Coupled Mode Equations and Gap Solitons for the 2D Gross-Pitaevskii equation with a non-separable periodic potential

摘要

Gap solitons near a band edge of a spatially periodic nonlinear PDE can beformally approximated by solutions of Coupled Mode Equations (CMEs). Here westudy this approximation for the case of the 2D Periodic NonlinearSchr\"{o}dinger / Gross-Pitaevskii Equation with a non-separable potential offinite contrast. We show that unlike in the case of separable potentials [T.Dohnal, D. Pelinovsky, and G. Schneider, J. Nonlin. Sci. {\bf 19}, 95--131(2009)] the CME derivation has to be carried out in Bloch rather than physicalcoordinates. Using the Lyapunov-Schmidt reduction we then give a rigorousjustification of the CMEs as an asymptotic model for reversible non-degenerategap solitons and even potentials and provide $H^s$ estimates for thisapproximation. The results are confirmed by numerical examples including somenew families of CMEs and gap solitons absent for separable potentials.