Asymptotic breathing pulse in optical transmission systems with dispersion compensation

We study a nonlinear process of the formation of a breathing solitary wave in the optical transmission systems with periodic amplification and dispersion compensation. Results of our numerical simulations demonstrate remarkably stable asymptotic propagation of such breathing pulse over long distances. We have derived approximate equations describing pulse amplitude and width oscillations and found that results obtained by this approach are in good agreement with the results of direct numerical modeling on the short and middle distances. It is shown that asymptotic averaged pulses have a form typically close to a Gaussian shape. We have found numerically that an input pulse evolves asymptotically into a stable breathing structure. After the first stage of propagation, the input pulse emits radiation that spreads due to dispersion. The asymptotic structure that is formed realizes a balance between the main pulse and the radiative tail.