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 1D 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:

• Gap solitons in periodic potentials
• Coupled-mode equations
• Justification of asymptotic approximations
• Lyapunov–Schmidt reductions
• Fourier analysis

Keywords:

• 2000 Mathematics Subject Classifications:
• 41A60
• 35C20
• 35Q55
• 35L60

Acknowledgements

DP thanks D. Agueev and W. Craig for their help at the early stage of this project. The work of D. Pelinovsky is supported by the Humboldt Research Foundation. The work of G. Schneider is partially supported by the Graduiertenkolleg 1294 “Analysis, simulation and design of nano-technological processes” sponsored by the Deutsche Forschungsgemeinschaft (DFG) and the Land Baden-Württemberg.