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.