Abstract. 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 Schrödinger 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