Bifurcation of Gap Solitons in Coupled Mode Equations in $d$ Dimensions

Abstract

We consider a system of first order coupled mode equations in $\mathbb{R}^d$ describing the envelopes of wavepackets in nonlinear periodic media. Under the assumptions of a spectral gap and a generic assumption on the dispersion relation at the spectral edge, we prove the bifurcation of standing gap solitons of the coupled mode equations from the zero solution. The proof is based on a Lyapunov-Schmidt decomposition in Fourier variables and a nested Banach fixed point argument. The reduced bifurcation equation is a perturbed stationary nonlinear Schr\"odinger equation. The existence of solitary waves follows in a symmetric subspace thanks to a spectral stability result. A numerical example of gap solitons in $\mathbb{R}^2$ is provided.