Justification of the coupled-mode approximation for a nonlinear elliptic problem with a periodic potential

Abstract

Coupled-mode systems are used in physical literature to simplify the nonlinear Maxwell and Gross-Pitaevskii equations with a small periodic potential and to approximate localized solutions called gap solitons by analytical expressions involving hyperbolic functions. We justify the use of the one-dimensional stationary coupled-mode system for a relevant elliptic problem by employing the method of Lyapunov--Schmidt reductions in Fourier space. In particular, existence of periodic/anti-periodic and decaying solutions is proved and the error terms are controlled in suitable norms. The use of multi-dimensional stationary coupled-mode systems is justified for analysis of bifurcations of periodic/anti-periodic solutions in a small multi-dimensional periodic potential.

18 pages, no figures

References in corpus (4)

• Wannier functions analysis of the nonlinear Schrödinger equation with a periodic potential
• Triviality of Bloch and Bloch-Dirac bundles
• Translationally invariant nonlinear Schrodinger lattices
• Coupled-mode equations and gap solitons in a two-dimensional nonlinear elliptic problem with a separable periodic potential

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