# On the management fourth-order Schr\"{o}dinger-Hartree equation

## TL;DR

This paper studies the well-posedness and dispersion management limits of a fourth-order nonlinear Schrödinger-Hartree equation with variable dispersion coefficients, providing new insights into its mathematical properties and limiting behaviors.

## Contribution

It establishes local and global well-posedness results for the equation with variable dispersion and analyzes the scaling limit leading to an averaged dispersion model.

## Key findings

- Proved local and global well-posedness in $H^s$-spaces.
- Analyzed the scaling limit of fast dispersion management.
- Demonstrated convergence to an averaged dispersion model.

## Abstract

We consider the Cauchy problem associated to the fourth-order nonlinear Schr\"{o}dinger-Hartree equation with variable dispersion coefficients. The variable dispersion coefficients are assumed to be continuous or periodic and piecewise constant in time functions. We prove local and global well-posedness results for initial data in $H^s$-spaces. We also analyze the scaling limit of fast dispersion management and the convergence to a model with averaged dispersions.