EFFECTIVE LOW-DIMENSIONAL POLYNOMIAL EQUATIONS FOR BOSE-EINSTEIN CONDENSATES

In the first part of this article we briefly overview the fundamentals of Bose-Einstein condensation and survey a series of results concerning the effective equations which describe the dynamics of elongated and oblate Bose-Einstein conden- sates. In the second part of the paper we show how one can construct effective one- and two-dimensional polynomial Schr¨ odinger equations which describe the longitudinal (transversal) dynamics of high-density cigar-shaped (pancake-shaped) Bose-Einstein condensates. These equations do not account for the interplay between the radial and the transversal modes of the condensate, but can accurately describe the dynamics of ef- fectively longitudinal (transversal) nonlinear waveforms through the rescaled effective nonlinearity in the case of cigar-shaped (pancake-shaped) condensates.