We prove the global existence of analytic solutions to the Cauchy problem for nonlinear Schrödinger equations in two dimensions, where the nonlinearity behaves as a cubic power at the origin and the Cauchy data are small and decay exponentially at infinity.
"Analytic smoothing effect for nonlinear Schrödinger equation in two space dimensions." Osaka J. Math. 51 (3) 609 - 619, July 2014.