Accelerating and abruptly autofocusing matter waves

We predict that classes of coherent matter waves can self-accelerate without the presence of an external potential. Such Bose-Einstein condensates can follow arbitrary power-law trajectories and can also take the form of diffraction-free Airy waves. We also show that suitably engineered radially symmetric matter waves can abruptly autofocus in space and time. We suggest different schemes for the preparation of the condensate using laser beams to imprint an amplitude or a phase pattern onto the matter wave. Direct and Fourier space generation of such waves is discussed using continuous and binary masks as well as magnetic mirrors and lenses. We study the effect of interactions and find that independently of the type and strength of the nonlinearity, the dynamics are associated with the generation of accelerating matter waves. In the case of strong attractive interactions, the acceleration is increased while the radiation reorganizes itself in the form of soliton(s).