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

Riemann-Hilbert problems are fundamental objects of study within complex analysis. Many problems in differential equations and integrable systems, probability and random matrix theory, and asymptotic analysis can be solved by reformulation as a Riemann-Hilbert problem.

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