# Mean-field equations for cigar- and disk-shaped Bose and Fermi superfluids

## TL;DR

This paper derives effective 1D and 2D mean-field equations for cigar- and disk-shaped Bose and Fermi superfluids from 3D equations, accurately capturing their dynamics and stationary properties.

## Contribution

It introduces simplified mean-field equations for anisotropic superfluids, enabling efficient analysis of their behavior with high accuracy compared to full 3D models.

## Key findings

- Derived analytic nonlinear terms for reduced equations.
- Achieved excellent agreement with 3D solutions for stationary and dynamic properties.
- Validated equations through comparison with full 3D simulations.

## Abstract

Starting from the three-dimensional (3D) time-dependent nonlinear Gross-Pitaevskii equation for a Bose-Einstein condensate (BEC) and density functional (DF) equation for a Fermi superfluid at the unitarity and Bardeen-Cooper-Schrieffer (BCS) limits, we derive effective one- (1D) and two-dimensional (2D) mean-field equations, respectively, for the dynamics of a trapped cigar- and disk-shaped BEC and Fermi superfluid by using the adiabatic approximation. The reduced 1D and 2D equations for a cigar- and disk-shaped Fermi superfluid have simple analytic nonlinear terms and at unitarity produce results for stationary properties and non-stationary breathing oscillation and free expansion in excellent agreement with the solution of the full 3D equation.