Modified scattering for a dispersion-managed nonlinear Schr\"odinger equation

TL;DR

This paper proves sharp decay and modified scattering for a 1D dispersion-managed cubic nonlinear Schrödinger equation with small initial data, extending techniques from prior work on standard NLS.

Contribution

It establishes sharp decay and modified scattering results for a dispersion-managed NLS in the strong dispersion regime, using adapted techniques from previous NLS studies.

Findings

Proves sharp $L^infty$ decay for the equation.

Establishes modified scattering behavior for small initial data.

Works with an averaged dispersion model in the strong regime.

Abstract

We prove sharp $L^\infty$ decay and modified scattering for a one-dimensional dispersion-managed cubic nonlinear Schr\"odinger equation with small initial data chosen from a weighted Sobolev space. Specifically, we work with an averaged version of the dispersion-managed NLS in the strong dispersion management regime. The proof adapts techniques from Hayashi-Naumkin and Kato-Pusateri, which established small-data modified scattering for the standard $1d$ cubic NLS.