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

TL;DR

This paper develops a WKB-type method for analyzing the direct spectral transform of the defocusing Davey-Stewartson II equation in the semiclassical limit, combining analytical proofs with numerical evidence to understand spectral behavior.

Contribution

It introduces a novel WKB-type approach for the spectral transform and provides numerical methods and insights into the behavior of solutions for large and small spectral parameters.

Findings

The WKB method is formally valid for large spectral parameter k.

Numerical evidence supports the accuracy of the method in the semiclassical limit.

Explicit solutions of the eikonal problem are obtained for specific potentials.

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 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.