This study successfully extracts the combined dark-bright and singular optical solitons to the three-component nonlinear Schrödinger equation by using the extended sinh-Gordon equation expansion method. The modulation instability analysis of the governing model is also presented. By using suitable values of the parameters, the 2-, 3-dimensional, and contour graphs to the reported results and the gain spectrum for the modulation instability analysis are plotted.

In this manuscript, the application of the extended sinh-Gordon equation expansion method to the Davey-Stewartson equation and the (2+1)-dimensional nonlinear complex coupled Maccari system is presented. The Davey-Stewartson equation arises as a result of multiple-scale analysis of modulated nonlinear surface gravity waves propagating over a horizontal seabed and the (2+1)-dimensional nonlinear complex coupled Maccari equation describes the motion of the isolated waves, localized in a small part of space, in many fields such as hydrodynamic, plasma physics, nonlinear optics. We successfully construct some soliton, singular soliton and singular periodic wave solutions to these two nonlinear complex models. The 2D, 3D and contour graphs to some of the obtained solutions are presented.