  We present the first detailed numerical study of the semiclassical limit of
the Davey-Stewartson II equations both for the focusing and the defocusing
variant. We concentrate on rapidly decreasing initial data with a single hump.
The formal limit of these equations for vanishing semiclassical parameter
$\epsilon$, the semiclassical equations, are numerically integrated up to the
formation of a shock. The use of parallelized algorithms allows to determine
the critical time $t_{c}$ and the critical solution for these $2+1$-dimensional
shocks. It is shown that the solutions generically break in isolated points
similarly to the case of the $1+1$-dimensional cubic nonlinear Schr\"odinger
equation, i.e., cubic singularities in the defocusing case and square root
singularities in the focusing case. For small values of $\epsilon$, the full
Davey-Stewartson II equations are integrated for the same initial data up to
the critical time $t_{c}$. The scaling in $\epsilon$ of the difference between
these solutions is found to be the same as in the $1+1$ dimensional case,
proportional to $\epsilon^{2/7}$ for the defocusing case and proportional to
$\epsilon^{2/5}$ in the focusing case. We document the Davey-Stewartson II
solutions for small $\epsilon$ for times much larger than the critical time
$t_{c}$. It is shown that zones of rapid modulated oscillations are formed near
the shocks of the solutions to the semiclassical equations. For smaller
$\epsilon$, the oscillatory zones become smaller and more sharply delimited to
lens shaped regions. Rapid oscillations are also found in the focusing case for
initial data where the singularities of the solution to the semiclassical
equations do not coincide.
