Split-step solitons
  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. 
Rodislav Driben and Boris A. Malomed (Department of Interdisciplinary   Studies, Faculty of Engineering, Tel Aviv University...