  We consider a time-dependent non-linear Schr\"odinger equation in one
dimension (1D) with a fifth-order interaction term and external harmonic
confinement, as a model for both (i) a Bose gas with hard-core contact
interactions in local-density approximation, and (ii) a spin-polarized Fermi
gas in the collisional regime. We evaluate analytically in the Thomas-Fermi
limit the density fluctuation profiles and the collective excitation
frequencies, and compare the results for the low-lying modes with those
obtained from the numerical solution of the Schr\"odinger equation. We find
that the excitation frequencies are multiples of the harmonic-trap frequency
even in the strong-coupling Thomas-Fermi regime. This result shows that the
hydrodynamic and the collisionless collective spectra coincide in the
harmonically confined 1D Fermi gas, as they do for sound waves in its
homogeneous analogue. It also shows that in this case the local-density theory
reproduces the exact collective spectrum of the hard-core Bose gas under
harmonic confinement.
