Norm inflation for a higher-order nonlinear Schrödinger equation with a derivative on the circle

Abstract

We consider a periodic higher-order nonlinear Schrödinger 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 $s\in \mathbb{R}$.

16 pages