This paper shows out optical soliton solutions of the twin-core couplers with parabolic law nonlinearity with optical metamaterials parameters. It is used the improved new sub-ODE method (SOM) to found dark optical soliton, trigonometric function solutions, bright optical soliton and Jacobian elliptic function (JEF) solutions. Besides, physical explanation of the obtained bright and dark optical soliton solutions have been done and the infuence of some parameters of the model have been set out. As the two-core coupler is favorable for modulation instability (MI), it has been studied the steady state of the results. More importantly, we have checked the efects of the optical metamaterials parameters on the formation of the Modulation bands in normal and anomalous dispersive regime. The results obtained in this paper set out Weierstrass Elliptic Function solutions compared to Mirzazadeh et al. (J Nonlinear Opt Phys Mater 24:1550017, 2015), Vega-Guzman et al. (Acta Phys Pol A 133:167–178, 2018) and will be helpful in the feature in optical birefringent fbers.
