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

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