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Starting from the 3D Gross-Pitaevskii equation and using a variationalapproach, we derive an effective 1D wave-equation that describes the axialdynamics of a Bose condensate confined in an external potential withcylindrical symmetry. The trapping potential is harmonic in the transversedirection and generic in the axial one. Our equation, that is a time-dependentnon-polynomial nonlinear Schr\"odinger equation (1D NPSE), can be used to modelcigar-shaped condensates, whose dynamics is essentially 1D. We show that 1DNPSE gives much more accurate results than all other effective equationsrecently proposed. By using 1D NPSE we find analytical solutions for bright anddark solitons, which generalize the ones known in the literature. We deducealso an effective 2D non-polynomial Schr\"odinger equation (2D NPSE) thatmodels disc-shaped Bose condensates confined in an external trap that isharmonic along the axial direction and generic in the transverse direction. Inthe limiting cases of weak and strong interaction, our approach gives rise toSchr\"odinger-like equations with different polynomial nonlinearities.Comment: 7 pages, 5 figures, to be published in Phys. Rev.

Effective wave equations for the dynamics of cigar-shaped and disk-shaped Bose condensates