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

Abstract

In the present paper, we consider the Cauchy problem of a system of quadratic derivative nonlinear Schrödinger 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 regularity by using $U^2$ space and $V^2$ space which are applied to prove the well-posedness and the scattering for KP-II equation at the scaling critical regularity by Hadac, Herr and Koch (2009).

35 pages

References in corpus (4)

• Nonlinear wave interactions for the Benjamin-Ono equation
• Global well-posedness of the Benjamin-Ono equation in H^1(R)
• Null structure in a system of quadratic derivative nonlinear Schrödinger equations
• Global existence without decay for quadratic Klein-Gordon equations

Cited by in corpus (10)

• On Schrödinger systems with cubic dissipative nonlinearities of derivative type
• Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity
• Sharp bilinear estimates and its application to a system of quadratic derivative nonlinear Schrödinger equations
• Well-posedness for a system of quadratic derivative nonlinear Schrödinger equations in almost critical spaces
• Well-posedness for a system of quadratic derivative nonlinear Schrödinger equations with radial initial data
• Small data global existence for a class of quadratic derivative nonlinear Schrödinger systems in two space dimensions
• Remarks on derivative nonlinear Schrödinger systems with multiple masses
• Well-posedness for a system of quadratic derivative nonlinear Schrödinger equations with low regularity periodic initial data
• Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in four and more spatial dimensions
• Well-posedness and ill-posedness for a system of periodic quadratic derivative nonlinear Schrödinger equations