Riemann-Hilbert Problems, Their Numerical Solution, and the Computation of Nonlinear Special Functions

Thomas Trogdon is an NSF Postdoctoral Fellow at the Courant Institute of Mathematical Sciences, New York University. He was awarded the 2014 SIAM Richard C. DiPrima Prize for his dissertation, which shares its title with this book. He has published in the fields of numerical analysis, approximation theory, optical physics, integrable systems, partial differential equations and random matrix theory. Sheehan Olver is a Senior Lecturer in the School of Mathematics and Statistics at the University of Sydney. Dr Olver was awarded the 2012 Adams Prize for his work on the numerical solution of Riemann–Hilbert problems. He has published in the fields of numerical analysis, approximation theory, integrable systems, oscillatory integrals, spectral methods and random matrix theory.

• Chapter 1: Classical Applications of Riemann–Hilbert ProblemsChapter 2: Riemann–Hilbert ProblemsChapter 3: Inverse Scattering and Nonlinear Steepest DescentChapter 4: Approximating FunctionsChapter 5: Numerical Computation of Cauchy TransformsChapter 6: The Numerical Solution of Riemann–Hilbert ProblemsChapter 7: Uniform Approximation Theory for Riemann–Hilbert ProblemsChapter 8: The Korteweg–de Vries and Modified Korteweg–de Vries EquationsChapter 9: The Focusing and Defocusing Nonlinear Schrödinger EquationsChapter 10: The Painlevé II TranscendentsChapter 11: The Finite-Genus Solutions of the Korteweg–de Vries EquationChapter 12: The Dressing Method and Nonlinear SuperpositionAppendix A: Function Spaces and Functional AnalysisAppendix B: Fourier and Chebyshev SeriesAppendix C: Complex AnalysisAppendix D: Rational ApproximationAppendix E: Additional KdV Results.