  We investigate the dynamics of travelling oscillating solitons of the cubic
NLS equation under an external spatiotemporal forcing of the form $f(x,t) = a
\exp[iK(t)x]$. For the case of time-independent forcing a stability criterion
for these solitons, which is based on a collective coordinate theory, was
recently conjectured. We show that the proposed criterion has a limited
applicability and present a refined criterion which is generally applicable, as
confirmed by direct simulations. This includes more general situations where
$K(t)$ is harmonic or biharmonic, with or without a damping term in the NLS
equation. The refined criterion states that the soliton will be unstable if the
"stability curve" $p(\v)$, where $p(t)$ and $\v(t)$ are the normalized momentum
and the velocity of the soliton, has a section with a negative slope. Moreover,
for the case of constant $K$ and zero damping we use the collective coordinate
solutions to compute a "phase portrait" of the soliton where its dynamics is
represented by two-dimensional projections of its trajectories in the
four-dimensional space of collective coordinates. We conjecture, and confirm by
simulations, that the soliton is unstable if a section of the resulting closed
curve on the portrait has a negative sense of rotation.
