We report results of systematic numerical simulations of the nonlinear Schrödinger equation whose dispersion coefficient is a periodic function of the propagation distance (formally, time). This is a model of propagation of a subpicosecond optical soliton in silica fibers with a variable cross section, which are now available for experiments. We demonstrate that, while-very broad solitons are practically stable in this model, only slowly decaying into radiation, the fundamental solitons whose dispersion length is comparable to the modulation period suffer a sudden splitting into a pair of secondary solitons, accompanied by a burst of radiation, after an initial period of steady evolution. The splitting takes place provided that the modulation amplitude exceeds a certain critical value. This remarkable fact, as well as the magnitude of the critical value, qualitatively agrees with predictions made recently for the same model on the basis of a variational approximation. Drawing a separatrix between the nonsplitting and splitting solitons in a parametric plane, we find a complicated structure. A set of stability islands is found inside the splitting regime. Furthermore, we demonstrate that, at large values of the modulation amplitude, there exists a stability "isthmus" between two large splitting areas, in which the soliton retrieves its stability. Unexpectedly, the soliton's stability is again restored at fairly large values of the modulation amplitude. In most cases, the (quasi-)stable soliton is actually a breather, which demonstrates persistent long-period shape oscillations.