Existence of Dark Solitons for the Generalized Nonlinear Schrödinger Equation with a Strong Generic Delay

This paper studies the solitary wave solutions to the generalized nonlinear Schrödinger (gNLS for short) equation with a strong generic delay and perturbation, appearing in the nonlinear optics. In the first part of the paper, we find the exact dark and bright solitons for the gNLS equation without any perturbation by applying the three-step dynamic system method, alongside the direct algebraic method. Building on this work, in the latter part of the paper we analyze the existence of dark and bright solitons for the gNLS equation with a strong generic delay by overcoming the loss of integrability. We prove that there exist two periodic wave solutions and two bright solitary solutions, respectively, when the wave speed is near a fixed number. These results can reveal some perturbed effects for the nonlinear Schrödinger equations.