The Numerical Solution of Riemann-Hilbert Problems
The method of nonlinear steepest descent has been a great success in the asymptotic analysis of oscillatory Riemann-Hilbert problems (RHPs) and makes possible a rigorous asymptotic analysis of nonlinear wave equations solvable by the inverse scattering method. Recently, the method of nonlinear steepest descent has been used to develop a numerical method for solving integrable PDEs such as KdV and NLS with uniform accuracy in the entire (x,t) plane. In this talk I will present some of these ideas and I will demonstrate that with a spectrally accurate discretization and a fast method for solving the related singular integral equation, it is not necessary to deform contours in order to achieve the efficient solution of oscillatory RHPs. I will use this method to implement numerically the inverse scattering transform for the defocusing NLS.