D.E. Pelinovsky and G. Schneider

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

Applicable Analysis 86, 1017-1036 (2007)

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.

justification of amplitude equations, gap solitons in periodic potentials, Gross-Pitaevskii equation, Lyapunov-Schmidt reductions, Fourier analysis