Four-wave solitons in waveguides with a cross-grating

A model of a two-dimensional optical waveguide with the Kerr nonlinearity and two transversal (cross-)Bragg gratings (BGs) is considered. Four waves trapped in the waveguide are coupled linearly by reflections on the cross-BG, and nonlinearly by self-phase-modulation, cross-phase-modulation, and four-wave-mixing. One-dimensional gap solitons (GSs) in the model are found by means of the variational approximation and numerical methods, the analytical and numerical results being in good agreement. The solitons fall into three distinct categories, which are identified as symmetric (S), anti-symmetric (anti-S), and asymmetric (aS) ones at the center of the bandgap, and those which are obtained by continuation of these three types in the general case. The stability of the GSs is studied in the spatial domain. All the solitons of the S and anti-S types are unstable (their instability modes are different), while the aS solitons have a well-defined stability region. The latter is identified by means of the Vakhitov-Kolokolov (VK) criterion, which is verified by direct simulations. It is demonstrated too that stable breathers, which are close to strongly asymmetric solitons, readily self-trap from a two-component input that corresponds to a physically relevant boundary condition in the spatial domain (the development of the instability of solitons of the S type gives rise to breathers of a different kind, in which the field periodically switches between two aS configurations that are mirror images to each other). Tilted spatial solitons of the aS type are found too; they are stable for relatively small values of the tilt.