  Nonlinear interactions in focusing media between traveling solitons and the
dispersive shocks produced by an initial discontinuity are studied using the
one-dimensional nonlinear Schrodinger equation. It is shown that, when solitons
travel from a region with nonzero background towards a region with zero
background, they always pass through the shock structure without generating
dispersive radiation. However, their properties (such as amplitude, velocity
and shape) change in the process. A similar effect arises when solitons travel
from a region with zero background towards a region with nonzero background,
except that, depending on its initial velocity, in this case the soliton may
remain trapped inside the shock-like structure indefinitely. In all cases, the
new soliton properties can be determined analytically. The results are
validated by comparison with numerical simulations.
