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.
Author(s): Thomas Trogdon, Sheehan Olver
Series: OT146 Other Titles in Applied Mathematics 146
Publisher: SIAM
Year: 2015
Language: English
Pages: 371
I Riemann–Hilbert Problems 1
1 Classical Applications of Riemann–Hilbert Problems 3
1.1 Error function: From integral representation to Riemann–Hilbert
problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Airy function: From differential equation to Riemann–Hilbert
problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Monodromy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Jacobi operators and orthogonal polynomials . . . . . . . . . . . . . . . 13
1.6 Spectral analysis of Schrödinger operators . . . . . . . . . . . . . . . . . 16
2 Riemann–Hilbert Problems 23
2.1 Precise statement of a Riemann–Hilbert problem . . . . . . . . . . . . 23
2.2 Hölder theory of Cauchy integrals . . . . . . . . . . . . . . . . . . . . . . 25
2.3 The solution of scalar Riemann–Hilbert problems . . . . . . . . . . . . 34
2.4 The solution of some matrix Riemann–Hilbert problems . . . . . . . 40
2.5 Hardy spaces and Cauchy integrals . . . . . . . . . . . . . . . . . . . . . . 44
2.6 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.7 Singular integral equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.8 Additional considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3 Inverse Scattering and Nonlinear Steepest Descent 87
3.1 The inverse scattering transform . . . . . . . . . . . . . . . . . . . . . . . 88
3.2 Nonlinear steepest descent . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
II Numerical Solution of Riemann–Hilbert Problems 107
4 Approximating Functions 109
4.1 The discrete Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . 109
4.2 Chebyshev series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.3 Mapped series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.4 Vanishing bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5 Numerical Computation of Cauchy Transforms 125
5.1 Convergence of approximation of Cauchy transforms . . . . . . . . . 126
5.2 The unit circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.3 Case study: Computing the error function . . . . . . . . . . . . . . . . . 130
5.4 The unit interval and square root singularities . . . . . . . . . . . . . . 131
5.5 Case study: Computing elliptic integrals . . . . . . . . . . . . . . . . . . 135
5.6 Smooth functions on the unit interval . . . . . . . . . . . . . . . . . . . . 136
5.7 Approximation of Cauchy transforms near endpoint singularities . . 144
6 The Numerical Solution of Riemann–Hilbert Problems 155
6.1 Projection methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.2 Collocation method for RH problems . . . . . . . . . . . . . . . . . . . 160
6.3 Case study: Airy equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.4 Case study: Monodromy of an ODE with three singular points . . . 168
7 Uniform Approximation Theory for Riemann–Hilbert Problems 173
7.1 A numerical Riemann–Hilbert framework . . . . . . . . . . . . . . . . 175
7.2 Solving an RH problem on disjoint contours . . . . . . . . . . . . . . . 177
7.3 Uniform approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
7.4 A collocation method realization . . . . . . . . . . . . . . . . . . . . . . . 187
III The Computation of Nonlinear Special Functions and Solutions of
Nonlinear PDEs 191
8 The Korteweg–de Vries and Modified Korteweg–de Vries Equations 193
8.1 The modified Korteweg–de Vries equation . . . . . . . . . . . . . . . . . 202
8.2 The Korteweg–de Vries equation . . . . . . . . . . . . . . . . . . . . . . . 209
8.3 Uniform approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
9 The Focusing and Defocusing Nonlinear Schrödinger Equations 231
9.1 Integrability and Riemann–Hilbert problems . . . . . . . . . . . . . . . 232
9.2 Numerical direct scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
9.3 Numerical inverse scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 237
9.4 Extension to homogeneous Robin boundary conditions on the half-
line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
9.5 Singular solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
9.6 Uniform approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
10 The Painlevé II Transcendents 253
10.1 Positive x, s 2 = 0, and 0 ≤ 1− s 1 s 3 ≤ 1 . . . . . . . . . . . . . . . . . . . . 256
10.2 Negative x, s 2 = 0, and 1− s 1 s 3 > 0 . . . . . . . . . . . . . . . . . . . . . . 258
10.3 Negative x, s 2 = 0, and s 1 s 3 = 1 . . . . . . . . . . . . . . . . . . . . . . . . 263
10.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
11 The Finite-Genus Solutions of the Korteweg–de Vries Equation 269
11.1 Riemann surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
11.2 The finite-genus solutions of the KdV equation . . . . . . . . . . . . . . 274
11.3 From a Riemann surface of genus g to the cut plane . . . . . . . . . . . 278
11.4 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
11.5 A Riemann–Hilbert problem with smooth solutions . . . . . . . . . . 284
11.6 Numerical computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
11.7 Analysis of the deformed and regularized RH problem . . . . . . . . . 297
11.8 Uniform approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
12 The Dressing Method and Nonlinear Superposition 303
12.1 A numerical dressing method for the KdV equation . . . . . . . . . . . 304
12.2 A numerical dressing method for the defocusing NLS equation . . . 315
IV Appendices 321
A Function Spaces and Functional Analysis 323
A.1 Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
A.2 Linear operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
A.3 Matrix-valued functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
B Fourier and Chebyshev Series 333
B.1 Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
B.2 Chebyshev series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
C Complex Analysis 345
C.1 Inferred analyticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
D Rational Approximation 347
D.1 Bounded contours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
D.2 Lipschitz graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
E Additional KdV Results 357
E.1 Comparison with existing numerical methods . . . . . . . . . . . . . . 357
E.2 The KdV g-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
Bibliography 363
Index 371
