In this paper we study various aspects of classical solutions to the affine Toda equations on a half-line with integrable boundary conditions. We begin by finding conditions that the theory has a stable vacuum by finding a Bogomolny bound on the energy, and analysing the possible singularities of the field at the boundary. Using these constraints and extensive numerical investigations we classify the vacuum configurations and reflection factors for Ar(1) Toda theories up to r = 5.
Bowcock, P., & Perkins, M. (2003). Aspects of classical backgrounds and scattering for affine Toda theory on a half-line. Journal of High Energy Physics, 2003(02), Article 016. https://doi.org/10.1088/1126-6708/2003/02/016