Optical solitary wave solutions for the higher order nonlinear Schrödinger equation with self-steepening and self-frequency shift effects

In this paper, we have obtained the optical solitary wave solutions for the nonlinear Schrödinger equation 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 was used to carry out the derivations of the solitons. The parametric conditions for the formation of soliton pulses were determined. Using the 1-soliton solution, a number of conserved quantities have been calculated for Hirota and Sasa-Satsuma cases. We have also constructed some periodic wave solutions of the higher order nonlinear Schrödinger equation by reducing it to quartic anharmonic oscillator equation and by using projective Ricatti equations. Moreover by using He's semi-inverse method, variational formulation was established to obtain exact soliton solutions. The 1-soliton solutions of time dependent form of this equation was also obtained. To visualize the propagation characteristics of dark-bright soliton solutions, few numerical simulations are given.