High precision numerical approach for Davey-Stewartson II type equations for Schwartz class initial data: Hybrid approach for DS II

Christian Klein, Ken McLaughlin, Nikola Stoilov

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

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.

Original languageEnglish (US)
Article number0864
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume476
Issue number2239
DOIs
StatePublished - Jul 2020
Externally publishedYes

Keywords

  • D-bar problems
  • Davey-Stewartson equations
  • Fourier spectral method

ASJC Scopus subject areas

  • Mathematics(all)
  • Engineering(all)
  • Physics and Astronomy(all)

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