Refined probabilistic local well-posedness for a cubic Schr\"odinger half-wave equation

TL;DR

This paper establishes probabilistic local well-posedness for a cubic Schr"odinger half-wave equation in quasilinear regimes, using a refined approach to overcome challenges posed by high-low frequency interactions.

Contribution

It adapts Bringmann's method to Schr"odinger equations, providing new probabilistic well-posedness results where traditional smoothing fails.

Findings

Probabilistic local well-posedness in quasilinear regimes.

Identification of high-low frequency interaction challenges.

Discussion of ill-posedness results for the equation.

Abstract

We obtain probabilistic local well-posedness in quasilinear regimes for the Schr\"odinger half-wave equation with a cubic nonlinearity. We need to use a refined ansatz because of the lack of probabilistic smoothing in the Picard's iterations, which is due to the high-low-low frequency interactions. The proof is an adaptation of the method of Bringmann on the derivative nonlinear wave equation to Schr\"odinger-type equations. In addition, we discuss ill-posedness results for this equation.