Lisa Wahlers | Searcharxiv

arXiv2021

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

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 Schrödinger 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.

math.AP↗

arXiv2018

Coupled Mode Equations and Gap Solitons in Higher Dimensions

We study waves-packets in nonlinear periodic media in arbitrary ($d$) spatial dimension, modeled by the cubic Gross-Pitaevskii equation. In the asymptotic setting of small and broad waves-packets with $N\in \mathbb{N}$ carrier Bloch waves the effective equations for the envelopes are first order coupled mode equations (CMEs). We provide a rigorous justification of the effective equations. The estimate of the asymptotic error is carried out in an $L^1$-norm in the Bloch variables. This translates to a supremum norm estimate in the physical variables. In order to investigate the existence of gap solitons of the $d$-dimensional CMEs, we discuss spectral gaps of the CMEs. For $N=4$ and $d=2$ a family of time harmonic gap solitons is constructed formally asymptotically and numerically. Moving gap solitons have not been found for $d>1$ and for the considered values of $N$ due to the absence of a spectral gap in the standard moving frame variables.

math.AP↗