  We investigate (theoretically and numerically) the dynamics of a soliton
moving in an asymmetrical potential well with a finite barrier. For large
values of the width of the well, the width of the barrier and/or the height of
the barrier, the soliton behaves classically. On the other hand, we obtain the
conditions for the existence of soliton tunneling with sub-barrier kinetic
energies. We apply these results to the study of soliton propagation in
disordered systems.
