Boundary Lax pairs for the Toda field theories

Based on the recent formulation of a general scheme to construct boundary Lax pairs, we develop this systematic construction for the affine Toda field theories (ATFT). We work out explicitly the first two models of the hierarchy, i.e. the sine-Gordon () and the models. The 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.