Trogdon T., Olver S. Riemann-Hilbert Problems, Their Numerical Solution, and the Computation of Nonlinear Special Functions

Philadelphia: SIAM, 2015. — 371 p.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,
a discussion of applications to integrable systems, differential equations, and special function theory, and
six fundamental examples and five more sophisticated examples of the analytical and numerical Riemann–Hilbert method, each of mathematical or physical significance or both.Riemann–Hilbert Problems
Classical Applications of Riemann–Hilbert Problems
Error function: From integral representation to Riemann–Hilbert problem
Elliptic integrals
Airy function: From differential equation to Riemann–Hilbert problem
Monodromy
Jacobi operators and orthogonal polynomials
Spectral analysis of Schrödinger operators
Riemann–Hilbert Problems
Precise statement of a Riemann–Hilbert problem
Hölder theory of Cauchy integrals
The solution of scalar Riemann–Hilbert problems
The solution of some matrix Riemann–Hilbert problems
Hardy spaces and Cauchy integrals
Sobolev spaces
Singular integral equations
Additional considerations
Inverse Scattering and Nonlinear Steepest Descent
The inverse scattering transform
Nonlinear steepest descentNumerical Solution of Riemann–Hilbert Problems
Approximating Functions
The discrete Fourier transform
Chebyshev series
Mapped series
Vanishing bases
Numerical Computation of Cauchy Transforms
Convergence of approximation of Cauchy transforms
The unit circle
Case study: Computing the error function
The unit interval and square root singularities
Case study: Computing elliptic integrals
Smooth functions on the unit interval
Approximation of Cauchy transforms near endpoint singularities
The Numerical Solution of Riemann–Hilbert Problems
Projection methods
Collocation method for RH problems
Case study: Airy equation
Case study: Monodromy of an ODE with three singular points
Uniform Approximation Theory for Riemann–Hilbert Problems
A numerical Riemann–Hilbert framework
Solving an RH problem on disjoint contours
Uniform approximation
A collocation method realizationThe Computation of Nonlinear Special Functions and Solutions of
Nonlinear PDEs
The Korteweg–de Vries and Modified Korteweg–de Vries Equations
The modified Korteweg–de Vries equation
The Korteweg–de Vries equation
Uniform approximation
The Focusing and Defocusing Nonlinear Schrödinger Equations
Integrability and Riemann–Hilbert problems
Numerical direct scattering
Numerical inverse scattering
Extension to homogeneous Robin boundary conditions on the half-line
Singular solutions
Uniform approximation
The Painlevé II Transcendents
Positive x, s 2 = 0, and 0 ≤ 1− s 1 s 3 ≤ 1
Negative x, s 2 = 0, and 1− s 1 s 3 > 0
Negative x, s 2 = 0, and s 1 s 3 = 1
Numerical results
The Finite-Genus Solutions of the Korteweg–de Vries Equation
Riemann surfaces
The finite-genus solutions of the KdV equation From a Riemann surface of genus g to the cut plane Regularization
A Riemann–Hilbert problem with smooth solutions
Numerical computation
Analysis of the deformed and regularized RH problem Uniform approximation
The Dressing Method and Nonlinear Superposition
A numerical dressing method for the KdV equation
A numerical dressing method for the defocusing NLS equation
Appendices
Function Spaces and Functional Analysis
Banach spaces
Linear operators
Matrix-valued functions
Fourier and Chebyshev Series
Fourier series
Chebyshev series
Complex Analysis
Inferred analyticity
Rational Approximation
Bounded contours
Lipschitz graphs
Additional KdV Results
Comparison with existing numerical methods
The KdV g-function