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

摘要

We address a two-dimensional nonlinear elliptic problem with afinite-amplitude periodic potential. For a class of separable symmetricpotentials, we study the bifurcation of the first band gap in the spectrum ofthe linear Schr\"{o}dinger operator and the relevant coupled-mode equations todescribe this bifurcation. The coupled-mode equations are derived by therigorous analysis based on the Fourier--Bloch decomposition and the ImplicitFunction Theorem in the space of bounded continuous functions vanishing atinfinity. Persistence of reversible localized solutions, called gap solitons,beyond the coupled-mode equations is proved under a non-degeneracy assumptionon the kernel of the linearization operator. Various branches of reversiblelocalized solutions are classified numerically in the framework of thecoupled-mode equations and convergence of the approximation error is verified.Error estimates on the time-dependent solutions of the Gross--Pitaevskiiequation and the coupled-mode equations are obtained for a finite-timeinterval.