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

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

2007-07-25

arxiv.org/abs/0707.3731v2

Mathematics

Analysis of PDEs

32 pages, 16 figures

Scientific paper

10.1007/s00332-008-9027-9

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 Schr\"{o}dinger 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.

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