# Inverse scattering and global well-posedness in one and two space dimensions

## TL;DR

This paper provides a comprehensive, self-contained overview of inverse scattering methods for the defocusing cubic nonlinear Schrödinger equation in one dimension and the defocusing Davey-Stewartson equation in two dimensions, highlighting recent advances.

## Contribution

It offers a detailed, revised exposition of inverse scattering techniques for specific nonlinear PDEs in low dimensions, integrating classical and recent research developments.

## Key findings

- Complete treatment of inverse scattering for 1D cubic NLS
- Extension of inverse scattering methods to 2D Davey-Stewartson equation
- Inclusion of recent advances by Nachman, Regev, and Tataru

## Abstract

These notes are a considerably revised and expanded version of expository lectures given at the Fields Institute Workshop on "Nonlinear Dispersive Partial Differential Equations and Inverse Scattering" in August 2017. We give a complete and self-contained treatment of inverse scattering for the defocussing cubic NLS in one-dimension, following the 2003 paper of Deift and Zhou, and the defocussing Davey-Stewartson equation in two space dimensions, following the work of Perry and more recent work of Nachman, Regev, and Tataru.