High precision numerical approach for the Davey-Stewartson II equation for Schwartz class initial data

TL;DR

This paper introduces a high-precision numerical method for solving the DS II equation with Schwartz class initial data, achieving machine precision by regularizing the nonlocal singular symbol, surpassing standard Fourier methods.

Contribution

The paper develops a regularized Fourier-based numerical approach for DS II that attains machine precision, applicable to both integrable and non-integrable cases, improving accuracy over existing methods.

Findings

Achieves machine precision accuracy for DS II equations.

Effectively handles nonlocal, nonlinear Schrödinger equations.

Provides test cases including a doubly periodic solution.

Abstract

We present an efficient high-precision numerical approach for the Davey-Stewartson (DS) II equation, treating initial data from the Schwartz class of smooth, rapidly decreasing functions. As with previous approaches, the presented code uses discrete Fourier transforms for the spatial dependence and Driscoll's composite Runge-Kutta method for the time dependence. Since the DS equation is a nonlocal, nonlinear Schr\"odinger equation with a singular symbol for the nonlocality, standard Fourier methods in practice only reach accuracies of the order of $10^{-6}$ or less for typical examples. This was previously demonstrated for the defocusing case by comparison with a numerical approach for DS via inverse scattering. By applying a regularization to the singular symbol, originally developed for D-bar problems, the presented code is shown to reach machine precision. The code can treat integrable and non-integrable DS II equations. Moreover, it has the same %complexity numerical complexity as existing codes for DS II. Several examples for the integrable defocusing DS II equation are discussed as test cases. In an appendix by C.~Kalla, a doubly periodic solution to the defocusing DS II equation is presented, providing a test for direct DS codes based on Fourier methods.