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.
Cite
@article{arxiv.0704.2121,
title = {Justification of the coupled-mode approximation for a nonlinear elliptic problem with a periodic potential},
author = {Dmitry Pelinovsky and Guido Schneider},
journal= {arXiv preprint arXiv:0704.2121},
year = {2009}
}