Optical soliton to multi-core (coupling with all the neighbors) directional couplers and modulation instability

n this research paper under consideration is the nonlinear multi-core coupler (coupling with all neighbors) with the parabolic law nonlinearity and optical metamaterials parameters. The integration scheme, namely the New sub-ODE method, is used to extract bright, dark, trigonometric function solutions, Jacobian elliptic function solutions and Weierstrass elliptic function solutions. To better understand the physical phenomena of the gained solutions, some physical explanation of the presented solutions under suitable values for the physical parameters via the 3D, 2D and contour simulations is given. The confrontation between the nonlinear and dispersion terms has permitted to study the behavior of the modulation instability (MI) gain spectra. The results obtained in this article are more general than those contained in the following papers [41-43] and will help to explain the virtue of the nonlinear optical birefringent.