Spectral approach to the scattering map for the semi-classical defocusing Davey-Stewartson II equation

TL;DR

This paper introduces a numerical spectral method for solving semi-classical D-bar problems related to the inverse scattering of the defocusing Davey-Stewartson II equation, enabling efficient computation for rapidly decreasing potentials.

Contribution

It develops a spectral approach using Fourier transforms and regularization techniques to solve D-bar equations numerically in the semi-classical regime.

Findings

Effective numerical solution for small semi-classical parameters

Implementation of fixed point and GMRES methods for algebraic equations

Demonstration on examples with rapidly decreasing potentials

Abstract

The inverse scattering approach for the defocusing Davey-Stewartson II equation is given by a system of D-bar equations. We present a numerical approach to semi-classical D-bar problems for real analytic rapidly decreasing potentials. We treat the D-bar problem as a complex linear second order integral equation which is solved with discrete Fourier transforms complemented by a regularization of the singular parts by explicit analytic computation. The resulting algebraic equation is solved either by fixed point iterations or GMRES. Several examples for small values of the semi-classical parameter in the system are discussed.