Coupled mode equations and gap solitons for the 2D Gross–Pitaevskii equation with a non-separable periodic potential

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 Schrödinger/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, G. Schneider, Coupled-mode equations and gap solitons in a two-dimensional nonlinear elliptic problem with a separable periodic potential, J. Nonlinear Sci. 19 (2009) 95–131] 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 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.