We introduce a system of nonlinear coupled-mode equations (CMEs) for Bragg gratings (BGs) where the Bragg reflectivity periodically switches off and on as a function of the evolution variable. The model may be realized in a planar waveguide with the Kerr nonlinearity, where the grating is represented by an array of parallel dashed lines (grooves), aligned with the propagation direction. In the temporal domain, a similar system can be derived for matter waves trapped in a rocking optical lattice. Using systematic simulations, we construct families of gap solitons (GSs) in this system, starting with inputs provided by exact GS solutions in the averaged version of the CMEs. Four different regimes of the dynamical behavior are identified: fully stable, weakly unstable, moderately unstable, and completely unstable solitons. The analysis is reported for both quiescent and moving solitons (in fact, they correspond to untilted and tilted beams in the spatial domain). Weakly and moderately unstable GSs spontaneously turn into persistent breathers (the moderate instability entails a small spontaneous change of the breather's velocity). Stability regions for the solitons and breathers are identified in the parameter space. Collisions between stably moving solitons and breathers always appear to be elastic.
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