  We study matter-wave dark solitons in atomic Bose-Einstein condensates at
finite temperatures, under the effect of linear and periodic potentials. Our
model, namely a dissipative Gross-Pitaevskii equation, is treated analytically
by means of dark soliton perturbation theory, which results in a Newtonian
equation of motion for the dark soliton center. This reduced model, which
incorporates an effective washboard potential and an anti-damping term,
constitutes an example of an anti-damped Josephson junction. We present a
qualitative (local and global) analysis of the equation of motion. For
sufficiently small wavenumbers of the periodic potential and weak linear
potentials, the results are found to be in good agreement with pertinent ones
obtained via a Bogoliubov-de Gennes analysis and direct numerical simulations.
