Dark and Bright Solitary Wave Solutions of the Higher Order Nonlinear Schrödinger Equation With Self-Steepening and Self-Frequency Shift Effects

In this paper, we obtain the exact bright and dark soliton solutions for the nonlinear Schrödinger equation (NLSE) which describes the propagation of femtosecond light pulses in optical fibers in the presence of self-steepening and a self-frequency shift terms. The solitary wave ansatz method is used to carry out the derivations of the solitons. The parametric conditions for the formation of soliton pulses are determined. Using the one-soliton solution, a number of conserved quantities have been calculated for Hirota and Sasa–Satsuma cases and finally, we have constructed some periodic wave solutions by reducing the higher order nonlinear Schrödinger equation (HNLS) to quartic anharmonic oscillator equation. The obtained exact solutions may be useful to understand the mechanism of the complicated nonlinear physical phenomena which are related to wave propagation in a HNLS model equation.