Well-posedness and blowup for the dispersion-managed nonlinear Schr\"odinger equation

TL;DR

This paper studies the nonlinear Schrödinger equation with periodic dispersion management, establishing key estimates, scattering results for small data, and demonstrating blowup solutions in three dimensions.

Contribution

It provides the first global-in-time Strichartz estimates for the linear part and shows both scattering and blowup phenomena in the nonlinear setting.

Findings

Established global-in-time Strichartz estimates for the linear equation.

Proved small-data scattering for the 3D cubic equation.

Demonstrated existence of blowup solutions with piecewise constant dispersion.

Abstract

We consider the nonlinear Schr\"odinger equation with periodic dispersion management. We first establish global-in-time Strichartz estimates for the underlying linear equation with suitable dispersion maps. As an application, we establish a small-data scattering result for the $3d$ cubic equation. Finally, we use a virial argument to demonstrate the existence of blowup solutions for the $3d$ cubic equation with piecewise constant dispersion map.