Physics Letters A 327 (2004) 296–311 www.elsevier.com/locate/pla Four-wave solitons in waveguides with a cross-grating Ilya M. Merhasin ∗ , Boris A. Malomed Department of Interdisciplinary Studies, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel Received 8 August 2003; received in revised form 6 January 2004; accepted 12 May 2004 Available online 28 May 2004 Communicated by A.R. Bishop Abstract 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.  2004 Elsevier B.V. All rights reserved. PACS: 05.45.Yv; 42.65.Tg; 03.75.Lm 1. Introduction Bragg gratings (BGs) are important ingredients of various optical systems, especially in fiber optics [1]. In combination with the Kerr nonlinearity, fiber BGs give rise to gap solitons (GSs for brevity, which are called this way because they are found in the gap of the grating’s linear spectrum). These solitons were first predicted theoretically [2], and then observed in fiber gratings [3]. BGs can also be combined with the Kerr nonlinearity in the spatial domain, which implies writing a quasi- one-dimensional grating—for instance, in the form of a system of parallel scratches—on a surface of a planar * Corresponding author. E-mail address: merkhasi@post.tau.ac.il (I.M. Merhasin). 0375-9601/$ – see front matter  2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.05.037