Norm inflation for a higher-order nonlinear Schr\"odinger equation with a derivative on the circle

TL;DR

This paper demonstrates that the higher-order nonlinear Schrödinger equation with derivative nonlinearity on the circle exhibits norm inflation, leading to ill-posedness in all Sobolev spaces.

Contribution

It proves norm inflation and ill-posedness for a class of higher-order nonlinear Schrödinger equations with derivative nonlinearities on the circle.

Findings

Norm inflation occurs in a subspace of Sobolev spaces.

The Cauchy problem is ill-posed in all Sobolev spaces.

Ill-posedness holds for any regularity index s in R.

Abstract

We consider a periodic higher-order nonlinear Schr\"odinger equation with the nonlinearity $u^k \partial_x u$, where $k$ is a natural number. We prove the norm inflation in a subspace of the Sobolev space $H^s(\mathbb{T})$ for any $s \in \mathbb{R}$. In particular, the Cauchy problem is ill-posed in $H^s(\mathbb{T})$ for any $s \in \mathbb{R}$.