Non-existence of solutions for the periodic cubic NLS below $L^2$

TL;DR

This paper proves that the cubic nonlinear Schrödinger equation on a circle has no solutions for initial data in certain negative Sobolev spaces below L^2, using a novel a priori bound and Fourier analysis techniques.

Contribution

It establishes the non-existence of solutions for initial data in negative Sobolev spaces below L^2 for the cubic NLS on the circle, extending understanding of solution regularity thresholds.

Findings

No solutions exist for initial data in H^s or s in (-1/8, 0)

The proof uses a short time Fourier restriction norm method

Provides a new non-existence result below L^2 regularity

Abstract

We prove non-existence of solutions for the cubic nonlinear Schr\"odinger equation (NLS) on the circle if initial data belong to $H^s(\mathbb{T}) \setminus L^2(\mathbb{T})$ for some $s \in (-\frac18, 0)$. The proof is based on establishing an a priori bound on solutions to a renormalized cubic NLS in negative Sobolev spaces via the short time Fourier restriction norm method.