Trapped Bose-Einstein Condensates: Role of Dimensionality

We analyse systematically, from the viewpoint of the nonlinear physics of solitary waves, the effect of the spatial dimension (D = 1,2,3) on the structure and stability of the Bose-Einstein condensates (BECs) trapped in an external anisotropic parabolic potential. While for the positive scattering length the stationary ground-state solutions of the Gross-Pitaevskii equation are shown to be always stable independently of the spatial dimension, for the negative scattering length the ground-state condensate is stable only in the 1D and 2D cases, whereas in the 3D case it becomes unstable. A direct link between nonlinear modes of BECs and (bright and dark) solitary waves of the nonlinear Schrodinger equation is demonstrated for all the dimensions.