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

TL;DR

This paper analyzes a 2D nonlinear elliptic problem with a separable periodic potential, deriving coupled-mode equations to describe bifurcations in the spectrum and studying the existence and classification of gap solitons.

Contribution

It rigorously derives coupled-mode equations for a 2D nonlinear elliptic problem with a separable potential and proves the persistence of gap solitons beyond these equations.

Findings

Bifurcation of the first band gap is characterized.

Coupled-mode equations are derived and validated.

Numerical classification of localized solutions is performed.

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 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.