  $S$-matrices associated to the vector representations of the quantum groups
for the classical Lie algebras are constructed. For the $a_{m-1}$ and $c_m$
algebras the complete $S$-matrix is found by an application of the bootstrap
equations. It is shown that the simplest form for the $S$-matrix which
generalizes that of the Gross-Neveu model is not consistent for the
non-simply-laced algebras due to the existence of unexplained singularities on
the physical strip. However, a form which generalizes the $S$-matrix of the
principal chiral model is shown to be consistent via an argument which uses a
novel application of the Coleman-Thun mechanism. The analysis also gives a
correct description of the analytic structure of the $S$-matrix of the
principle chiral model for $c_m$.
