Basic models for generating solitons in one and two dimensions (1D and 2D), such as the Gross-Pitaevsky (GP) equation for the Bose-Einstein condensate (BEC), are Galilean invariants that allow the generation of families is characterized by of a moving soliton from a stationary soliton. A difficult problem is to find a model that admits stable self-accelerating (SA) motion of solitons. The SA mode is known in linear systems in the form of Airy waves, but is a poorly localized state. In this brief review, we present a two-component BEC model that enables the prediction of SA solitons. In one system, pairs of interacting 1D solitons with opposite signs of effective mass are created in a binary BEC trapped in an optical lattice potential. In that case, opposing interaction forces acting on solitons with positive and negative masses produce equal accelerations, but total momentum is conserved. The second model is based on his GP system of equations of two atomic components resonantly coupled by a microwave field. The latter model predicts 1D and 2D stable SA solitons with vortex rings to generate accurate transformations to the acceleration reference frame.
