Wave propagation in periodic structures like, for instance, photonic crystals, typically features spectral gaps, i.e., frequency regions in which linear waves cannot propagate. In dimensions higher than one such gaps occur only for high enough contrast of the periodicity. In the presence of nonlinearity like, e.g., the Kerr nonlinearity of some optical media, waves can propagate in these gaps and a special class of these are exponentially localized gap solitons. We study gap solitons in 2D in the context of the periodic nonlinear Schroedinger equation (PNLS) and present analysis of an asymptotic approximation of stationary gap solitons in neighborhoods of edges of spectral gaps. We present a general derivation of their asymptotic models, so called Coupled Mode Equations (CMEs). CMEs offer an efficient tool for studying gap solitons as they have constant coefficients and the independent variables are slow compared to the original system. We justify CMEs rigorously via the Bloch transform and Lyapunov-Schmidt reduction producing $H^s$ estimates on the accuracy of the approximation. Due to a persistence result we also prove existence of gap solitons of the PNLS based on existence of reversible non-degenerate solutions of the CMEs.

Finally, we present numerical computations of several examples of gap solitons to corroborate our analysis and verify the convergence rate of the approximation error.

References:
[1] T. Dohnal, D. Pelinovsky and G. Schneider, ``Coupled-mode equations and gap solitons in a two-dimensional nonlinear elliptic problem with a separable periodic potential'', J. Nonlin. Sci. (2008) online.
[2] T. Dohnal and H. Uecker, ``Coupled Mode Equations and Gap Solitons for the 2D Gross-Pitaevskii equation with a non-separable periodic potential'', Physica D (2008) submitted.