Traveling wave solutions with modulation instability of coupled nonlinear Schrödinger equations via two analytical approaches

This paper investigates the soliton pulses of various kinds and their behavior as they traverse a birefringent optical fiber. Coupled nonlinear Schrödinger equations (CNLSEs) is a nonlinear model to describe light propagation within these fibers that accounts for coherent and incoherent nonlinear coupling phenomena. In particular, this study investigates several optical solitons subjected to third-order nonlinearity and group-velocity dispersion, which is physically significant regarding elliptical core optical fiber. This paper offers dark, bright, singular, periodic, and periodic-singular solitons to CNLSEs by employing the \(\phi ^6\)-expansion technique and the \(G'/(bG'+G+a)\)-expansion technique. It also provides 2D and 3D graphical illustrations of these solutions using fixed parameter values to showcase their physical significance. The modulation instability of the proposed model is also examined in this paper. The methods used are more accessible, effective, and computationally efficient. These methods can be applied to tackle complex problems obtained in mathematical physics, engineering, and optical fiber technology.