# Global well-posedness for a $L^2$-critical nonlinear higher-order Schr\"odinger equation

## TL;DR

This paper establishes the global well-posedness of a critical nonlinear higher-order Schrödinger equation in the $L^2$ space for certain initial data regularities, extending understanding of such equations' long-term behavior.

## Contribution

It proves global well-posedness for a class of $L^2$-critical higher-order Schrödinger equations with specific initial data regularity conditions.

## Key findings

- Global well-posedness proven for the equation in the specified function space.
- Initial data in $H^eta$ with $eta > rac{k(4k-1)}{14k-3}$ suffices for global solutions.
- Extension of well-posedness results to higher-order, critical nonlinear Schrödinger equations.

## Abstract

We prove the global well-posedness for a $L^2$-critical defocusing cubic higher-order Schr\"odinger equation, namely \[ i\partial_t u + \Lambda^k u = -|u|^2 u, \] where $\Lambda=\sqrt{-\Delta}$ and $k\geq 3, k \in \mathbb{Z}$ in $\mathbb{R}^k$ with initial data $u_0 \in H^\gamma, \gamma>\gamma(k):=\frac{k(4k-1)}{14k-3}$.