Averaging for the dispersion-managed NLS

TL;DR

This paper proves that solutions to the dispersion-managed nonlinear Schrödinger equation exhibit averaging behavior over time in the fast dispersion management regime, with results depending on the average dispersion and initial data size.

Contribution

It establishes global-in-time averaging results for the $L^2$-critical dispersion-managed NLS, including cases with nonzero average dispersion and positive dispersion maps.

Findings

Averaging holds for nonzero average dispersion with any subcritical data.

Averaging holds for positive dispersion maps with $L^2$ data.

Results are valid in the fast dispersion management regime.

Abstract

We establish global-in-time averaging for the $L^2$-critical dispersion-managed nonlinear Schr\"odinger equation in the fast dispersion management regime. In particular, in the case of nonzero average dispersion, we establish averaging with any subcritical data, while in the case of a strictly positive dispersion map, we obtain averaging for data in $L^2$.