Spectral approach to D-bar problems
This work provides a high-accuracy numerical tool for solving D-bar problems, which are central to inverse scattering and integrable systems, but the method is domain-specific and incremental.
The paper presents the first numerical method for D-bar problems that achieves spectral convergence for real analytic rapidly decreasing potentials. The method is demonstrated on the Davey-Stewartson II equations, enabling testing of direct PDE solutions.
We present the first numerical approach to D-bar problems having spectral convergence for real analytic rapidly decreasing potentials. The proposed method starts from a formulation of the problem in terms of an integral equation which is solved with Fourier techniques. The singular integrand is regularized analytically. The resulting integral equation is approximated via a discrete system which is solved with Krylov methods. As an example, the D-bar problem for the Davey-Stewartson II equations is solved. The result is used to test direct numerical solutions of the PDE.