iant solutions and a great insight on nonlinear propagation, but at the end, soliton systems never passed from laboratories to production, even if a great number of field tests were done. The practical result of all this research was however the discovery that, even if math- ematical solitons are too complex to be implemented in a practical system, the soliton prin- ciple can be exploited in any case. In this way, the present day ultra-long-haul systems were born. In these systems, SPM is used to partially compensate chromatic dispersion to greatly reduce pulse broadening without creating perfect phase matching between spectrally adjacent channels. 4.2.7.2 Kerr-Induced Cross-Phase Modulation When a comb of amplitude-modulated signals at different wavelengths is injected into the fiber, the expression of the phase of one of them becomes, in the approximation of weak nonlinear effect: (4.41) Besides the phase modulation imposed by SPM, another phase modulation term exists due to the presence of the channels at other wavelengths. In practical cases in which different channels are quite near (e.g., at 100 GHz spacing, that in wavelength units is a spacing of about 0.84 nm around 1.55 um), N2 is almost the same for all the channels and the weight of XPM is doubled with respect to the weight of SPM. However, due to the time dependence of the intensity Ik(t), XPM is effective only when the intensity of the signal under consideration and the intensity of the interfering signal are superimposed in time. In a WDM transmission, due to the frequency difference of the different channels, there is a shift of one channel with respect to the other whose speed depends on the frequency distance of the considered channels. This shift, also called walk- off, reduces the efficiency of XPM and all the effects requiring phase matching. Moreover, the effect of chromatic dispersion also causes a shift of one channel with respect to the other due to the different group velocity, implying that higher the dispersion less efficient is the XP