The Bullough–Dodd model coupled to matter fields

The Bullough–Dodd model is an important two-dimensional integrable field theory which finds applications in physics and geometry. We consider a conformally invariant extension of it, and study its integrability properties using a zero curvature condition based on the twisted Kac–Moody algebra A2(2). The one- and two-soliton solutions as well as the breathers are constructed explicitly. We also consider integrable extensions of the Bullough–Dodd model by the introduction of spinor (matter) fields. The resulting theories are conformally invariant and present local internal symmetries. All the one-soliton solutions, for two examples of those models, are constructed using a hybrid of the dressing and Hirota methods. One model is of particular interest because it presents a confinement mechanism for a given conserved charge inside the solitons.