Space-time analytic smoothing effect for the nonlinear Schrödinger equations with nonlinearity of exponential type

Abstract

. In this paper, we consider the global Cauchy problem for the nonlinear Schr¨odinger equations with nonlinearity of exponential type in higher space dimensions n (cid:2) 2 . In particular, we study the global existence of the solutions to the Cauchy problem with small data in the framework of intersection of Sobolev and weighted Lebesgue space: H n / 2 ∩ F H n / 2 . More precisely, we show that if data decay exponentially in H n / 2 ∩ F H n / 2 then for any time t (cid:3) = 0 , solutions are real-analytic in both space and time variables and have analytic continuation.