Diverse bistable dark novel explicit wave solutions of cubic–quintic nonlinear Helmholtz model

This paper examines three different recent computational schemes (extended simplest equation (ESE) method, modified Kudryashov (MKud) method, and modified Khater (MKha) method) for obtaining novel solitary wave solutions of cubic–quintic nonlinear Helmholtz (CQ–NLH) model. This model is considered as a general model of the well-known Schrödinger equation where it takes into account the effects of backward scattering that are neglected in the more common nonlinear Schrödinger model. Many distinct wave solutions are explained in the different formulas, such as trigonometric, rational, and hyperbolic formulas. These solutions are described in some precise sketches in two- and three-dimensional. The methods’ performance is explained to demonstrate their effectiveness and power.