Averaged dynamics of optical pulses described by a nonlinear Schrödinger equation with periodic coefficients

A nonlinear Schrödinger equation with periodic coefficients, as it appears, e.g., in nonlinear optics, is considered. The high-frequency, variable part of the dispersion may be even much larger than the mean value. The ratio of the length of the dispersion map to the period of a solution is assumed as one small parameter. The second one corresponds to the integral over the variable part of the dispersion. For the averaged dynamics, we propose a procedure based on the Bogolyubov method. As a result, we obtain the asymptotic equation in the dominating order, as well as with the next corrections. The equation is valid for all combinations of the small parameters. The explicit forms of the coefficients are presented for a two-step dispersion map with an exponential loss function. The forms of the bright and black soliton solutions are discussed. The results are compared to those from other averaging methods, namely, the multiple-scale method and the method based on Lie transformations.