Modulation instability analysis in a birefringence FBG structure driven by compete nonlinearities and trapping potential

In this paper, we investigate modulation instability (MI) in a coupled nonlinear Schrödinger (NLS) system incorporating quintic nonlinearity and gain/loss parameters, using linear stability analysis (LSA). The model is designed to describe the dynamics of nonlinear wave propagation in non-Kerr birefringent media and fiber Bragg grating (FBG) structures. By employing two-dimensional and three-dimensional representations, we analyze the influence of various system parameters – including nonlinear coupling terms and amplification coefficients – on the shape, bandwidth, and location of the MI gain spectra. The results provide insights into how these parameters control the onset and evolution of instability in periodic nonlinear media. This theoretical prediction offers valuable guidance for the experimental realization of MI in nonlinear periodic structures, such as fiber Bragg gratings and photonic crystal fibers, and contributes to the advancement of instability-based applications in nonlinear photonics.