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

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.

Numerical study of Davey-Stewartson I systems

“…In this paper, we have presented a detailed numerical study of integrable DS I equations with trivial boundary conditions at infinity for initial data from the Schwartz class of rapidly decreasing smooth functions. As in [20] we have presented a hybrid approach based on a Fourier spectral method with an analytic (up to the use of the error function) regularisation of the singular Fourier symbols. With this approach, it was possible to identify a localized stationary solution to DS I which was shown to be exponentially localized as the analytically known dromion for radiative boundary conditions.…”

“…An interesting question to be studied in the future is whether dromions also exist for non-integrable generalisations of DS I and DS II, and whether a blow-up is still observed in such cases. A first study of these questions for DS II was presented in [23] and should also be redone with the methods of [20].…”

“…the singular Fourier symbol will not lead to an exponentially decreasing numerical error if the Fourier transform is approximated via an FFT. Thus we use a hybrid approach, a combination of numerical and analytical techniques, similar to the approach in [20] for the DS II equation. Concretely we write (17)…”

Numerical study of Davey–Stewartson I systems

“…Thus, we use a hybrid approach, a combination of numerical and analytical techniques, similar to the approach in Ref. 29 for the DS II equation. Concretely, we write…”

“…In Refs. 27–29, we have shown how to regularize terms of the type () arising in the context of D‐bar equations with a hybrid approach: we subtract a singular term for which the Fourier transform can be analytically found. The term is chosen in a way that what is left is smooth within finite numerical precision, and that its Fourier transform can be numerically computed (we work here with double precision which is roughly of the order of $$10^{-16}$$).…”