  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 through one
cell of the link is described by an analytically derived map. 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, SSM supports stable pulses
which almost exactly coincide with fundamental solitons of the averaged 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; splitting can be suppressed by appropriately chirping the initial
pulse. 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 the soliton's dispersion length, the soliton rapidly
rearranges itself into another soliton, with nearly the same area but
essentially 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.
