  One-dimensional nonlinear Schr\"odinger equations are derived to describe the
axial effective dynamics of cigar-shaped atomic repulsive Bose-Einstein
condensates trapped with anharmonic transverse potentials. The accuracy of
these equations in the perturbative, Thomas-Fermi, and crossover regimes were
verified numerically by comparing the ground-state profiles, transverse
chemical potentials and oscillation patterns with those results obtained for
the full three-dimensional Gross-Pitaevskii equation. This procedure allows us
to derive different patterns of 1D nonlinear models by the control of the
transverse confinement.
