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

• Published in: Applicable analysis Vol. 86; no. 8; pp. 1017 - 1036
• Main Authors: Pelinovsky, Dmitry, Schneider, Guido
• Format: Journal Article
• Language: English
• Published: Taylor & Francis Group 01.08.2007
• Subjects: 2000 Mathematics Subject Classifications; Coupled-mode equations; Fourier analysis; Gap solitons in periodic potentials; Justification of asymptotic approximations; Lyapunov-Schmidt reductions
• ISSN: 0003-6811; 1563-504X
• DOI: 10.1080/00036810701493850

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.