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

**Olga Assainova and Christian Klein
**

*Department of Mathematics, University of Bourgogne, Dijon, France*

**Kenneth McLaughlin**

*Department of Mathematics, Colorado State University*

**Peter D. Miller**

*Department of Mathematics, University of Michigan*

### Abstract:

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 Schrödinger 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. We provide a rigorous semiclassical analysis of the solution for real radial potentials at *k = 0*, which yields an asymptotic formula for the reflection coefficient at *k = 0* and suggests an annular structure for the solution that may be exploited when *k* is small but nonzero. The numerics also suggest that for some potentials the reflection coefficient converges pointwise as ε tends to zero to a limiting function that is supported in the domain of *k*-values on which the eikonal problem does not have a global solution. It is expected that singularities of the eikonal function play a role similar to that of turning points in the one-dimensional theory.