  We derive rigorously one- and two-dimensional mean-field equations for cigar-
and pancake-shaped Bose-Einstein condensates (BEC) with higher order
interactions (HOI). We show how the higher order interaction modifies the
contact interaction of the strongly confined particles. Surprisingly, we find
that the usual Gaussian profile assumption for the strongly confining direction
is inappropriate for the cigar-shaped BEC case, and a Thomas-Fermi type profile
should be adopted instead. Based on the derived mean field equations, the
Thomas-Fermi densities are analyzed in presence of the contact interaction and
HOI. For both box and harmonic traps in one, two and three dimensions, we
identify the analytical Thomas-Fermi densities, which depend on the competition
between the contact interaction and the HOI.
