  Soliton propagation dynamics under the presence of a complex potential are
investigated. A large variety of qualitatively different potentials, including
periodic, semi-infinite periodic and localized potentials, is considered. Cases
of both symmetric and non-symmetric potentials are studied in terms of their
effect on soliton dynamics. The rich set of dynamical features of soliton
propagation include dynamical trapping, periodic and non-periodic soliton mass
variation and non-reciprocal scattering dynamics. These features are
systematically investigated with the utilization of an effective particle phase
space approach which is shown in remarkable agreement with direct numerical
simulations. The generality of the results enables the consideration of
potential applications where the inhomogeneity of the gain and loss is
appropriately engineered in order to provide desirable soliton dynamics.
