Table Of ContentVersion4.6
February2003
NotforCirculation
ThomasKappeler
Universita¨tZu¨rich,Institutfu¨rMathematik
Winterthurerstrasse190,CH-8057Zu¨rich
[email protected]
Ju¨rgenPo¨schel
MathematischesInstitutA,Universita¨tStuttgart
Pfaffenwaldring57,D-70569Stuttgart
[email protected]
Thomas Ju¨rgen
Kappeler Po¨schel
KdV
&
KAM
Finalstversion(cid:13)c Kappeler&Po¨schel
In
memory
of
JU¨RGENMOSER
teacher
mentor
friend
Preface
Thisbookisconcernedwithtwoaspectsofthetheoryofintegrablepartialdifferential
equations. The first aspect is a normal form theory for such equations, which we
exemplify by the periodic Korteweg de Vries equation – undoubtedly one of the
mostimportantnonlinear,integrablepdes.Thismakesforthe‘KdV’partofthetitle
ofthebook.
The second aspect is a theory for Hamiltonian perturbations of such pdes. Its
prototypeisthesocalledKAMtheory,developedforfinitedimensionalsystemsby
Kolmogorov, Arnold and Moser. This makes for the ‘KAM’ part of the title of the
book.
Tobemorespecific,ourstartingpointistheperiodicKdVequationconsidered
as an infinite dimensional, integrable Hamiltonian system admitting a complete set
of independent integrals in involution. We show that this leads to a single, global,
realanalyticsystemof Birkhoffcoordinates–the cartesian versionofaction-angle
coordinates –, such that the KdV Hamiltonian becomes a function of the actions
alone. In fact, these coordinates work simultaneously for all Hamiltonians in the
KdVhierarchy.
While the existence of global Birkhoff coordinates is a special feature of KdV,
localBirkhoffcoordinatesmaybeconstructedviaourapproachformanyintegrable
pdesanywhereinphasespace.Specificallythisholdstrueforthedefocusingnonlin-
earSchro¨dingerequation,forwhichparallelresultsweredevelopedin[51].
The global coordinates make it evident that all solutions of the periodic KdV
equation are periodic, quasi-periodic, or almost-periodic in time. It also provides a
convenienthandletostudysmallHamiltonianperturbations,byapplyingasuitable
generalizationoftheKAMtheorytopartialdifferentialoperators.Tochecktheperti-
nentnondegeneracyconditions,weconstructBirkhoffnormalformsuptoordersix
togainsufficientcontrolovertheKdVfrequenciesasfunctionsoftheactions.Infact,
theseBirkhoffnormalformsarejustthefirsttermsinthepowerseriesexpansionof
theKdVHamiltonianinBirkhoffcoordinates.
Finally,wedescribethesetup,assumptionsandconclusionsofageneralinfinite
dimensional KAM theorem, that is applicable here and goes back to Kuksin. The
situationdiffersfrommoreconventionalapplicationsofKAMtopdesinthattheper-
X Preface
turbationsaregivenbyunboundedoperators.Thisisonlypartiallycompensatedby
asmoothingeffectofthesmalldivisors.Inaddition,onehastomodifytheiteration
schemeandusenormalformswhichalsodependonangularvariables.
Only recently, monographs on KAM theory for integrable pdes appeared, by
Bourgain [17], Craig [29], and Kuksin [75]. Of these, the first two choose a dif-
ferentapproach,settingupafunctionalequationandapplyingaLyapunov-Schmidt
decompositionschemepioneeredbyCraig&Wayne[28].Thelatteremploysanor-
malformtheoryforLax-integrablepdesnearfinitedimensionaltori,whichisbased
on the Its-Matveev formula. In contrast, the normal form theory presented in this
bookwithitsglobalfeaturesgoesmuchfurther.Itallowsustoobtainaperturbation
theory for KdV from an abstract KAM theorem of a particularly simple form, and
touseBirkhoffnormalformstochecktherelevantnondegeneracyconditions.More-
over,thisnormalformmightturnouttobeusefulforotherlongtimestabilityresults
forperturbedintegrablepdessuchasNekhoroshevestimates.
This book is not only intended for the handful of specialists working at the in-
tersection of integrable partial differential equations and Hamiltonian perturbation
theory,butalsoresearchersfartherawayfromthesefields.Infact,itisourintention
toreachouttograduatestudentsaswell.Itisforthisreasonthatfirstofall,wehave
included a chapter on the classical theory, describing the finite dimensional back-
groundofintegrableHamiltoniansystemsandtheirperturbationtheoryaccordingto
thetheoryinitiatedbyKolmogorov,ArnoldandMoser.
Secondly, we made the book self-contained, omitting only those proofs which
canbefoundinwellknowntextbooks.Wethereforeincludednumerousappendices
–someofthem,wehope,ofindependentinterest–ontopicsfromcomplexanalysis
onHilbertspaces,spectraltheoryofSchro¨dingeroperators,Riemannsurfacetheory,
representationofholomorphicdifferentials,andcertainaspectsoftheKdVequation
suchastheKdVhierarchyandnewformulasfortheKdVfrequencies.
Thirdly, we wrote the book in a modular manner, where each of its five main
chapters–chaptersIItoVI–aswellasitsappendicesmaybereadindependentlyof
eachother.Everychapterhasitsownintroduction,andthenotationisexplained.Asa
result,thereissomenaturalrepetitionandoverlapamongthem.Moreover,theresults
of these chapters are summarized in the very first chapter, titled “The Beginning”,
andheretoowetookthelibertytoquotefromtheintroductionstothelaterchapters.
Weconsidertheserepetitionsabenefitforthereaderratherthananuisance,sinceit
allowshim,orher,toperusethematerialinanonlinearmanner.
Thisbooktookmanyyearstocomplete,andduringthislongtimewebenefitted
from discussions and collaborations with many friends and colleagues. We would
like to thank all of them, in particular Benoˆıt Gre´bert, with whom we developped
parallelresultsforthedefocusingnonlinearSchro¨dingerequationin[51],andJu¨rg
Kramer,forhiscontributiontothenondegeneracyresultforthefirstKdVHamilton-
ian.MostofallweareindebtedtoJu¨rgenMoser,whoinitiatedthisjointeffortand
neverfailedtoencourageusaslongashewasabletodoso.Wededicatethisbook
tohim.
ThesecondauthoralsogratefullyacknowledgesthehospitalityoftheForschungs-
institut at the ETH Zu¨rich and the Institute of Mathematics at the University of
Preface XI
Zu¨richduringmanyperiodsofourcollaborativeefforts,aswellasthesupportofthe
Deutsche Forschungsgemeinschaft, while the first author gratefully acknowledges
thesupportoftheSwissNationalScienceFoundatonandoftheEuropeanResearch
TrainingNetworkHPRN-CT-1999-00118.
FinallywewouldliketothankJulesHobbesforhisnevertiringTEXpertisefrom
theveryfirstlinesthroughmany,manyrevisionsuptothefinal,press-readyoutput,
andJu¨rgenJostandSpringerVerlagfortheirpleasantcooperationtomakethisbook
happen.
Last, but not least we thank our families for their patience and support during
thesemanyyears.
Zu¨rich/Stuttgart
February14/16,2003 TK/JP
Contents
ChapterI TheBeginning
1 Overview 1
ChapterII ClassicalBackground
2 HamiltonianFormalism 19
3 LiouvilleIntegrableSystems 27
4 BirkhoffIntegrableSystems 34
5 KAMTheory 39
ChapterIII BirkhoffCoordinates
6 BackgroundandResults 51
7 Actions 63
8 Angles 69
9 CartesianCoordinates 74
10 OrthogonalityRelations 85
11 TheDiffeomorphismProperty 91
12 TheSymplectomorphismProperty 102
ChapterIV PerturbedKdVEquations
13 TheMainTheorems 111
14 BirkhoffNormalForm 118
15 GlobalCoordinatesandFrequencies 127
16 TheKAMTheorem 133
17 ProofoftheMainTheorems 139
XIV Contents
ChapterV TheKAMProof
18 SetUpandSummaryofMainResults 145
19 TheLinearizedEquation 152
20 TheKAMStep 160
21 IterationandConvergence 165
22 TheExcludedSetofParameters 171
ChapterVI Kuksin’sLemma
23 Kuksin’sLemma 177
ChapterVII BackgroundMaterial
A Analyticity 187
B Spectra 194
C KdVHierarchy 207
ChapterVIII Psi-FunctionsandFrequencies
D ConstructionofthePsi-Functions 211
E ATraceFormula 223
F Frequencies 227
ChapterIX BirkhoffNormalForms
G TwoResultsonBirkhoffNormalForms 233
H BirkhoffNormalFormofOrder6 240
I Kramer’sLemma 248
J NondegeneracyoftheSecondKdVHamiltonian 252
ChapterX SomeTechnicalities
K SymplecticFormalism 257
L InfiniteProducts 260
M AuxiliaryResults 262
References 267
Index 275
Notations
