Modulational instability in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">PT</mml:mi></mml:math> -symmetric Bragg grating structures with saturable nonlinearity

We investigate the nontrivial characteristics of modulational instability (MI) in a system Bragg gratings with saturable nonlinearity. also introduce an equal amount gain and loss into existing system, which gives rise to additional degree freedom, due concept $\mathcal{PT}$ symmetry. obtain nonlinear dispersion relation model discover that such relations for both conventional $\mathcal{PT}$-symmetric cases contradict Kerr systems by not displaying typical signature loop formation either upper branch or lower curve drawn against wave number detuning parameter. employ standard linear stability analysis study MI dynamics continuous waves perturbed infinitesimal perturbation. The main objective this paper is twofold: spectrum at top bottom photonic band gap followed comprehensive carried out anomalous normal regimes. As result, driven nonlinearity yields variety spectra, include sidebands, monotonically increasing gain, emergence single Stokes wave-number regions, so on. In particular, we observe remarkably peculiar spectrum, caused predominantly parameter, though perturbation boosts former. address impact all physical parameters considered proposed model, coupling coefficient, on phenomenon different regimes ranging from unbroken one broken greater detail.