Well- and ill-posedness of the Cauchy problem for semi-linear Schr\"odinger equations on the torus

TL;DR

This paper investigates the conditions under which the Cauchy problem for semi-linear Schrödinger equations on the torus is well-posed or ill-posed in Sobolev spaces, providing a complete characterization based on polynomial nonlinearities.

Contribution

It establishes a necessary and sufficient condition on polynomial nonlinearities for well-posedness in Sobolev spaces on the torus, and demonstrates ill-posedness via non-existence of solutions.

Findings

Characterizes well-posedness for s > 5/2 based on polynomial nonlinearities

Identifies conditions leading to ill-posedness and non-existence of solutions

Uses energy estimates and gauge transformation techniques

Abstract

We consider the Cauchy problem for semi-linear Schr\"odinger equations on the torus $\mathbb T$. We establish a necessary and sufficient condition on the polynomial nonlinearity for the Cauchy problem to be well-posed in the Sobolev space $H^s(\mathbb T)$ for $s>\frac 52$. For the well-posedness, we use the energy estimates and the gauge transformation. For the ill-posedness, we prove the non-existence of solutions to the Cauchy problem.