  Fragmentation of filaments into dense cores is thought to be an important
step in forming stars. The bar-mode instability of spherically collapsing cores
found in previous linear analysis invokes a possibility of re-fragmentation of
the cores due to their ellipsoidal (prolate or oblate) deformation. To
investigate this possibility, here we perform three-dimensional
self-gravitational hydrodynamics simulations that follow all the way from
filament fragmentation to subsequent core collapse. We assume the gas is
polytropic with index \gamma, which determines the stability of the bar-mode.
For the case that the fragmentation of isolated hydrostatic filaments is
triggered by the most unstable fragmentation mode, we find the bar mode grows
as collapse proceeds if \gamma < 1.1, in agreement with the linear analysis.
However, it takes more than ten orders-of-magnitude increase in the central
density for the distortion to become non-linear. In addition to this fiducial
case, we also study non-fiducial ones such as the fragmentation is triggered by
a fragmentation mode with a longer wavelength and it occurs during radial
collapse of filaments and find the distortion rapidly grows. In most of
astrophysical applications, the effective polytropic index of collapsing gas
exceeds 1.1 before ten orders-of-magnitude increase in the central density.
Thus, supposing the fiducial case of filament fragmentation, re-fragmentation
of dense cores would not be likely and their final mass would be determined
when the filaments fragment.
