The University of Arizona

The Numerical Solution of Riemann-Hilbert Problems

The Numerical Solution of Riemann-Hilbert Problems

Series: Mathematical Physics and Probability Seminar
Location: Math 402
Presenter: Joseph Gibney, University of Arizona

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.

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