  The Bose-stimulated self-organization of a quasi-two dimensional
non-equilibrium Bose-Einstein condensate in an in-plane potential is proposed.
We obtained the solution of the nonlinear, driven-dissipative Gross-Pitaevskii
equation for a Bose-Einstein condensate trapped in an external asymmetric
parabolic potential within the method of the spectral expansion. We found that,
in sharp contrast to previous observations, the condensate can spontaneously
acquire a soliton-like shape for spatially homogenous pumping. This condensate
soliton performs oscillatory motion in a parabolic trap and, also, can
spontaneously rotate. Stability of the condensate soliton in the spatially
asymmetric trap is analyzed. In addition to the nonlinear dynamics of
non-equilibrium Bose-Einstein condensates of ultra-cold atoms, our findings can
be applied to the condensates of quantum well excitons and cavity polaritons in
semiconductor heterostructure, and to the condensates of photons.
