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T. Dohnal, D. Pelinovsky, and G. Schneider

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

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Journal of Nonlinear Science 19, 95–131 (2009)

**Abstract:**

We address a two-dimensional nonlinear elliptic problem with a finiteamplitude
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 coupledmode
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 nondegeneracy
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 approximated by solutions
of the coupled-mode equations are obtained for a finite-time interval.

**Keywords**:

justification of amplitude equations, gap solitons in periodic potentials,
Gross-Pitaevskii equation, Fourier-Bloch decomposition, separable two-dimensional potentials