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We derive rigorously one- and two-dimensional mean-field equations for cigar-and pancake-shaped Bose-Einstein condensates (BEC) with higher orderinteractions (HOI). We show how the higher order interaction modifies thecontact interaction of the strongly confined particles. Surprisingly, we findthat the usual Gaussian profile assumption for the strongly confining directionis inappropriate for the cigar-shaped BEC case, and a Thomas-Fermi type profileshould be adopted instead. Based on the derived mean field equations, theThomas-Fermi densities are analyzed in presence of the contact interaction andHOI. For both box and harmonic traps in one, two and three dimensions, weidentify the analytical Thomas-Fermi densities, which depend on the competitionbetween the contact interaction and the HOI.

Mean-field regime and Thomas-Fermi approximations of trapped Bose-Einstein condensates with higher order interactions in one and two dimensions