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

371 Pages·2015·6.531 MB·English

by  Olver, Sheehan, Trogdon, Thomas D

Riemann–Hilbert Problems,
Their Numerical Solution,
and the Computation
of Nonlinear Special
Functions
OT146_Trogdon_FM.indd 1 11/12/2015 2:20:33 PM
Riemann–Hilbert Problems,
Their Numerical Solution,
and the Computation
of Nonlinear Special
Functions
Thomas Trogdon
New York University
New York, New York
Sheehan Olver
The University of Sydney
New South Wales, Australia
Society for Industrial and Applied Mathematics
Philadelphia
OT146_Trogdon_FM.indd 3 11/12/2015 2:20:33 PM
Copyright © 2016 by the Society for Industrial and Applied Mathematics
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Library of Congress Cataloging-in-Publication Data
Trogdon, Thomas D.
Riemann–Hilbert problems, their numerical solution, and the computation of nonlinear
special functions / Thomas Trogdon, New York University, New York, New York, Sheehan
Olver, The University of Sydney, New South Wales, Australia.
pages cm. -- (Other titles in applied mathematics ; 146)
Includes bibliographical references and index.
ISBN 978-1-611974-19-5
1. Riemann–Hilbert problems. 2. Differentiable dynamical systems. I. Olver, Sheehan. II.
Title.
QA379.T754 2016
515’.353--dc23
2015032776
is a registered trademark.
OT146_Trogdon_FM.indd 4 11/12/2015 2:20:33 PM
To Karen and Laurel
t
OT146_Trogdon_FM.indd 5 11/12/2015 2:20:33 PM
Contents
Preface xi
NotationandAbbreviations xv
I Riemann–HilbertProblems 1
1 ClassicalApplicationsofRiemann–HilbertProblems 3
1.1 Error function: From integral representation to Riemann–Hilbert
problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Ellipticintegrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Airyfunction: FromdifferentialequationtoRiemann–Hilbert
problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Monodromy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Jacobioperatorsandorthogonalpolynomials . . . . . . . . . . . . . . . 13
1.6 SpectralanalysisofSchrödingeroperators . . . . . . . . . . . . . . . . . 16
2 Riemann–HilbertProblems 23
2.1 PrecisestatementofaRiemann–Hilbertproblem . . . . . . . . . . . . 23
2.2 HöldertheoryofCauchyintegrals . . . . . . . . . . . . . . . . . . . . . . 25
2.3 ThesolutionofscalarRiemann–Hilbertproblems . . . . . . . . . . . . 34
2.4 ThesolutionofsomematrixRiemann–Hilbertproblems . . . . . . . 40
2.5 HardyspacesandCauchyintegrals . . . . . . . . . . . . . . . . . . . . . . 44
2.6 Sobolevspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.7 Singularintegralequations . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.8 Additionalconsiderations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3 InverseScatteringandNonlinearSteepestDescent 87
3.1 Theinversescatteringtransform . . . . . . . . . . . . . . . . . . . . . . . 88
3.2 Nonlinearsteepestdescent . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
II NumericalSolution ofRiemann–Hilbert Problems 107
4 ApproximatingFunctions 109
4.1 ThediscreteFouriertransform . . . . . . . . . . . . . . . . . . . . . . . . 109
4.2 Chebyshevseries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.3 Mappedseries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.4 Vanishingbases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
vii
viii Contents
5 NumericalComputationofCauchyTransforms 125
5.1 ConvergenceofapproximationofCauchytransforms . . . . . . . . . 126
5.2 Theunitcircle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.3 Casestudy: Computingtheerrorfunction . . . . . . . . . . . . . . . . . 130
5.4 Theunitintervalandsquarerootsingularities . . . . . . . . . . . . . . 131
5.5 Casestudy: Computingellipticintegrals . . . . . . . . . . . . . . . . . . 135
5.6 Smoothfunctionsontheunitinterval . . . . . . . . . . . . . . . . . . . . 136
5.7 ApproximationofCauchytransformsnearendpointsingularities . . 144
6 TheNumericalSolutionofRiemann–Hilbert Problems 155
6.1 Projectionmethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.2 CollocationmethodforRHproblems . . . . . . . . . . . . . . . . . . . 160
6.3 Casestudy: Airyequation . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.4 Casestudy: MonodromyofanODEwiththreesingularpoints . . . 168
7 UniformApproximation TheoryforRiemann–HilbertProblems 173
7.1 AnumericalRiemann–Hilbertframework . . . . . . . . . . . . . . . . 175
7.2 SolvinganRHproblemondisjointcontours . . . . . . . . . . . . . . . 177
7.3 Uniformapproximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
7.4 Acollocationmethodrealization . . . . . . . . . . . . . . . . . . . . . . . 187
III The Computation of Nonlinear Special Functions and Solutions of
NonlinearPDEs 191
8 TheKorteweg–deVriesandModifiedKorteweg–deVriesEquations 193
8.1 ThemodifiedKorteweg–deVriesequation . . . . . . . . . . . . . . . . . 202
8.2 TheKorteweg–deVriesequation . . . . . . . . . . . . . . . . . . . . . . . 209
8.3 Uniformapproximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
9 TheFocusingandDefocusingNonlinearSchrödingerEquations 231
9.1 IntegrabilityandRiemann–Hilbertproblems . . . . . . . . . . . . . . . 232
9.2 Numericaldirectscattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
9.3 Numericalinversescattering . . . . . . . . . . . . . . . . . . . . . . . . . . 237
9.4 ExtensiontohomogeneousRobinboundaryconditionsonthehalf-
line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
9.5 Singularsolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
9.6 Uniformapproximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
10 ThePainlevé IITranscendents 253
10.1 Positivex,s =0,and0≤1−s s ≤1 . . . . . . . . . . . . . . . . . . . . 256
2 1 3
10.2 Negativex,s =0,and1−s s >0 . . . . . . . . . . . . . . . . . . . . . . 258
2 1 3
10.3 Negativex,s =0,ands s =1 . . . . . . . . . . . . . . . . . . . . . . . . 263
2 1 3
10.4 Numericalresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
11 TheFinite-GenusSolutions oftheKorteweg–deVriesEquation 269
11.1 Riemannsurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
11.2 Thefinite-genussolutionsoftheKdVequation . . . . . . . . . . . . . . 274
11.3 FromaRiemannsurfaceofgenus g tothecutplane . . . . . . . . . . . 278
11.4 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
11.5 ARiemann–Hilbertproblemwithsmoothsolutions . . . . . . . . . . 284
Contents ix
11.6 Numericalcomputation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
11.7 AnalysisofthedeformedandregularizedRHproblem . . . . . . . . . 297
11.8 Uniformapproximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
12 TheDressingMethodandNonlinearSuperposition 303
12.1 AnumericaldressingmethodfortheKdVequation . . . . . . . . . . . 304
12.2 AnumericaldressingmethodforthedefocusingNLSequation . . . 315
IV Appendices 321
A FunctionSpacesandFunctionalAnalysis 323
A.1 Banachspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
A.2 Linearoperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
A.3 Matrix-valuedfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
B FourierandChebyshevSeries 333
B.1 Fourierseries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
B.2 Chebyshevseries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
C ComplexAnalysis 345
C.1 Inferredanalyticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
D RationalApproximation 347
D.1 Boundedcontours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
D.2 Lipschitzgraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
E AdditionalKdVResults 357
E.1 Comparisonwithexistingnumericalmethods . . . . . . . . . . . . . . 357
E.2 TheKdV g-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
Bibliography 363
Index 371
Preface
Thisbook grew out ofthe collaboration oftheauthors, which began in theSpring
of2010,andthefirstauthor’sPhDdissertation. Thesecondauthordeveloped muchof
thetheoryinPartIIduringhisJuniorResearchFellowshipatSt. John’sCollegeinOx-
ford,applyingittothePainlevéIIequationinthenonasymptoticregime. Theauthors,
togetherwithBernardDeconinck,thendevelopedthemethodologypresentedinChap-
ter 8 fortheKorteweg–deVries(KdV) equation. Theaccuracy thatis observed begged
foranexplanation, leadingtotheframeworkinChapter 7. Aroundthesametime,the
approachesforthenonlinearSchrödinger(NLS)equations,thePainlevéIItranscendents,
andthefinite-genussolutionsoftheKdVequationweredeveloped. Theseapplications,
alongwiththeoriginalKdVmethodology,makeupPartIII.Motivatedbythedifficultyin
findingacomprehensive,beginninggraduate-level referenceforRiemann–Hilbertprob-
lems (RH problems) that included the L2 theory of singular integrals, the first author
compiledmuchofPartIduringhisPhDstudiesattheUniversityofWashington.
Centraltothephilosophyofthisbookisthequestion,Whatdoesitmeanto“solve”
an equation? Themostbasicansweristoestablishexistence—theequationhasatleast
one solution —and uniqueness—theequation hasone andonly onesolution. A more
concreteansweristhatanequationissolvedifitssolutioncanbe evaluated,typicallyby
approximationinareliableandefficientway. Thiscanbeaccomplishedviaasymptotics:
thesolutiontotheequationisgivenbyanapproximationthatimproveswithaccuracyin
certain parameter regimes. Otherwise, thesolution canbeevaluated bynumerics, ase-
quenceofapproximationsthatconvergetothetruesolution. Inthecaseoflinearpartial
differentialequations(PDEs),standardsolutionsaregivenasintegralrepresentationsob-
tainedvia,say,FourierseriesorGreen’sfunctions. Integralrepresentationsarepreferable
totheoriginalPDEbecausetheysatisfyallthepropertiesofa“solution”totheequation:
1. Existenceanduniquenessgenerallyfollowdirectlyfromthewell-understoodintegra-
tiontheory.
2. Asymptoticsareachievableviathemethodofstationaryphaseorthemethodofsteepest
descent.
3. Numericsforintegralshaveawell-developedtheory,andtheintegralrepresentations
canbereadilyevaluatedviaquadrature.
Inplaceofintegralrepresentations,fundamentalintegrablenonlinearordinarydiffer-
entialequations(ODEs)andPDEshaveanRHproblem representation. RHproblems
areboundary-valueproblemsforpiecewise(orsectionally)analyticfunctionsinthecom-
plexplane. OurgoalistosolvetheseintegrablenonlinearODEsandPDEsbyutilizing
theirRHproblemrepresentationsinamanneranalogoustointegralrepresentations. In
somecases,weuseRHproblemstoestablishexistenceanduniqueness,aswellasderive
xi
xii Preface
asymptotics. But mostimportantly, wewanttobeabletoaccurately evaluatethesolu-
tionsinsideandoutsideofasymptoticregimeswithaunifiednumericalapproach.
Thestringentrequirementsweputintoourdefinitionofa“solution”forceallsolu-
tions we find to be in a very particular category: nonlinear special functions. A special
function is shorthand for a mathematical function which arises in many physical, bio-
logical,orcomputationalcontextsorinavarietyofmathematicalsettings. Anonlinear
specialfunctionisaspecialfunctionarisingfromafundamentallynonlinearsetting. For
centuries, mathematicians have been studying special functions. An important feature
thatseparatesspecialfunctionsfromotherelementaryfunctionsisthattheygenerically
takeatranscendental1form.
Thecatenary,discoveredbyLeibniz,Huygens,andBernoulliinthe1600s,describes
theshapeofafreelyhangingropeintermsofthehyperboliccousin ofthecosinefunc-
tion. Thestudyofspecialtranscendentalfunctionscontinuedwiththediscoveryofthe
Airy and Bessel functions which share similar but more complicated series representa-
tionswhencomparedtothehyperboliccosinefunction. Theseseriesrepresentationsare
oftenderivedusingadifferentialequationthatissatisfiedbythegivenfunction. Sucha
derivationsucceedsinmanycaseswhenthedifferentialequationislinear.
The19thcenturywasagoldenageforspecialfunctiontheory. Techniquesfromthe
field of complex analysis were invoked to study the so-called elliptic functions. These
functionsareofafundamentallynonlinearnature: ellipticfunctionsaresolutionsofnon-
lineardifferential equations. Theearly20thcentury marked thework ofPaul Painlevé
andhiscollaboratorsinidentifyingtheso-calledPainlevétranscendents.ThePainlevétran-
scendentsare solutionsof nonlineardifferential equationsthat possess important prop-
erties in the complex plane. Independent of their mathematical properties, which are
describedatlengthin[52],thePainlevétranscendentshavefounduseintheasymptotic
study ofwater wavemodels [4,32,35] and in statisticalmechanics and random matrix
theory[109,120].
ThroughthestudyofRHproblems, wediscussclassical special functions(theAiry
function,ellipticfunctions,andtheerrorfunction),canonicalnonlinearspecialfunctions
(ellipticfunctionsandthePainlevéIItranscendents),andsomenoncanonicalspecialfunc-
tions (solutions of integrable nonlinear PDEs) which we advocate for inclusion in the
pantheonofnonlinearspecialfunctionsbasedonthestructurewedescribe.
Wenowpresentthelayoutofthebooktoguidethereader. Acomprehensivetableof
notationsisgivenafterthePreface,andoptionalsectionsaremarkedwithanasterisk.
• PartIcontainsanintroductiontotheappliedtheoryofRHproblems. Chapter1
contains a survey of applications in which RH problems arise. Then Chapter 2
describes the classical development of the theory of Cauchy integrals of Hölder
continuousfunctions. ThistheoryisusedtoexplicitlysolvemanyscalarRHprob-
lems. Lebesgue andSobolev spaces areused todevelop thetheory ofsingular in-
tegralequationsinordertodealwiththematrix,ornoncommutative,case. Some
oftheseresultsarenew,whilemanyothersarecompiledfromamultitudeofrefer-
ences. Finally,themethodofnonlinearsteepestdescentdevelopedbyP.Deiftand
X.ZhouisreviewedinasimplifiedforminChapter3. Onfirstreading,manyof
theproofsinthispartcanbeomitted.
• PartIIcontainsadetaileddevelopmentofthenumericalmethodologyusedtoap-
proximate the solutions of RH problems. While there is certainly some depen-
denceofPartIIonPartI,forthemorenumericallyinclined,itcanbereadmostly
1Inthiscontext,transcendentalmeansthatthefunctioncannotbeexpressedasafinitenumberofalgebraic
steps,includingrationalpowers,appliedtoavariableorvariables[63].