  The defocusing Davey-Stewartson II equation has been shown in numerical
experiments to exhibit behavior in the semiclassical limit that qualitatively
resembles that of its one-dimensional reduction, the defocusing nonlinear
Schr\"odinger equation, namely the generation from smooth initial data of
regular rapid oscillations occupying domains of space-time that become
well-defined in the limit. As a first step to study this problem analytically
using the inverse-scattering transform, we consider the direct spectral
transform for the defocusing Davey-Stewartson II equation for smooth initial
data in the semiclassical limit. The direct spectral transform involves a
singularly-perturbed elliptic Dirac system in two dimensions. We introduce a
WKB-type method for this problem, prove that it makes sense formally for
sufficiently large values of the spectral parameter $k$ by controlling the
solution of an associated nonlinear eikonal problem, and we give numerical
evidence that the method is accurate for such $k$ in the semiclassical limit.
Producing this evidence requires both the numerical solution of the
singularly-perturbed Dirac system and the numerical solution of the eikonal
problem. The former is carried out using a method previously developed by two
of the authors and we give in this paper a new method for the numerical
solution of the eikonal problem valid for sufficiently large $k$. For a
particular potential we are able to solve the eikonal problem in closed form
for all $k$, a calculation that yields some insight into the failure of the WKB
method for smaller values of $k$. Informed by numerical calculations of the
direct spectral transform we then begin a study of the singularly-perturbed
Dirac system for values of $k$ so small that there is no global solution of the
eikonal problem.
