Numerical scattering for the defocusing Davey-Stewartson II equation for initial data with compact support

TL;DR

This paper develops spectral algorithms using polar coordinates and Chebychev and Fourier methods to numerically solve the scattering problem for the defocusing Davey-Stewartson II equation with compactly supported initial data, demonstrating spectral convergence.

Contribution

It introduces two novel spectral algorithms for the d-bar problem in DS II, improving efficiency and accuracy for compact support initial data.

Findings

Spectral convergence achieved with exponential error decay.

Effective for small spectral parameter modulus $k$.

Provides asymptotic formula for large $|k|$ reflection coefficient.

Abstract

In this work we present spectral algorithms for the numerical scattering for the defocusing Davey-Stewartson (DS) II equation with initial data having compact support on a disk, i.e., for the solution of d-bar problems. Our algorithms use polar coordinates and implement a Chebychev spectral scheme for the radial dependence and a Fourier spectral method for the azimuthal dependence. The focus is placed on the construction of complex geometric optics (CGO) solutions which are needed in the scattering approach for DS. We discuss two different approaches: The first constructs a fundamental solution to the d-bar system and applies the CGO conditions on the latter. This is especially efficient for small values of the modulus of the spectral parameter $k$. The second approach uses a fixed point iteration on a reformulated d-bar system containing the spectral parameter explicitly, a price paid to have simpler asymptotics. The approaches are illustrated for the example of the characteristic function of the disk and are shown to exhibit spectral convergence, i.e., an exponential decay of the numerical error with the number of collocation points. An asymptotic formula for large $|k|$ is given for the reflection coefficient.