Nonlinear interactions in focusing media between traveling solitons and the dispersive shocks produced by an initial discontinuity are studied using the one-dimensional nonlinear Schrödinger equation. It is shown that, when solitons travel from a region with nonzero background toward 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 toward a region with nonzero background, except that, depending on its initial velocity, in this case the soliton may remain trapped inside the shocklike structure indefinitely. In all cases, the new soliton properties can be determined analytically. The results are validated by comparison with numerical simulations.
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