Abstract
In periodic media gap solitons with frequencies inside a spectral gap but close to a spectral band can be formally approximated by a slowly varying envelope ansatz. The ansatz is based on the linear Bloch waves at the edge of the band and on effective coupled mode equations (CMEs) for the envelopes. We provide a rigorous justification of such CME asymptotics in two-dimensional photonic crystals described by the Kerr nonlinear Maxwell system. We use a Lyapunov–Schmidt reduction procedure and a nested fixed point argument in the Bloch variables. The theorem provides an error estimate in between the exact solution and the envelope approximation. The results justify the formal and numerical CME-approximation in Dohnal and Dörfler, [2013 Multiscale Model. Simul. 11 162–191].
Highlights
Maxwell’s equations in Kerr nonlinear dielectric materials without free charges are described by μ0∂tH = −∇ × E, ǫ0∂tD = ∇ × H, ∇ · D = ∇ · H = 0, (1.1)where E = (E1, E2, E3) and H = (H1, H2, H3) are the electric and the magnetic field respectively, D = (D1(E), D2(E), D3(E)) is the electric displacement field and ǫ0 and μ0 are the permittivity and the permeability of the free space, respectively
We model a two dimensional photonic crystal and assume that the dielectric function ǫ : R2 → R and the cubic electric susceptibility χ(3) : R2 → R3×3×3×3 are periodic and ǫ is positive
The coupled mode equations (CMEs) can be seen as the effective bifurcation system of the Lyapunov-Schmidt decomposition
Summary
For any ω in a spectral gap of the linear problem (L(E) − ω2ǫ(·))u = 0 equation (1.6) is expected to have localized solutions u with u(x1, x2) → 0 as |(x1, x2)| → ∞, called gap solitons. For several examples with the coefficients ∇2ωn∗ (k(j)) and Iαj ,β,γ obtained from actual Bloch waves of the corresponding Maxwell problem, localized solutions were found numerically in [9]. The CMEs can be seen as the effective bifurcation system of the Lyapunov-Schmidt decomposition This approach has been used, e.g., for wave packets of the Gross-Pitaevskii equation with periodic coefficients in [11, 14, 15, 10]. We prove estimates on a linear inhomogeneous problem on the periodicity cell
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