Modulation Instability, Dark and Singular Soliton for Weakly Nonlocal Schrodinger Equation

The optical soliton solution of nonlinear complex models holds significant importance in nonlinear optics and communication systems. Considering nonlinear complex models, often described by equations like the nonlinear Schrodinger equation (NLSE), plays a crucial role in defining the balance between dispersive and nonlinear effects, enabling the formation and maintenance of solitons over long distances. This stability is crucial for signal integrity in optical communication systems. The investigation of optical soliton solutions from nonlinear complex models is sometimes complicated. With this in mind, we employed an effective method of extended Tanh function with a Riccatti differential equation to retrieve the dark, singular and periodic wave solutions for the weakly nonlocal nonlinear Schrodinger equation with parabolic law. The obtained solutions were verified by back-substitution in the original equations, with the aid of a Mathematica to affirm the robustness of the chosen approach. Respective 2D and 3D graphs for some of the obtained results was portrayed by choosing suitable values of the parameters that were involved. An analysis of instability that results in the modulation of the steady-state as a result of co-action between the nonlinear and dispersive effects was performed on the proposed model where the condition for stable wave under small perturbation was obtained and presented. The gain spectrum plot for the modulation instability was portrayed.