Norm inflation for the derivative nonlinear Schr\"odinger equation

TL;DR

This paper demonstrates that the derivative nonlinear Schrödinger equation (DNLS) exhibits norm inflation in Sobolev spaces below the critical regularity, highlighting ill-posedness in these function spaces.

Contribution

It introduces a novel approach using a ternary-quinary tree expansion to prove ill-posedness for DNLS below the critical regularity, controlling derivative loss via quintic nonlinearity.

Findings

Proves norm inflation in Sobolev spaces below critical regularity.

Shows sharp ill-posedness results for DNLS.

Controls derivative loss through a new analytical technique.

Abstract

In this note, we study the ill-posedness problem for the derivative nonlinear Schr\"odinger equation (DNLS) in the one-dimensional setting. More precisely, by using a ternary-quinary tree expansion of the Duhamel formula we prove norm inflation in Sobolev spaces below the (scaling) critical regularity for the gauged DNLS. This ill-posedness result is sharp since DNLS is known to be globally well-posed in $L^2(\mathbb{R})$. The main novelty of our approach is to control the derivative loss from the cubic nonlinearity by the quintic nonlinearity with carefully chosen initial data.