# The derivative nonlinear Schr\"odinger equation on the half line

## TL;DR

This paper investigates the well-posedness and regularity of the derivative nonlinear Schrödinger equation on the half line, establishing new results on local and global solutions, smoothing effects, and boundary condition handling.

## Contribution

It introduces a novel approach to handle boundary conditions via a fixed point argument, improving understanding of DNLS on the half line and related equations.

## Key findings

- Almost sharp local well-posedness established
- Global well-posedness for small data proven
- Enhanced smoothing estimates for the quintic Schrödinger equation obtained

## Abstract

We study the initial-boundary value problem for the derivative nonlinear Schr\"odinger (DNLS) equation. More precisely we study the wellposedness theory and the regularity properties of the DNLS equation on the half line. We prove almost sharp local wellposedness, nonlinear smoothing, and small data global wellposedness in the energy space. One of the obstructions is that the crucial gauge transformation we use replaces the boundary condition with a nonlocal one. We resolve this issue by running an additional fixed point argument. Our method also implies almost sharp local and small energy global wellposedness, and an improved smoothing estimate for the quintic Schr\"odinger equation on the half line. In the last part of the paper we consider the DNLS equation on $\R$ and prove smoothing estimates by combining the restricted norm method with a normal form transformation.