Justification of the Asymptotic Coupled Mode Approximation of Out-of-Plane Gap Solitons in Maxwell Equations

TL;DR

This paper rigorously justifies the use of coupled mode equations to approximate out-of-plane gap solitons in two-dimensional Maxwell systems within photonic crystals, providing error estimates and validating previous formal approaches.

Contribution

It offers a rigorous mathematical proof for the validity of coupled mode equation asymptotics in 2D photonic crystals with Kerr nonlinearity, including error bounds.

Findings

Error estimates in $H^2(\

between exact solutions and approximations.

Validation of formal and numerical CME-approximations from prior work.

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 $H^2(\mathbb R^2)$ between the exact solution and the envelope approximation. The results justify the formal and numerical CME-approximation in [Dohnal and D\"orfler, Multiscale Model. Simul., p. 162-191, 11 (2013)].