Global existence and asymptotic behavior of small solutions to nonlinear Schrödinger equations in 3D

Abstract

We study the global existence and asymptotic behavior in time of small solutions to nonlinear Schrödinger equations with quadratic nonlinearities, \begin{eqnarray*} \left\{ \begin{array}{ll} i \partial_{t} u + \frac{1}{2}\Delta u = \mathcal{N}(u,\bar u), & (t, x) \in {\mbox{\bf{R}}} \times {\mbox{\bf{R}}}^3, \\ u(0, x) = u_{0}, & x \in {\mbox{\bf{R}}}^3, \end{array} \right. \label{0.1} \end{eqnarray*} where the initial data $u_{0}$ are sufficiently small in a suitable norm, $\bar{u}$ is the complex conjugate of $u$. The nonlinear term $\mathcal{N}$ is a smooth quadratic function in the neighborhood of the origin with respect to $u$ and $\bar u$ and does not contain the term $|u|^2$. Our purpose in this paper is to show there exists a unique final state $u_{+}$ such that $$ \big \|u(t)-e^{\frac{i t}{2} \Delta} u_{+} \big \|_{L^2} \le C t^{-\frac{5}{4}}, \ \ \text{ for small $u_{0}.$}