Search results for "D-Bar problems"
showing 3 items of 3 documents
Spectral approach to the scattering map for the semi-classical defocusing Davey–Stewartson II equation
2019
International audience; The inverse scattering approach for the defocusing Davey–Stewartson II equation is given by a system of D-bar equations. We present a numerical approach to semi-classical D-bar problems for real analytic rapidly decreasing potentials. We treat the D-bar problem as a complex linear second order integral equation which is solved with discrete Fourier transforms complemented by a regularization of the singular parts by explicit analytic computation. The resulting algebraic equation is solved either by fixed point iterations or GMRES. Several examples for small values of the semi-classical parameter in the system are discussed.
On the semiclassical limit of the defocusing Davey-Stewartson II equation
2018
Inverse scattering is the most powerful tool in theory of integrable systems. Starting in the late sixties resounding great progress was made in (1+1) dimensional problems with many break-through results as on soliton interactions. Naturally the attention in recent years turns towards higher dimensional problems as the Davey-Stewartson equations, an integrable generalisation of the (1+1)-dimensionalcubic nonlinear Schrödinger equation. The defocusing Davey-Stewartson II equation, in its semi-classical limit has been shown in numerical experiments to exhibit behavior that qualitatively resembles that of its one-dimensional reduction, namely the generation of a dispersive shock wave: smooth i…
High precision numerical approach for Davey–Stewartson II type equations for Schwartz class initial data
2020
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 10−6or less for typical examples. This was previously demonstrated for the defocusing integrable case by comparison with a numerical approach for …