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By applying the standard adiabatic approximation and using the accurateanalytical expression for the corresponding local chemical potential obtainedin our previous work [Phys. Rev. A \textbf{75}, 063610 (2007)] we derive aneffective 1D equation that governs the axial dynamics of mean-fieldcigar-shaped condensates with repulsive interatomic interactions, accountingaccurately for the contribution from the transverse degrees of freedom. Thisequation, which is more simple than previous proposals, is also more accurate.Moreover, it allows treating condensates containing an axisymmetric vortex withno additional cost. Our effective equation also has the correct limit in boththe quasi-1D mean-field regime and the Thomas-Fermi regime and permits one toderive fully analytical expressions for ground-state properties such as thechemical potential, axial length, axial density profile, and local soundvelocity. These analytical expressions remain valid and accurate in between theabove two extreme regimes. Following the same procedure we also derive aneffective 2D equation that governs the transverse dynamics of mean-fielddisk-shaped condensates. This equation, which also has the correct limit inboth the quasi-2D and the Thomas-Fermi regime, is again more simple andaccurate than previous proposals. We have checked the validity of our equationsby numerically solving the full 3D Gross-Pitaevskii equation.

Effective mean-field equations for cigar-shaped and disk-shaped Bose-Einstein condensates