On a quadratic nonlinear Schrödinger equation: sharp well-posedness and ill-posedness

Abstract

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

26 pages