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D.E. Pelinovsky and G. Schneider

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

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Applicable Analysis 86, 1017-1036 (2007)

**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.

**Keywords**:

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