{align} \frac{\mathrm{d}}{\mathrm{d} t}\mathcal {D}(x_0)= \tilde{\mathcal {L}}_t^{(2)}(x_0)\mathcal {D}(x_0)-\mathcal {D}(x_0) \tilde{ \mathcal {L}}_t^{(1)}(x_0)\,, \end{align}$$where the matrices $$\tilde{ \mathcal {L}}_t^{(i)}(x_0)$$ are the time components of the Lax pair evaluated at the defect location and are derived by demanding analyticity at the defect, i.e. $$ { \mathcal {L}}_t^{(i)}(x_0^\pm )\rightarrow \tilde{ \mathcal {L}}_t^{(i)}(x_0)$$.…”

Liouville integrable defects: the non-linear Schrödinger paradigm

“…They are evaluated separately in the right bulk (x 0 , A) and the left bulk (−A, x 0 ) and on the defect point -from left and right. As in the boundary integrable systems [21] it is required that V (±) (x ± 0 ) →Ṽ (±) (x 0 ) in order to avoid singular contributions from the zero curvature condition for the Lax pair U, V:…”

“…Construction of the time-like Lax operators also becomes then a very non trivial operation (see e.g. [21]). …”

The sine-Gordon model with integrable defects revisited

“…[29]). The generic expressions for the bulk left and right theories as well as the defect points are given as [1,32]:…”