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

2009/10/23 by Li, Yongsheng, Wu, Yifei
#35Q55 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.0910.4537

Abstract

We study the initial value problem of the quadratic nonlinear Schrödinger equation iu_t+u_xx=uu, where u:\R× \R→ \C. We prove that it's locally well-posed in H^s(\R) when s≥ -\dfrac14 and ill-posed when s< -\dfrac14, which improve the previous work in \citeKPV. Moreover, we consider the problem in the following space, H^s,a(\R)=u:‖u‖_H^s,a\triangleq (∫ (|ξ|^sχ_\|ξ|>1\+|ξ|^aχ_\|ξ|≤ 1\)²|u(ξ)|² dξ)^1/2\dfrac12. It remains the cases on the line segment: a=\dfrac12, -\dfrac12≤ s≤ 0 open in this paper.

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