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 condensates. In the second part of the paper we show how one can construct effective oneand 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 effectively longitudinal (transversal) nonlinear waveforms through the rescaled effective nonlinearity in the case of cigar-shaped (pancake-shaped) condensates.