  The T-duality symmetries of a family of two-dimensional massive integrable
field theories defined in terms of asymmetric gauged Wess-Zumino-Novikov-Witten
actions modified by a potential are investigated. These theories are examples
of massive non-linear sigma models and, in general, T-duality relates two
different dual sigma models perturbed by the same potential. When the
unperturbed theory is self-dual, the duality transformation relates two
perturbations of the same sigma model involving different potentials. Examples
of this type are provided by the Homogeneous sine-Gordon theories, associated
with cosets of the form G/U(1)^r where G is a compact simple Lie group of rank
r. They exhibit a duality transformation for each element of the Weyl group of
G that relates two different phases of the model. On-shell, T-duality provides
a map between the solutions to the equations of motion of the dual models that
changes Noether soliton charges into topological ones. This map is carefully
studied in the complex sine-Gordon model, where it motivates the construction
of Bogomol'nyi-like bounds for the energy that provide a novel characterisation
of the already known one-solitons solutions where their classical stability
becomes explicit.
