Dark solitons in an extended nonlinear Schrödinger equation with higher-order odd and even terms

An extended nonlinear Schrödinger equation that includes terms accounting for higher-order odd (third order) and even (fourth order) terms is investigated. The model contains a variety of higher-order dispersion and nonlinear terms that are important for applications in fiber optics, the Heisenberg spin chain, plasma physics and ocean waves. We derive a nonlinear structural equation with fifth-degree nonlinear term describing the evolution of the wave amplitude in the nonlinear media by means of the coupled amplitude-phase formulation. We find the exact analytical dark soliton solution of the model in the presence of all the physical parameters. The results show that these structures characteristically exist due to a balance among higher-order nonlinear and dispersive effects of different nature. Particular cases are also discussed. We also investigate numerically the parametric stability of the solitary dark wave solution. We further calculate the power required for generating dark solitons in the nonlinear media together with their width.