A Study of the Direct Spectral Transform for the Defocusing Davey-Stewartson II Equation in the Semiclassical Limit

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.