On the 1D Cubic Nonlinear Schrodinger Equation in an Almost Critical Space

TL;DR

This paper establishes local well-posedness for the 1D cubic nonlinear Schrödinger equation in a broad class of modulation spaces, extending to an almost critical endpoint space with almost global results using advanced restriction estimates.

Contribution

It proves local well-posedness in modulation spaces for all subcritical cases and extends to an almost critical endpoint space with almost global well-posedness, introducing a new endpoint restriction estimate.

Findings

Proved local well-posedness in $M_{2,p}$ for all $2 \\le p < \\infty$.

Extended results to the endpoint space $M_{2,\\infty}$ with almost global well-posedness.

Developed a new endpoint restriction estimate for the analysis.

Abstract

We obtain the local well-posedness of the one dimensional cubic nonlinear Schr\"odinger Equation for initial data in the modulation space $M_{2, p}$ for all $2\le p<\infty$, which covers all the subcritical cases from the viewpoint of scaling. Moreover, in order to approach the endpoint space $M_{2,\infty}$, we will prove the almost global well-posedness in some Orlicz-type space, which is a natural generalisation of $M_{2,p}$ for large $p$. The new ingredient is an endpoint version of the two dimensional restriction estimate.