  We investigate the dynamics of Bose-condensed bright solitons made of
alkali-metal atoms with negative scattering length and under harmonic
confinement in the transverse direction. Contrary to the 1D case, the 3D bright
soliton exists only below a critical attractive interaction which depends on
the extent of confinement. Such a behavior is also found in multi-soliton
condensates with box boundary conditions. We obtain numerical and analytical
estimates of the critical strength beyond which the solitons do not exist. By
using an effective 1D nonpolynomial nonlinear Schr\"odinger equation (NPSE),
which accurately takes into account the transverse dynamics of cigar-like
condensates, we numerically simulate the dynamics of the "soliton train"
reported in a recent experiment (Nature {\bf 417} 150 (2002)). Then, analyzing
the macroscopic quantum tunneling of the bright soliton on a Gaussian barrier
we find that its interference in the tunneling region is strongly suppressed
with respect to non-solitonic case; moreover, the tunneling through a barrier
breaks the solitonic nature of the matter wave. Finally, we show that the
collapse of the soliton is induced by the scattering on the barrier or by the
collision with another matter wave when the density reaches a critical value,
for which we derive an accurate analytical formula.
