TY - JOUR

T1 - High precision numerical approach for Davey-Stewartson II type equations for Schwartz class initial data

AU - Klein, Christian

AU - McLaughlin, Ken

AU - Stoilov, Nikola

N1 - Funding Information:
Data accessibility. This article has no additional data. Authors’ contributions. All authors have contributed equally to the writing of codes, testing them and studying examples, and have also equally contributed to the writing of the paper. Competing interests. We declare we have no competing interest. Funding. This work was partially supported by the ANR-FWF project ANuI - ANR-17-CE40-0035, the isite BFC project NAANoD, EIPHI Graduate School (contract ANR-17-EURE-0002) and by the European Union Horizon 2020 research and innovation program under the Marie Sklodowska-Curie RISE 2017 grant agreement no. 778010 IPaDEGAN. K.M. was supported in part by the National Science Foundation under grant DMS-1733967.
Publisher Copyright:
© 2020 The Author(s).

PY - 2020/7

Y1 - 2020/7

N2 - 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.

AB - 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.

KW - D-bar problems

KW - Davey-Stewartson equations

KW - Fourier spectral method

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U2 - 10.1098/rspa.2019.0864

DO - 10.1098/rspa.2019.0864

M3 - Article

AN - SCOPUS:85094633495

SN - 1364-5021

VL - 476

JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

IS - 2239

M1 - 0864

ER -