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Motivated by numerous experiments on Bose-Einstein condensed atoms which havebeen performed in tight trapping potentials of various geometries (elongatedand/or toroidal/annular), we develop a general method which allows us to reducethe corresponding three-dimensional Gross-Pitaevskii equation for the orderparameter into an effectively one-dimensional equation, taking into account theinteractions (i.e., treating the width of the transverse profile variationally)and the curvature of the trapping potential. As an application of our model weconsider atoms which rotate in a toroidal trapping potential. We evaluate thestate of lowest energy for a fixed value of the angular momentum within variousapproximations of the effectively one-dimensional model and compare our resultswith the full solution of the three-dimensional problem, thus getting evidencefor the accuracy of our model.

Dimensional reduction in Bose-Einstein condensed clouds of atoms confined in tight potentials of any geometry and any interaction strength