Well-posedness and scattering for a system of quadratic derivativenonlinear Schrödinger equations with low regularity initial data

Abstract

In the present paper, we consider the Cauchy problem of a system of quadratic derivative nonlinearSchrödinger equations which was introduced by M. Colin and T. Colin (2004) as a model of laser-plasma interaction.The local existence of the solution of the system in the Sobolev space $H^s$ for $s > d/2+3$ is proved by M. Colin and T. Colin.We prove the well-posedness of the system with low regularity initial data.For some cases, we also prove the well-posedness and the scattering at the scaling critical regularityby using $U^2$ space and $V^2$ space which are applied to provethe well-posedness and the scattering for KP-II equation at the scaling critical regularity by Hadac, Herr and Koch (2009).