Fig. 1. Schematic view of the system. The top grating provides Fano resonance and the bottom grating provides Bragg scattering between the guided resonant modes. The guided modes have the wave vectors ±β, k ₀ is the wave vector and θ is the incident angle of the pump wave.

Fig. 2. (a),(b) Dependence of the normalized energy density J_norm =Γ²·(|A ₊|²+|A _-|²)/I ² on the frequency detuning δ for q=-1 and Γ=7.55·10^-4. The dashed line corresponds to the linear case with pump I=5.5·10^-6 and the solid line corresponds to the nonlinear case with pump I=5.5·10^-5. Bistability appears in the nonlinear case, see full lines. Modulational instability on the positive slope can appear also; this region is situated between points A and B in (b). (c) and (d) show the behavior of the reflection coefficients on the pump frequency detuning for the linear (dashed line) and nonlinear (solid line) cases.

Fig. 3. The same as Fig. 2, but for q=0.

Fig. 4. The dependence of the energy density W=|A ₊|²+|A _-|² of the spatially uniform solution and on the maximum of W=|A ₊|²+|A _-|² of the bright LS on the pump amplitude I. Frequency detuning δ=0.9978, wave vector detuning q=0 and dissipation Γ=7.55·10^-4. The upper branch of the spatially uniform state is modulationally unstable.

Fig. 5. The dependences of the energy density W=|A ₊|²+|A _-|² in bright (a) and dark (b) solitons are shown for the case with I=4.4·10^-5, Γ=7.55·10^-4. For bright solitons the pump frequency is δ=0.9977 and for dark solitons the frequency is δ=-1.00238.