Analyticity and global existence of small solutions to some nonlinear Schrödinger equations

Abstract

In this paper we will study the nonlinear Schrödinger equations:

$$\begin{gathered} i\partial _t u + \tfrac{1}{2}\Delta u = \left| u \right|^2 u, (t,x) \in \mathbb{R} \times \mathbb{R}_x^n , \hfill \\ u(0,x) = \phi (x), x \in \mathbb{R}_x^n \hfill \\ \end{gathered} $$

. It is shown that the solutions of (*) exist and are analytic in space variables fort∈ℝ∖{0} if φ(x) (∈H ^2n+1,2(ℝ ⁿ _x )) decay exponentially as |x|→∞ andn≧2.