We report results of a systematic study of one-dimensional four-wave movingsolitons in a recently proposed model of the Bragg cross-grating in planaroptical waveguides with the Kerr nonlinearity; the same model applies to afiber Bragg grating (BG) carrying two polarizations of light. We concentrate onthe case when the system's spectrum contains no true bandgap, but onlysemi-gaps (which are gaps only with respect to one branch of the dispersionrelation), that nevertheless support soliton families. Solely zero-velocitysolitons were previously studied in this system, while current experimentscannot generate solitons with the velocity smaller than half the maximum groupvelocity. We find the semi-gaps for the moving solitons in an analytical form,and demonstrated that they are completely filled with (numerically found)solitons. Stability of the moving solitons is identified in direct simulations.The stability region strongly depends on the frustration parameter, whichcontrols the difference of the present system from the usual model for thesingle BG. A completely new situation is possible, when the velocity intervalfor stable solitons is limited not only from above, but also from below.Collisions between stable solitons may be both elastic and strongly inelastic.Close to their instability border, the solitons collide elastically only iftheir velocities c1 and c2 are small; however, collisions between more robustsolitons are elastic in a strip around c1=-c2.
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