  In this paper, we study ill-posedness of cubic fractional nonlinear
Schr\"odinger equations. First, we consider the cubic nonlinear half-wave
equation (NHW) on $\mathbb R$. In particular, we prove the following
ill-posedness results: (i) failure of local uniform continuity of the solution
map in $H^s(\mathbb R)$ for $s\in (0,\frac 12)$, and also for $s=0$ in the
focusing case; (ii) failure of $C^3$-smoothness of the solution map in
$L^2(\mathbb R)$; (iii) norm inflation and, in particular, failure of
continuity of the solution map in $H^s(\mathbb R)$, $s<0$. By a similar
argument, we also prove norm inflation in negative Sobolev spaces for the cubic
fractional NLS. Surprisingly, we obtain norm inflation above the scaling
critical regularity in the case of dispersion $|D|^\beta$ with $\beta>2$.
