  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.
