Propagation of optical pulses in nonlinear systems with varying dispersion

Optical fiber systems with periodically varying dispersion are the most perspective candidates for high capacity communication lines in the future. An opportunity to assemble the system from pieces of fiber with a dispersion of opposite sign gives an ample freedom to a designer. If systems are completely linear it is possible to design an ideal line, transmitting the pulses to an arbitrary long distance with zero average distortion. In reality, due to the presence of noise, the pulse amplitude cannot be too small and the nonlinear effects are inevitable. In the majority of optical materials the main nonlinear effect is the increasing of the refraction coefficient in the presence of an electromagnetic field (focusing nonlinearity). This effect can be used for compensation of the dispersion spreading and formation of optical solitons pulses propagating without average changing of their shape. Now the concept of ”optical solitons” is accepted by a very broad community of engineers and applied mathematicians. It is important to stress that the concept of optical soliton was elaborated initially for media with constant positive dispersion, ∂ω/∂k > 0, ∂k/∂ω < 0. In this case focusing nonlinearity plays a positive role. In fibers with constant negative dispersion, ∂ω/∂k < 0, ∂k/∂ω > 0, focusing nonlinearity plays a negative role, enhancing the dispersion spreading. In such media usual ”bright” solitons do not exist. We do not discuss in this article ”dark” solitons, propagating on the background of a monochromatic wave of finite amplitude in defocusing media. What happens in a media with periodically varying dispersion? It is unclear a priori. The nonlinearity which decreases the dispersion spreading in the legs with positive dispersion, enhance the