  Using a Fourier spectral method, we provide a detailed numerically
investigation of dispersive Schr\"odinger type equations involving a fractional
Laplacian. By an appropriate choice of the dispersive exponent, both mass and
energy sub- and supercritical regimes can be computed in one spatial dimension,
only. This allows us to study the possibility of finite time blow-up versus
global existence, the nature of the blow-up, the stability and instability of
nonlinear ground states, and the long time dynamics of solutions. The latter is
also studied in a semiclassical setting. Moreover, we numerically construct
ground state solutions to the fractional nonlinear Schr\"odinger equation.
