Global existence of solutions for a subcritical nonlinear Schrödinger equation

We consider the one dimensional nonlinear Schrödinger equation {iut+12uxx=N(u,u¯),x∈R,t>1,u(1,x)=u0(x),x∈R, where the nonlinearity N(u,u¯)=|u|−2γu3=u3−2γu¯−2γ, the exponent γ>0γ>0 is sufficiently small. Our purpose in this paper is to prove a global existence in time of small solutions.