On a quadratic nonlinear Schr\"odinger equation: sharp well-posedness and ill-posedness

TL;DR

This paper investigates the well-posedness of a quadratic nonlinear Schrödinger equation in various Sobolev spaces, establishing sharp thresholds for existence and uniqueness of solutions, and extends previous results with refined conditions.

Contribution

It improves the understanding of well-posedness thresholds for the quadratic NLS in standard and weighted Sobolev spaces, including new sharp bounds.

Findings

Well-posed in H^s for s ≥ -1/4

Ill-posed in H^s for s < -1/4

Extended results to H^{s,a} spaces with specific conditions

Abstract

We study the initial value problem of the quadratic nonlinear Schr\"odinger equation $$ iu_t+u_{xx}=u\bar{u}, $$ where $u:\R\times \R\to \C$. We prove that it's locally well-posed in $H^s(\R)$ when $s\geq -\dfrac{1}{4}$ and ill-posed when $s< -\dfrac{1}{4}$, which improve the previous work in \cite{KPV}. Moreover, we consider the problem in the following space, $$ H^{s,a}(\R)={u:\|u\|_{H^{s,a}}\triangleq (\displaystyle\int (|\xi|^s\chi_{\{|\xi|>1\}}+|\xi|^a\chi_{\{|\xi|\leq 1\}})^2|\hat{u}(\xi)|^2 d\xi)^{{1/2}}<\infty} $$ for $s\leq 0, a\geq 0$. We establish the local well-posedness in $H^{s,a}(\R)$ when $s\geq -\dfrac{1}{4}-\dfrac{1}{2}a$ and $a<\dfrac{1}{2}$. Also we prove that it's ill-posed in $H^{s,a}(\R)$ when $s<-\dfrac{1}{4}-\dfrac{1}{2}a$ or $a>\dfrac{1}{2}$. It remains the cases on the line segment: $a=\dfrac{1}{2}$, $-\dfrac{1}{2}\leq s\leq 0$ open in this paper.