  We investigate bright and dark solitons with anomalous or normal dispersion
and under transverse harmonic confinement. In matter waves, positive atomic
mass implies anomalous dispersion (kinetic spreading) while negative mass gives
normal dispersion (kinetic shrinking). We find that, contrary to the strictly
one-dimensional case, the axial and transverse profiles of these solitons
crucially depend on the strength of the nonlinearity and on their dispersive
properties. In particular, we show that, like bright solitons with anomalous
dispersion, also dark solitons with normal dispersion disappear at a critical
axial density. Our predictions are useful for the study of atomic matter waves
in Bose-Einstein condensates and also for optical bullets in inhomogeneous Kerr
media.
