  We present numerical simulations of the defocusing nonlinear Schrodinger
(NLS) equation with an energy supercritical nonlinearity. These computations
were motivated by recent works of Kenig-Merle and Kilip-Visan who considered
some energy supercritical wave equations and proved that if the solution is {a
priori} bounded in the critical Sobolev space (i.e. the space whose homogeneous
norm is invariant under the scaling leaving the equation invariant), then it
exists for all time and scatters.
  In this paper, we numerically investigate the boundedness of the
$H^2$-critical Sobolev norm for solutions of the NLS equation in dimension five
with quintic nonlinearity.
  We find that for a class of initial conditions, this norm remains bounded,
the solution exists for long time, and scatters.
