Periodic and solitary waves in an inhomogeneous optical waveguide with third-order dispersion and self-steepening nonlinearity

Abstract

We demonstrate the formation of periodic waves and envelope solitons in dispersive optical media having a Kerr nonlinear response under the influence of third-order dispersion and self-steepening effect. The stability properties of the bright- and dark-soliton solutions are proved using the stability criterion based on the theory of nonlinear dispersive waves. Regimes for the modulation instability of a continuous wave signal propagating inside the dispersive optical medium are also investigated. The results show that the gain spectrum depends crucially on the self-steepening parameter, while the third-order dispersion has no effect on the modulation instability condition. A similarity transformation is presented to reduce the generalized extended nonlinear Schr\"odinger equation with distributed coefficients which models the pulse evolution in the presence of the inhomogeneities of media to the related constant-coefficient one. The propagation behaviors of self-similar bright and dark solitons are discussed in a periodic distributed fiber system and an exponential dispersion decreasing fiber.