  Basic models which give rise to one- and two-dimensional (1D and 2D)
solitons, such as the Gross-Pitaevskii (GP) equations for Bose-Einstein
condensates (BECs), feature the Galilean invariance, which makes it possible to
generate families of moving solitons from quiescent ones. A challenging problem
is to find models admitting stable self-accelerating (SA) motion of solitons.
SA modes are known in linear systems in the form of Airy waves, but they are
poorly localized states. This brief review presents two-component BEC models
which make it possible to predict SA solitons. In one system, a pair of
interacting 1D solitons with opposite signs of the effective mass is created in
a binary BEC trapped in an optical-lattice potential. In that case, opposite
interaction forces, acting on the solitons with positive and negative masses,
produce equal accelerations, while the total momentum is conserved. The second
model is based on a system of GP equations for two atomic components, which are
resonantly coupled by a microwave field. The latter model produces an exact
transformation to an accelerating references frame, thus predicting 1D and 2D
stable SA solitons, including vortex rings.
