Nonlinear Sciences – Pattern Formation and Solitons
Scientific paper
2008-10-24
Nonlinear Sciences
Pattern Formation and Solitons
corrections of v.5: 1-assumption A.1 strengthened; 2-powers of epsilon fixed in (4.25); 3-\hat{R}_j estimated in L_{s-2}^2 ins
Scientific paper
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 Schr\"{o}dinger / 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. {\bf 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.
Dohnal Tomas
Uecker Hannes
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