Global Existence of Small Solutions to the Quadratic Nonlinear Schrödinger Equations in Two Space Dimensions

Abstract

We study a global existence in time of small solutions to the quadratic nonlinear Schrodinger equation in two space dimensions, \begin{equation} \left\{ \begin{array}{lc} i\partial _{t}u+\frac{1}{2}\Delta u=\mathcal{N}(u), & \quad (t,x)\in \mathbf{R}\times \mathbf{R}^{2}, u(0,x)=u_{0}(x), & x\in \mathbf{R}^{2}, \end{array} \right. \label{A} \end{equation} where \mathcal{N}(u)=\sum_{j,k=1}^{2}\left( \lambda _{jk}(\partial _{x_{j}}u)(\partial _{x_{k}}u)+\mu _{jk}(\partial _{x_{j}}\bar{u})(\partial _{x_{k}}\bar{u})\right) , $$ $\lambda _{jk},\mu _{jk}\in \mathbf{C}$. We prove that if the initial data $u_{0}$ satisfy some analyticity and smallness conditions in a suitable norm, then the solution of the Cauchy problem (0.1) exists globally in time. Furthermore we prove the existence of the usual scattering states.