Well-posedness of dispersion managed nonlinear Schr\"odinger equations

TL;DR

This paper establishes local and global well-posedness for the dispersion managed nonlinear Schrödinger equation across various nonlinearities and dispersions, and proves orbital stability of ground states when the average dispersion is non-negative.

Contribution

It provides the first proof of orbital stability for saturated nonlinearities and extends well-posedness results to a broad class of nonlinearities and dispersions.

Findings

Proved local and global well-posedness on L^2 and H^1 spaces.

Established orbital stability of ground states for non-negative average dispersion.

Covered both saturated and non-saturated nonlinear polarizations.

Abstract

We prove local and global well-posedness results for the Gabitov-Turitsyn or dispersion managed nonlinear Schr\"odinger equation with a large class of nonlinearities and arbitrary average dispersion on $L^2(\mathbb{R})$ and $H^1(\mathbb{R})$. Moreover, when the average dispersion is non-negative, we show that the set of ground states is orbitally stable. This covers the case of non-saturated and saturated nonlinear polarizations and yields, for saturated nonlinearities, the first proof of orbital stability.