NLS approximation for wavepackets in periodic cubically nonlinear wave problems in $\mathbb{R}^d$

TL;DR

This paper rigorously proves that wavepacket dynamics in nonlinear periodic structures can be effectively approximated by the nonlinear Schrödinger equation, with detailed error estimates and numerical validation in two dimensions.

Contribution

It provides a rigorous proof of NLS approximation for wavepackets in periodic nonlinear wave equations, including error bounds and regularity conditions.

Findings

NLS effectively approximates wavepacket dynamics in periodic nonlinear media

Error estimates are established in an $L^1$-type norm

Numerical example confirms theoretical results in two dimensions

Abstract

The dynamics of single carrier wavepackets in nonlinear wave problems over periodic structures can be often formally approximated by the constant coefficient nonlinear Schr\"odinger equation (NLS) as an effective model for the wavepacket envelope. We provide a detailed proof of this approximation result for the Gross-Pitaevskii equation (GP) and a semilinear wave equation, both with periodic coefficients in $\mathbb{N}\ni d$ spatial dimensions and with cubic nonlinearities. The proof is carried out in Bloch expansion variables with estimates in an $L^1$-type norm, which translates to an estimate of the supremum norm of the error. The regularity required from the periodic coefficients in order to ensure a small residual and a small error is discussed. We also present a numerical example in two spatial dimensions confirming the approximation result and presenting an approximate traveling solitary wave in the GP with periodic coefficients.