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