  In the case of tight transverse confinement (cigar-shaped trap) the
three-dimensional (3D) nonlinear Schr\"odinger equation, describing superfluid
Fermi atoms at unitarity (infinite scattering length $|a|\to \infty$), is
reduced to an effective one-dimensional form by averaging over the transverse
coordinates. The resultant effective equation is a 1D nonpolynomial Schrodinger
equation, which produces results in good agreement with the original 3D one. In
the limit of small and large fermion number $N$ the nonlinearity is of simple
power-law type. A similar reduction of the 3D theory to a two-dimensional form
is also performed for a tight axial confinement (disk-shaped trap). The
resultant effective 2D nonpolynomial equation also produces results in
agreement with the original 3D equation and has simple power-law nonlinearity
for small and large $N$. For both cigar- and disk-shaped superfluids our
nonpolynomial Schr\"odinger equations are quite attractive for phenomenological
application.
