On the Boundary of the Dispersion - Managed Soliton Existence

Propagation of optical pulse in nonlinear media with varying dispersion is both fundamental [1] and important applied problem [2–8] because the dispersion-managed (DM) system, which is a system with periodic dispersion variation along an optical fiber, is one of the most prospective candidate for ultrafast high-bit-rate optical communication lines. Lossless propagation of optical pulse in DM fiber is described by a nonlinear Schrödinger equation (NLS) with periodically varying dispersion d (z): (1) where u is the envelope of the optical pulse, z is the propagation distance, and all quantities are dimension-less. Consider a two-step periodic dispersion map: d (z) = d 0 + , where = d 1 for 0 < z + nL < L 1 and = d 2 for L 1 < z + nL < L 1 + L 2 ; d 0 is the path-averaged dispersion; d 1 and d 2 are the amplitudes of dispersion variation subject to the condition d 1 L 1 + d 2 L 2 = 0; L ≡ L 1 + L 2 is the dispersion compensation period; and n is the arbitrary integer number. Equation (1) also describes the pulse propagation in a fiber with losses compensated by periodically placed amplifiers, if the distance between amplifiers is much less than L. In a linear regime, in which the nonlinear term in Eq. (1) is negligible, the periodical variation of dispersion is a way to overcome pulse broadening due to the chromatic dispersion, provided that the residual dispersion d 0 is small enough. However, in a real optical fiber, the nonlinear term in Eq. (1) is important because the optical pulse amplitude should be large enough to get high signal/noise ratio. One of the fascinating features of the DM system is the numerical observation of a space-breathing soliton-like structure, which is called the DM soliton, for both positive and negative residual dispersion d 0 [9]. This observation is in sharp contrast with the system described by the NLS with the constant dispersion, where the stable soliton propagation is pos-iu z d z ()u tt u 2 u + + 0, = d ˜ z () d ˜ z () d ˜ z () sible only for the positive dispersion [10], because the nonlinearity can continuously compensate the positive dispersion only. In the DM soliton, the balance between the nonlinearity and dispersion is achieved, on average, over the dispersion period …