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

  • Salasnich L
  • Parola A
  • Reatto L
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Starting from the three-dimensional (3D) Gross-Pitaevskii equation
and using a variational approach, we derive an effective 1D wave
equation that describes the axial dynamics of a Bose condensate confined
in an external potential with cylindrical symmetry. The trapping
potential is harmonic in the transverse direction and generic in
the axial one. Our equation, that is a time-dependent nonpolynomial
nonlinear Schrödinger equation (1D NPSE), can be used to model cigar-shaped
condensates, whose dynamics is essentially 1D. We show that 1D NPSE
gives much more accurate results than all other effective equations
recently proposed. By using 1D NPSE we find analytical solutions
for bright and dark solitons, which generalize the ones known in
the literature. We deduce also an effective 2D nonpolynomial Schrödinger
equation (2D NPSE) that models disk-shaped Bose condensates confined
in an external trap that is harmonic along the axial direction and
generic in the transverse direction. In the limiting cases of weak
and strong interaction, our approach gives rise to Schrödinger-like
equations with different polynomial nonlinearities.

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