W-shaped profile and multiple optical soliton structure of the coupled nonlinear Schrödinger equation with the four-wave mixing term and modulation instability spectrum.

• The Coupled Nonlinear Schrödinger Equation (CNLSE) throughout Kerr law nonlinearity is considered. • The linear stability analysis is used. • The modulation instability analysis is presented. • The obtained results are showed by 2D and 3D Figures. We investigate diverse optical solitons and other solution for the Coupled Nonlinear Schrödinger Equation (CNLSE) having Four-Wave Mixing (FWM) and Kerr nonlinearity term. We stress the effects of different parameters of the model on the obtained optical soliton by employing the New Generalized Extended Direct Algebraic Method (NGEDAM). The solutions contain W-shaped profile of the bright optical soliton, dark optical soliton and diverse other solutions including the combined complex optical soliton. Interestingly, W-shaped optical soliton and dark optical soliton own a good argument for long communication and big data transmission originating from the self-steepening, Kerr nonlinearity effects and FWM. We also, emphasize the Modulation Instability (MI) gain spectra to narrow the impact of the FWM in birefringence. Thereafter, we involve the ellipticity angle effects in normal and anomalous dispersion regime on the MI gain spectra. Parametric conditions for the existence of the obtained optical soliton solutions and MI gain are given. [ABSTRACT FROM AUTHOR]