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

Abstract

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, 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.

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 instead of L_s^2; 4-error estimate in Thm 4.9 fixed; 5- reversibility analysis in the persistence step corrected, evenness of V added as an assumption for the persistence step

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• Surface vortex solitons
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• Localized modes in two-dimensional Schroedinger lattices with a pair of nonlinear sites
• Justification of the Asymptotic Coupled Mode Approximation of Out-of-Plane Gap Solitons in Maxwell Equations
• Vortex families near a spectral edge in the Gross-Pitaevskii equation with a two-dimensional periodic potential
• Bifurcation of gap solitons in periodic potentials with a sign-varying nonlinearity coefficient
• Families of solitons in Bragg supergratings
• Bifurcation of nonlinear bound states in the periodic Gross-Pitaevskii equation with PT-symmetry
• Minimal energy solutions for repulsive nonlinear Schrödinger systems