solitary wave solutions for a higher order nonlinear schrödinger equation

we consider a higher order nonlinear schrödinger equation with third- and fourth-order dispersions, cubic–quintic nonlinearities, self steepening, and self-frequency shift effects. this model governs the propagation of femtosecond light pulses in optical fibers. in this paper, we investigate general analytic solitary wave solutions and derive explicit bright and dark solitons for the considered model. the derived analytical dark and bright wave solutions are expressed in terms of the model coefficients. these exact solutions are useful to understand the mechanism of the complicated nonlinear physical phenomena which are related to wave propagation in a higher-order nonlinear and dispersive schrödinger system.