Nonlinear Equations with Small Parameter

358 Pages·2015·2.156 MB·English

by  Glebov, Sergei;Kiselev, Oleg M.;Tarkhanov, Nikolai

SergeyG.Glebov,OlegM.Kiselev,NikolaiN.Tarkhanov
NonlinearEquationswithSmallParameter
De Gruyter Series in Nonlinear
Analysis and Applications
EditorinChief
JürgenAppell,Würzburg,Germany
Editors
CatherineBandle,Basel,Switzerland
AlainBensoussan,Richardson,Texas,USA
AvnerFriedman,Columbus,Ohio,USA
MikioKato,Nagano,Japan
WojciechKryszewski,Torun,Poland
UmbertoMosco,Worcester,Massachusetts,USA
LouisNirenberg,NewYork,USA
SimeonReich,Haifa,Israel
AlfonsoVignoli,Rome,Italy
KatrinWendland,Freiburg,Germany
Volume 23/1
Sergey G. Glebov
Oleg M. Kiselev
Nikolai N. Tarkhanov
Nonlinear Equations
with Small Parameter
Volume 1: Oscillations and Resonances
MathematicsSubjectClassification2010
34–02,34A34,34C15,34D05,34D20,34E05,34E10,34E15,34E20,70K30
Authors
Prof.Dr.SergeyGlebov Prof.Dr.NikolaiTarkhanov
ChairofMathematics UniversitätPotsdam
UfaStatePetroleum InstitutfürMathematik
TechnologicalUniversity AmNeuenPalais10
FacultyofGeneralScientificDiscipline 14469Potsdam
KosmonavtovSt.1 Germany
Ufa450062 [email protected]
RussianFederation
[email protected]
Prof.Dr.OlegM.Kiselev
RussianAcademyofSciences
UFAScientificCentre
InstituteofMathematics
Chernyshevskystr.112
Ufa450008
RussianFederation
[email protected]
ISBN978-3-11-033554-5
e-ISBN(PDF)978-3-11-033568-2
e-ISBN(EPUB)978-3-11-038272-3
Set-ISBN978-3-11-033569-9
ISSN0941-813X
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Preface
Thebookpresentsnewmethodsofconstructionofglobalasymptoticsofsolutionsto
nonlinear equations withsmall parameter. They allow one to match asymptotics of
variouspropertieswitheachotherintransitionregionsandtogetunifiedformulas
forconnectionofcharacteristicparametersofapproximatesolutions.
This approach underlies modern asymptotic methods and gives a deep insight
intocrucialnonlinearphenomenaofnaturalsciences.Thesearebeginningsofchaos
in dynamical systems, incipient solitary and shock waves, oscillatory processes in
crystals, engineering constructions and quantum systems. Apart from independent
interest, the approximate solutions serve as a foolproof basis for testing numerical
algorithms.
The book is naturally divided into two volumes. In the first volume we discuss
asymptotic methods mostly by the example of ordinary differential equations. The
secondvolumeisdevotedtoapplicationsofasymptoticmethodsinpartialdifferential
equations.
Prerequisites for reading this book are some acquaintance with the elementary
theoryofordinarydifferentialequationsandagoodknowledgeofclassicalanalysis.
The main concepts and methods will be comprehensible for graduate students of
physicsandengineeringsciences.
Weuseandpresentmanyideasgoingbacktodiversemathematicians.Wetriedto
describethehistoricaldevelopmenttothebestofourknowledge.However,wehave
probablyfailedatmanyinstances.
Thisbookgrewoutoftheresultsofajointproject“Nonlineardifferentialequa-
tionswithsmallparameter”oftheauthorssupportedbytheGermanResearchSociety
(DFG)andRussianFoundationforBasicResearch(RFFI)in2007–2014andpartially
by Russian Scientific Foundation (14-11-00078) in 2014–2016. O.M. Kiselev was also
gratefultohiswifeIrinaforhersupportwhileworkingonthisbook.Thehelpofall
theseinstitutions,friendsandcolleagueswasinvaluable.

Contents
Introduction XV
1 Asymptoticexpansionsandseries 1
1.1 DefinitionsofAsymptoticSeriesandExamples 1
1.1.1 AnExampleofDivergentSeries 1
1.1.2 OrderOperators 4
1.1.3 CalibrationSequence.AsymptoticSeries 5
1.1.4 Problems 8
1.2 SummationofAsymptoticSeries 9
1.2.1 AsymptoticRepresentationofFunctions 9
1.2.2 TheoremontheUniquenessofAsymptoticExpansion 10
1.2.3 TheoremonExistenceofaFunctionwiththeGiven
AsymptoticExpansion 11
1.2.4 Problems 12
1.3 LaplaceMethodandGammaFunction 13
1.3.1 AsymptoticExpansionofIntegralwhenSubintegralFunction
ExponentDoesNotContainExtrema 13
1.3.2 AsymptoticExpansionofIntegral,WhenIntegrandExponent
ContainsExtrema 13
1.3.3 DerivationofIntegralFormulaforGammaFunction 15
1.3.4 Moivre–StirlingFormula 17
1.3.5 Problems 17
1.4 FresnelIntegralandStationaryPhaseMethod 17
1.4.1 RiemannLemma 18
1.4.2 FresnelIntegralFormulae 18
1.4.3 LargeValuesofArgument 19
1.4.4 MethodofStationaryPhase 21
1.4.5 Problems 23
1.5 AiryFunctionandItsAsymptoticExpansion 24
1.5.1 Airy’sEquation 24
1.5.2 AnIntegralRepresentationofGeneralSolutionforAiry
Equation 24
1.5.3 Asymptotic Expansion for the Airy Function as
z→–∞ 28
1.5.4 Saddle-PointMethodandtheAiryFunctionAsymptotic
Expansionasz→∞ 29
1.5.5 Problem 30
1.6 FunctionsofParabolicCylinder 30
1.6.1 ParabolicCylinderEquation 30
1.6.2 IntegralRepresentation 31
VIII Contents
1.6.3 ConnectionFormulaeatDifferentValuesofParameter 33
1.6.4 ValuesatOriginofCoordinates 34
1.6.5 Problems 35
1.7 WKBMethod 35
1.7.1 ApplicationofWKBMethodforOrdinaryDifferentialEquation
oftheSecondOrder 35
1.7.2 JustificationofConstructedAsymptoticExpansion 38
1.7.3 Quasi-ClassicalAsymptoticExpansionandTransition
Points 41
1.7.4 EssentiallySingularPointsofDifferentialEquationandthe
StokesPhenomenon 42
1.7.5 Problems 43
2 Asymptoticmethodsforsolvingnonlinearequations 44
2.1 FastOscillatingAsymptoticExpansionsforWeakNonlinearCase 44
2.1.1 AsymptoticSubstitution 45
2.1.2 EquationsforLeading-OrderTermandFirst-OrderCorrection
Term 46
2.1.3 EquationforthenthTermofFormalSeriesandDomainof
Validity 48
2.1.4 JustificationofAsymptoticSeries 49
2.1.5 Problems 52
2.2 BoundaryLayerMethod 52
2.2.1 AsymptoticSolution 53
2.2.2 BoundaryLayer 54
2.2.3 JustificationofAsymptoticSolution 57
2.2.4 Problems 59
2.3 CatastrophesandRegularExpansions 59
2.3.1 A Formal Series with Respect to Powers of Small
Parameter 60
2.3.2 JustificationofAsymptoticSeries 62
2.3.3 Non-analyticPerturbation 64
2.3.4 MatchingoftheAsymptoticSeriesfortheRootofCubic
Equation 65
2.3.5 CompoundAsymptoticExpansionforRootofCubic
Equation 66
2.3.6 Problems 67
2.4 WeierstrassFunction 67
2.4.1 DifferentialEquation 67
2.4.2 DoublyPeriodicity 68
2.4.3 ARepresentationbySeries 69
2.4.4 Evenness 70
Contents IX
2.4.5 Liouville’sTheorem 70
2.4.6 Problem 71
2.5 JacobiEllipticFunctions 71
2.5.1 Sin-AmplitudeFunction 71
2.5.2 Periodicity 72
2.5.3 JacobiEllipticFunctions 73
2.5.4 RegularExpansionintheNeighbourhoodofZeroValueof
Argument 75
2.5.5 RegularExpansionintheNeighbourhoodofZeroValueof
Parameter 75
2.5.6 RegularExpansionintheNeighbourhoodofk=1 76
2.5.7 Problems 77
2.6 UniformAsymptoticBehaviourofJacobi-snNearaSingularPoint.TheLost
FormulafromHandbooksforEllipticFunctions 77
2.6.1 TheAsymptoticBehaviourofthePeriod 78
2.6.2 AsymptoticBehaviouronaRegularPartofTrajectory 80
2.6.3 AsymptoticBehaviourNearTurningPoint 82
2.6.4 UniformAsymptoticExpansion 83
2.7 Mathieu’sandLame’sFunctions 84
2.7.1 Hill’sEquationandFloquet’sTheory 84
2.7.2 Examples 86
2.7.3 MathieuFunctions 87
2.7.4 ConstructionofMathieu’sFunctions 88
2.7.5 LindemanformofMathieu’sEquation 90
2.7.6 SpecialCaseofLame’sEquation 90
2.7.7 DegenerateCase 92
2.7.8 Problems 92
3 Perturbationofnonlinearoscillations 93
3.1 RegularPerturbationTheoryforNonlinearOscillations 95
3.1.1 PropertiesofSolutionsforUnperturbedEquation 95
3.1.2 FormalAsymptoticExpansionforSolution 96
3.1.3 SolutionofNonlinearEquationforPrimaryTerm 97
3.1.4 HomogeneousLinearizedEquation 97
3.1.5 Non-homogeneousLinearizedEquation 99
3.1.6 FirstCorrectionforPerturbationTheory 100
3.1.7 SecondCorrectioninFormula(3.8) 101
3.1.8 CausesLeadingtotheGrowingofCorrections 101
3.1.9 Problems 102
3.2 FastandSlowVariables 102
3.2.1 Two-ScalingMethod 102
3.2.2 IsochronousOscillations 104
X Contents
3.2.3 AnAveragingofIsochronousOscillations 105
3.2.4 TranscendentEquationforParameter 107
3.2.5 EquationsforParametersofAveraging 108
3.2.6 Problems 109
3.3 Krylov–BogolyubovMethod 109
3.3.1 AsymptoticalSubstitution 111
3.3.2 Formula for Leading-Order Term of Asymptotic
Expansion 112
3.3.3 SolutionofLinearizedEquation 113
3.3.4 PeriodicSolutionfortheFirstCorrection 115
3.3.5 Problems 118
3.4 Higher-OrderTermsinKrylov–BogolyubovMethod 118
3.4.1 SecondCorrectionofPerturbationTheory 118
3.4.2 PeriodicSolutionofEquationfornthCorrection 122
3.5 IntervalofValidityforKrylov–Bogolyubov’sAnsatz 123
3.5.1 SmallNeighbourhoodsofaCentre 123
3.5.2 NeighbourhoodofSeparatrixandSaddle 124
3.5.3 AsymptoticSolutionofaCauchyProblem 124
4 Nonlinearoscillatorinpotentialwell 128
4.1 NonlinearOscillatorNearSeparatrix 131
4.1.1 ChangetoSimpleForm 132
4.1.2 QualitativeBehaviourandNumericalAnalysis 134
4.2 AsymptoticSolutionClosetoSeparatrix 135
4.2.1 ConstructionofGermAsymptoticExpansion 135
4.2.2 BehaviourofCorrectionTermsintheNeighbourhoodofthe
Separatrix 136
4.2.3 AsymptoticExpansionNearSaddlePoint 140
4.2.4 AsymptoticSolutionintheNeighbourhoodofLower
Separatrix 141
4.2.5 ParametersofEquationandCantorSet 144
4.3 OscillationswithExternalForceintoPotentialWell 145
4.3.1 AnAsymptoticProblemofaCapture 148
4.4 Non-resonantRegionsofParameter 149
4.4.1 AnEquationforAveragedAction 149
4.4.2 TheSubstitutionofKrylov–Bogolyubov 151
4.4.3 LinearizedEquation 152
4.4.4 ConstructionoftheFirstCorrectionTerm 154
4.4.5 ConstructionoftheSecondCorrectionTerm 156
4.4.6 ResonancesinHigher-OrderCorrectionTerms 159
4.5 AsymptoticsinResonantRegions 159
4.5.1 FormalDerivationofNonlinearResonanceEquation 159