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 -