We consider, numerically and analytically, a wave pulse passing a point where the dispersion coefficient changes its sign from focusing to defocusing. Simulations demonstrate that, in the focusing region, the pulse keeps a soliton-like shape until it is close to the zero-dispersion point, but then, after the passage of this point, the pulse decays into radiation if its energy is below a certain threshold, or, in the opposite case, it quickly rearranges itself into a new double-humped structure, with a minimum at the center, twin maxima propagating away from the center, and decaying tails. In the focusing region, the pulse distortion is correctly described by the well-known adiabatic approximation, provided that it has sufficient energy. In the defocusing region, we find analytically an exact reduction of the underlying nonlinear-Schrödinger equation with a linearly varying dispersion coefficient to an ordinary differential equation. Comparison with the numerical simulations suggests that the inner region of the double-humped structure is accurately represented by solutions of this ordinary differential equation. The separation between the maxima is thus predicted to grow nearly linearly with the propagation distance, which accords with the numerical results. The structure found in this work may be readily observed experimentally in dispersion-decreasing nonlinear optical fibers.