Coupled Mode Equations and Gap Solitons in Higher Dimensions

TL;DR

This paper rigorously derives coupled mode equations for wave packets in nonlinear periodic media across arbitrary dimensions and constructs gap solitons in specific cases, advancing understanding of wave phenomena in higher-dimensional systems.

Contribution

It provides a rigorous justification of coupled mode equations in higher dimensions and constructs gap solitons for specific cases, extending prior work.

Findings

Effective equations are justified with error estimates in $L^1$-norm.

A family of time harmonic gap solitons is constructed for $d=2$, $N=4$.

Moving gap solitons are not found for $d>1$ due to spectral gap issues.

Abstract

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.