Coupled-mode equations and gap solitons in a two-dimensional nonlinear elliptic problem with a separable periodic potential

Abstract

We address a two-dimensional nonlinear elliptic problem with a finite-amplitude periodic potential. For a class of separable symmetric potentials, we study the bifurcation of the first band gap in the spectrum of the linear Schrödinger operator and the relevant coupled-mode equations to describe this bifurcation. The coupled-mode equations are derived by the rigorous analysis based on the Fourier--Bloch decomposition and the Implicit Function Theorem in the space of bounded continuous functions vanishing at infinity. Persistence of reversible localized solutions, called gap solitons, beyond the coupled-mode equations is proved under a non-degeneracy assumption on the kernel of the linearization operator. Various branches of reversible localized solutions are classified numerically in the framework of the coupled-mode equations and convergence of the approximation error is verified. Error estimates on the time-dependent solutions of the Gross--Pitaevskii equation and the coupled-mode equations are obtained for a finite-time interval.

32 pages, 16 figures

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• Symmetry breaking in two-dimensional square grids: persistence and failure of the dimensional crossover
• Thermodynamic properties of separable square-wave potentials
• Justification of the Asymptotic Coupled Mode Approximation of Out-of-Plane Gap Solitons in Maxwell Equations
• Exponential asymptotics for line solitons in two-dimensional periodic potentials
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