T. Dohnal and H. Uecker
Gap solitons near a band edge of a spatially periodic nonlinear PDE can be formally approximated by solutions of Coupled Mode Equations (CMEs).
Here we study this approximation for the case of the 2D Periodic Nonlinear Schroedinger / Gross-Pitaevskii Equation with a non-separable potential
of finite contrast. We show that unlike in the case of separable potentials [T. Dohnal, D. Pelinovsky, and G. Schneider, J. Nonlin. Sci. 19,
95-131 (2009)] the CME derivation has to be carried out in Bloch rather than physical coordinates. Using the Lyapunov-Schmidt reduction we then give
a rigorous justification of the CMEs as an asymptotic model for reversible non-degenerate gap solitons and even potentials and provide $H^s$ estimates
for this approximation. The results are confirmed by numerical examples including some new families of CMEs and gap solitons absent for separable potentials.
see the paper:
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 238, 860-879 (2009).
(arXiv:0810.4499) Note: The arXiv version is a largely revised and corrected one.