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

This book, the most comprehensive one to date on the applied and computational theory of Riemann-Hilbert problems, includes an introduction to computational complex analysis, an introduction to the applied theory of Riemann-Hilbert problems from an analytical and numerical perspective, and a discussion of applications to integrable systems, differential equations, and special function theory. It also includes six fundamental examples and five more sophisticated examples of the analytical and numerical Riemann-Hilbert method, each of mathematical or physical significance or both.

Audience: This book is intended for graduate students and researchers interested in a computational or analytical introduction to the Riemann-Hilbert method.

Contents: Chapter 1: Classical Applications of Riemann-Hilbert Problems; Chapter 2: Riemann-Hilbert Problems; Chapter 3: Inverse Scattering and Nonlinear Steepest Descent; Chapter 4: Approximating Functions; Chapter 5: Numerical Computation of Cauchy Transforms; Chapter 6: The Numerical Solution of Riemann-Hilbert Problems; Chapter 7: Uniform Approximation Theory for Riemann-Hilbert Problems; Chapter 8: The Korteweg-de Vries and Modified Korteweg-de Vries Equations; Chapter 9: The Focusing and Defocusing Nonlinear Schrödinger Equations; Chapter 10: The Painlevé II Transcendents; Chapter 11: The Finite-Genus Solutions of the Korteweg-de Vries Equation; Chapter 12: The Dressing Method and Nonlinear Superposition; Appendix A: Function Spaces and Functional Analysis; Appendix B: Fourier and Chebyshev Series; Appendix C: Complex Analysis; Appendix D: Rational Approximation; Appendix E: Additional KdV Results