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

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

2007-04-17

arxiv.org/abs/0704.2121v1

Mathematics

Analysis of PDEs

18 pages, no figures

Scientific paper

10.1007/s00220-008-0640-0

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.

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