  We study Darboux transformations for the two boson (TB) hierarchy both in the
scalar as well as in the matrix descriptions of the linear equation. While
Darboux transformations have been extensively studied for integrable models
based on $SL(2,R)$ within the AKNS framework, this model is based on
$SL(2,R)\otimes U(1)$. The connection between the scalar and the matrix
descriptions in this case implies that the generic Darboux matrix for the TB
hierarchy has a different structure from that in the models based on $SL(2,R)$
studied thus far. The conventional Darboux transformation is shown to be quite
restricted in this model. We construct a modified Darboux transformation which
has a much richer structure and which also allows for multi-soliton solutions
to be written in terms of Wronskians. Using the modified Darboux
transformations, we explicitly construct one soliton/kink solutions for the
model.
