  We consider dynamical stabilization of Bose-Einstein condensates (BEC) by
time-dependent modulation of the scattering length. The problem has been
studied before by several methods: Gaussian variational approximation, the
method of moments, method of modulated Townes soliton, and the direct averaging
of the Gross-Pitaevskii (GP) equation. We summarize these methods and find that
the numerically obtained stabilized solution has different configuration than
that assumed by the theoretical methods (in particular a phase of the
wavefunction is not quadratic with $r$). We show that there is presently no
clear evidence for stabilization in a strict sense, because in the numerical
experiments only metastable (slowly decaying) solutions have been obtained. In
other words, neither numerical nor mathematical evidence for a new kind of
soliton solutions have been revealed so far. The existence of the metastable
solutions is nevertheless an interesting and complicated phenomenon on its own.
We try some non-Gaussian variational trial functions to obtain better
predictions for the critical nonlinearity $g_{cr}$ for metastabilization but
other dynamical properties of the solutions remain difficult to predict.
