  We show that when spatially localized gain landscape performs accelerated
motion in the transverse plane, i.e., when it rotates or oscillates around the
light propagation axis, the effective gain experienced by the light beam
considerably reduces with increase of the amplitude of oscillations or
frequency of rotation of the localized gain. In the presence of uniform
background losses and defocusing nonlinearity, such gain landscapes may support
dynamically oscillating gain-managed solitons, but if the amplitude of
oscillations or the frequency of rotation of the localized gain exceeds a
threshold, stable attractors disappear and any input beam decays.
