  We numerically investigate the long time dynamics of spatially periodic
breather solutions of the 1-D nonlinear Schr\"odinger equation under parametric
forcing of the form $f(x)=f_0 \exp(iKx)$ along with dissipation. In the absence
of dissipation, robust soliton-like excitations are observed that travel with
constant amplitude and velocity. With dissipation, these solitons lose energy
(and amplitude) yet gain speed - a characteristic not observed in an ordinary
soliton. Moreover, these novel solitons are found to be stable against random
perturbations.
