We consider a long fiber-optical link consisting of alternating dispersive and nonlinear segments, i.e., a split-step model (SSM), in which the dispersion and nonlinearity are completely separated. Passage of a soliton (localized pulse) through one cell of the link is described by an analytically derived map. Multiple numerical iterations of the map reveal that, at values of the system's stepsize (cell's size) L comparable to the pulse's dispersion length zD, SSM supports stable propagation of pulses which almost exactly coincide with fundamental solitons of the corresponding averaged nonlinear Schrodinger (NLS) equation. However, in contrast with the NLS equation, the SSM soliton is a strong attractor, i.e., a perturbed soliton rapidly relaxes to it, emitting some radiation. A pulse whose initial amplitude is too large splits into two solitons; however, splitting can be suppressed by appropriately chirping the initial pulse. On the other hand, if the initial amplitude is too small, the pulse turns into a breather, and, below a certain threshold, it quickly decays into radiation. If L is essentially larger than zD, the input soliton rapidly rearranges itself into another soliton, with nearly the same area but smaller energy. At L still larger, the pulse becomes unstable, with a complex system of stability windows found inside the unstable region. Moving solitons are generated by lending them a frequency shift, which makes it possible to consider collisions between solitons. Except for a case when the phase difference between colliding solitons is ≲0.05π, the interaction between them is repulsive. We also simulate collisions between solitons in two-channel SSM, concluding that the collisions are strongly inelastic: even if the solitons pass through each other, they suffer a large reduction of the amplitude.