  We consider the Lagrangian description of the soliton sector of the so-called
affine $\hat{sl}(3)$ Toda model coupled to matter (Dirac) fields (ATM). The
theory is treated as a constrained system in the contexts of the
Faddeev-Jackiw, the symplectic, as well as the master Lagrangian approaches. We
exhibit the master Lagrangian nature of the model from which generalizations of
the sine-Gordon (GSG) or the massive Thirring (GMT) models are derivable. The
GMT model describes $N_{f}=3$ [number of positive roots of $su(3)$] massive
Dirac fermion species with current-current interactions amongst all the U(1)
species currents; on the other hand, the GSG theory corresponds to $N_{b}=2$
[rank of the $su(3)$ Lie algebra] independent Toda fields (bosons) with a
potential given by the sum of three SG cosine terms. The dual description of
the model is further emphasized by providing some on shell relationships
between bilinears of the GMT spinors and the relevant expressions of the GSG
fields. In this way, in the {\bf first part} of the chapter, we exhibit the
strong/weak coupling phases and the (generalized) soliton/particle
correspondences of the model at the classical level. In the {\bf second part}
of the chapter we give a full Lie algebraic formulation of the duality at the
level of the equations of motion written in matrix form. The effective
off-critical $\hat{sl}(3)$ ATM action is written in terms of the
Wess-Zumino-Novikov-Witten (WZNW) action plus some kinetic terms for the
spinors and scalar-spinor interaction terms. Moreover, this theory still
presents a remarkable equivalence between the Noether and topological currents,
describes the soliton sector of the original model and turns out to be the
master Lagrangian describing the GMT and GSG models.
