  Self-accelerating waves in conservative systems, which usually feature slowly
decaying tails, such as Airy waves, have drawn great interest in studies of
quantum and classical wave dynamics. They typically appear in linear media,
while nonlinearities tend to deform and eventually destroy them. We
demonstrate, by means of analytical and numerical methods, the existence of
robust one- and two-dimensional (1D and 2D) self-accelerating tailless solitons
and solitary vortices in a model of two-component Bose-Einstein condensates,
dressed by a microwave (MW) field, whose magnetic component mediates long-range
interaction between the matter-wave constituents, with the feedback of the
matter waves on the MW field taken into account. In particular,
self-accelerating 2D solitons may move along a curved trajectory in the
coordinate plane. The system may also include the spin-orbit coupling between
the components, leading to similar results for the self-acceleration. The
effect persists if the contact cubic nonlinearity is included. A similar
mechanism may generate 1D and 2D self-accelerating solitons in optical media
with thermal nonlinearity.
