We present an efficient high-precision numerical approach for Davey–Stewartson (DS) II type equa- tions, 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 DS equations are non-local, nonlinear Schrödinger equations with a singular symbol for the non-locality, standard Fourier methods in practice only reach accuracy of the order of 10−6 or less for typical examples. This was previously demonstrated for the defocusing integrable case by comparison with a numerical approach for DS II 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 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.

中文翻译：

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用于 Schwartz 类初始数据的 Davey-Stewartson II 型方程的高精度数值方法

我们为 Davey-Stewartson (DS) II 型方程提出了一种有效的高精度数值方法，处理来自 Schwartz 类平滑、快速递减函数的初始数据。与以前的方法一样，所呈现的代码使用离散傅立叶变换来处理空间相关性，并使用 Driscoll 的复合 Runge-Kutta 方法来处理时间相关性。由于 DS 方程是非局部的非线性薛定谔方程，非局部性用奇异符号表示，标准傅立叶方法在实践中只能达到典型例子的 10-6 级或更低的精度。先前通过与通过逆散射的 DS II 数值方法进行比较，证明了散焦可积分情况。通过对最初为 D-bar 问题开发的奇异符号应用正则化，显示的代码可以达到机器精度。该代码可以处理可积和不可积的 DS II 方程。此外，它具有与 DS II 现有代码相同的数值复杂度。可积散焦 DS II 方程的几个例子作为测试案例进行了讨论。在 C. Kalla 的附录中，提出了散焦 DS II 方程的双周期解，提供了对基于傅立叶方法的直接 DS 码的测试。