Almost global existence for cubic nonlinear Schr\"odinger equations in one space dimension

We consider non-gauge-invariant cubic nonlinear Schr"odinger equations in one space dimension. We show that initial data of size $\epsilon$ in a weighted Sobolev space lead to solutions with sharp $L_x^{\mathrm{\infty }}$ decay up to time $\exp (C{\epsilon }^{-2})$. We also exhibit norm growth beyond this time for a specific choice of nonlinearity.