Continuous Symmetries and Integrability of Discrete Equations

520 Pages·2023·5.293 MB·English

by  Decio Levi, Pavel Winternitz, Ravil I. Yamilov

Volume 38
CRM
C
R
MONOGRAPH
M
SERIES
Centre de Recherches Mathématiques
Montréal
Continuous Symmetries
and Integrability of
Discrete Equations
Decio Levi
Pavel Winternitz
Ravil I. Yamilov
Continuous Symmetries
and Integrability of
Discrete Equations

Volume 38
CRM
C
R MONOGRAPH
M
SERIES
Centre de Recherches Mathématiques
Montréal
Continuous Symmetries
and Integrability of
Discrete Equations
Decio Levi
Pavel Winternitz
Ravil I. Yamilov
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Library of Congress Cataloging-in-Publication Data
Names: Levi,D.(Decio),author. |Winternitz,Pavel,author. |Yamilov,RavilI.,1957-2020,
author.
Title: Continuoussymmetriesandintegrabilityofdiscreteequations/DecioLevi,PavelWinter-
nitz,RavilI.Yamilov.
Description: Providence, Rhode Island : American Mathematical Society, [2022] | Series: CRM
monographseries/Centre de RecherchesMath´ematiques,Montr´eal, 1065-8599; volume 38|
Includesbibliographicalreferencesandindex.
Identifiers: LCCN2022037569|ISBN9780821843543(hardcover)|ISBN9781470472382(ebook)
Subjects: LCSH: Differential equations. | Symmetry (Mathematics) | Integral equations. | Dif-
ference equations. | Discrete mathematics. | AMS: Ordinary differential equations. | Partial
differential equations. | Dynamical systems and ergodic theory. | Difference and functional
equations. | Nonassociative rings and algebras – Lie algebras and Lie superalgebras – Kac-
Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras. | Topological
groups,Liegroups–Liegroups–Infinite-dimensionalLiegroupsandtheirLiealgebras: gen-
eralproperties.
Classification: LCCQA371.L3942022|DDC515/.38–dc23/eng20221024
LCrecordavailableathttps://lccn.loc.gov/2022037569
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Contents
Foreword xi
ListofFigures xiii
ListofTables xv
Preface xvii
Acknowledgment xxi
Chapter1. Introduction 1
1. Liepointsymmetriesofdifferentialequations,theirextensionsandapplications 2
2. Whatisalattice 10
2.1. 1-dimensionallattices 10
2.2. 2-dimensionallattices 10
2.3. Differentialanddifferenceoperatorsonthelattice 13
2.4. Gridsandlatticesinthedescriptionofdifferenceequations 14
2.4.1. Cartesianlattices 14
2.4.2. Galileiinvariantlattice 15
2.4.3. Exponentiallattice 15
2.4.4. Polarcoordinatesystems 16
2.5. Clairaut–Schwarz–Youngtheoremonthelatticesanditsconsequences 18
2.5.1. Commutativityandnoncommutativityofdifferenceoperators 18
3. Whatisadifferenceequation 20
3.1. Examples 22
4. Howdowefindsymmetriesfordifferenceequations 23
4.1. Examples 26
4.1.1. LiepointsymmetriesofthediscretetimeTodalattice 26
4.1.2. LiepointsymmetriesofDΔEs 28
4.1.3. LiepointsymmetriesoftheTodalattice 30
4.1.4. ClassificationofDΔEs 32
4.1.5. LiepointsymmetriesofthetwodimensionalTodaequation 34
4.2. LiepointsymmetriespreservingdiscretizationofODEs 35
4.3. GroupclassificationandsolutionofOΔEs 38
4.3.1. SymmetriesofsecondorderODEs 38
4.3.2. Symmetriesofthethree-pointdifferenceschemes 40
4.3.3. Lagrangianformalismandsolutionsofthree-pointOΔS 44
5. Whatweleaveoutonsymmetriesinthisbook 47
6. Outlineofthebook 48
v
vi CONTENTS
Chapter 2. Integrabilityandsymmetriesofnonlineardifferentialanddifference
equationsintwoindependentvariables 51
1. Introduction 51
2. IntegrabilityofPDEs 52
2.1. Introduction 52
2.2. AllyoueverwantedtoknowabouttheintegrabilityoftheKdVequationand
itshierarchy 53
2.2.1. TheKdVhierarchy: recursionoperator 57
2.2.2. TheBäcklundtransformations,DarbouxoperatorsandBianchiidentityfor
theKdVhierarchy 61
2.2.3. TheconservationslawsfortheKdVequation 64
2.2.4. ThesymmetriesoftheKdVhierarchy 65
2.2.5. Liealgebraofthesymmetries 66
2.2.6. RelationbetweenBäcklundtransformationsandisospectralsymmetries 68
2.2.7. SymmetryreductionsoftheKdVequation 70
2.3. The cylindrical KdV, its hierarchy and Darboux and Bäcklund
transformations 71
2.4. IntegrablePDEsasinfinite-dimensionalsuperintegrablesystems 74
2.5. IntegrabilityoftheBurgersequation,theprototypeoflinearizablePDEs 77
2.5.1. BäcklundtransformationandBianchiidentityfortheBurgershierarchyof
equations 79
2.5.2. SymmetriesoftheBurgersequation 81
2.5.3. SymmetryreductionbyLiepointsymmetries 81
2.6. Generalideasonlinearization 82
2.6.1. LinearizationofPDEsthroughsymmetries 83
3. IntegrabilityofDΔEs 86
3.1. Introduction 86
3.2. TheTodalattice,theTodasystem,theTodahierarchyandtheirsymmetries 87
3.2.1. SymmetriesfortheTodahierarchy 92
3.2.2. TheLiealgebraofthesymmetriesfortheTodasystemandTodalattice 93
3.2.3. Contractionofthesymmetryalgebrasinthecontinuouslimit 96
3.2.4. BäcklundtransformationsandBianchiidentitiesfortheTodasystemand
Todalattice 97
3.2.5. RelationbetweenBäcklundtransformationsandisospectralsymmetries 100
3.2.6. SymmetryreductionofageneralizedsymmetryoftheTodasystem 102
3.2.7. TheinhomogeneousTodalattices 103
3.3. Volterra hierarchy, its symmetries, Bäcklund transformations, Bianchi
identityandcontinuouslimit 106
3.3.1. Bäcklundtransformations 108
3.3.2. Infinitedimensionalsymmetryalgebra 109
3.3.3. Contractionofthesymmetryalgebrasinthecontinuouslimit 111
3.3.4. SymmetryreductionofageneralizedsymmetryoftheVolterraequation 112
3.3.5. InhomogeneousVolterraequations 113
3.4. Discrete Nonlinear Schrödinger equation, its symmetries, Bäcklund
transformationsandcontinuouslimit 113
3.4.1. ThedNLShierarchyanditsintegrability 114
3.4.2. LiepointsymmetriesofthedNLS 117
3.4.3. GeneralizedsymmetriesofthedNLS 118
CONTENTS vii
3.4.4. ContinuouslimitofthesymmetriesofthedNLS 120
3.4.5. Symmetryreductions 121
3.5. TheDΔEBurgers 127
3.5.1. Bäcklund transformations for the DΔE Burgers and its non linear
superpositionformula 128
3.5.2. SymmetriesfortheDΔEBurgers 129
4. IntegrabilityofPΔEs 129
4.1. Introduction 129
4.2. Discrete time Toda lattice, its hierarchy, symmetries, Bäcklund
transformationsandcontinuouslimit 131
4.2.1. ConstructionofthediscretetimeTodalatticehierarchy 131
4.2.2. Isospectralandnonisospectralgeneralizedsymmetriesforthediscrete
timeTodalattice 133
4.2.3. SymmetryreductionsforthediscretetimeTodalattice. 135
4.2.4. BäcklundtransformationsandsymmetriesforthediscretetimeTodalattice.135
4.3. DiscretetimeVolterraequation 136
4.3.1. ContinuouslimitofthediscretetimeVolterraequation 137
4.3.2. SymmetriesforthediscreteVolterraequation 137
4.4. LatticeversionofthepotentialKdV,itssymmetriesandcontinuouslimit 138
4.4.1. Introduction 138
4.4.2. Solution of the discrete spectral problem associated with the lpKdV
equation 140
4.4.3. SymmetriesofthelpKdVequation 142
4.5. LatticeversionoftheSchwarzianKdV 145
4.5.1. TheintegrabilityofthelSKdVequation 146
4.5.2. PointsymmetriesofthelSKdVequation 147
4.5.3. GeneralizedsymmetriesofthelSKdVequation 148
4.6. VolterratypeDΔEsandtheABSclassification 151
4.6.1. Thederivationofthe𝑄𝑉 equation 155
4.6.2. LaxpairandBäcklundtransformationsfortheABSequations 156
4.6.3. SymmetriesoftheABSequations 158
4.7. ExtensionoftheABSclassification: Bollresults. 162
4.7.1. Independentequationsonasinglecell 164
4.7.2. Independentequationsonthe2𝐷-lattice 166
4.7.3. Examples 168
4.7.4. Thenonautonomous𝑄 equation 175
V
4.7.5. SymmetriesofBollequations 177
4.7.6. Darboux integrability of trapezoidal 𝐻4 and 𝐻6 families of lattice
equations: firstintegrals[336,345] 188
4.7.7. Darboux integrability of trapezoidal 𝐻4 and 𝐻6 families of lattice
equations: generalsolutions[336,344] 196
4.8. Integrableexampleofquad-graphequationsnotintheABSorBollclass 201
4.9. ThecompletelydiscreteBurgersequation 203
4.10. ThediscreteBurgersequationfromthediscreteheatequation 204
4.10.1. SymmetriesofthenewdiscreteBurgers 205
4.10.2. SymmetryreductionforthenewdiscreteBurgersequation 208
4.11. LinearizationofPΔEsthroughsymmetries 210
4.11.1. Examples. 212
viii CONTENTS
4.11.2. NecessaryandsufficientconditionsforaPΔEtobelinear. 217
4.11.3. Four-pointlinearizablelatticeschemes 220
Chapter3. Symmetriesasintegrabilitycriteria 225
1. Introduction 225
2. ThegeneralizedsymmetrymethodforDΔEs 230
2.1. Generalizedsymmetriesandconservationlaws 231
2.2. Firstintegrabilitycondition 239
2.3. Formalsymmetriesandfurtherintegrabilityconditions 243
2.4. Formalconserveddensity 251
2.4.1. WhytheshapeofscalarS-integrableevolutionaryDΔEsaresymmetric 256
2.4.2. DiscussionofPDEsfromthepointofviewofTheorem34 258
2.4.3. DiscussionofPΔEsfromthepointofviewofTheorem34 259
2.5. Discussionoftheintegrabilityconditions 262
2.5.1. Derivationofintegrabilityconditionsfromtheexistenceofconservation
laws 262
2.5.2. Explicitformoftheintegrabilityconditions 263
2.5.3. Constructionofconservationlawsfromtheintegrabilityconditions 264
2.5.4. Leftandrightorderofgeneralizedsymmetries 265
2.6. Hamiltonianequationsandtheirproperties 266
2.7. DiscreteMiuratransformationsandmastersymmetries 269
2.8. Generalizedsymmetriesforsystemsoflatticeequations: Todatypeequations275
2.9. IntegrabilityconditionsforrelativisticTodatypeequations 281
3. Classificationresults 288
3.1. Volterratypeequations 288
3.1.1. Examplesofclassification 288
3.1.2. Listsofequations,transformationsandmastersymmetries 292
3.2. Todatypeequations 297
3.3. RelativisticTodatypeequations 301
3.3.1. NonpointconnectionbetweenLagrangianandHamiltonianequations,and
propertiesofLagrangianequations 302
3.3.2. Hamiltonianformofrelativisticlatticeequations 306
3.3.3. Lagrangianformofrelativisticlatticeequations 308
3.3.4. Relationsbetweenthepresentedlistsofrelativisticequations 310
3.3.5. Mastersymmetriesfortherelativisticlatticeequations 312
4. Explicitdependenceonthediscretespatialvariable𝑛andtime𝑡 316
4.1. Dependenceon𝑛inVolterratypeequations 316
4.1.1. Discussionofthegeneraltheory 316
4.1.2. Examples 320
4.2. Todatypeequationswithanexplicit𝑛and𝑡dependence 324
4.3. ExampleofrelativisticTodatype 328
5. Othertypesoflatticeequations 330
5.1. ScalarevolutionaryDΔEsofanarbitraryorder 330
5.2. Multi-componentDΔEs 335
6. Completelydiscreteequations 339
6.1. GeneralizedsymmetriesforPΔEsandintegrabilityconditions 339
6.1.1. Preliminarydefinitions 339
6.1.2. Derivationofthefirstintegrabilityconditions 342
6.1.3. Integrabilityconditionsforfivepointsymmetries 345
CONTENTS ix
6.2. TestingPΔEsfortheintegrabilityandsomeclassificationresults 350
6.2.1. Asimpleclassificationproblem 350
6.2.2. Furtherapplicationofthemethodtoexamplesandclassesofequations 353
7. LinearizabilitythroughchangeofvariablesinPΔEs 360
7.1. Three-pointPΔEslinearizablebylocalandnonlocaltransformations 362
7.1.1. Linearizabilityconditions. 363
7.1.2. Classificationofcomplexmultilinearequationsdefinedonathree-point
latticelinearizablebyone-pointtransformations 366
7.1.3. LinearizabilitybyaCole–Hopftransformation 369
7.1.4. Classificationofcomplexmultilinearequationsdefinedonthreepoints
linearizablebyCole-Hopftransformation 371
7.2. Nonlinearequationsonaquad-graphlinearizablebyone-point,two-point
andgeneralizedCole–Hopftransformations 372
7.2.1. Linearizationbyone-pointtransformations 372
7.2.2. Two-pointtransformations 374
7.2.3. Linearization by a generalized Cole–Hopf transformation to an
homogeneouslinearequation 379
7.2.4. Examples 383
7.3. Results on the classification of multilinear PΔEs linearizable by point
transformationonasquarelattice 392
7.3.1. Quad-graphPΔEslinearizablebyapointtransformation. 392
7.3.2. Classification of complex autonomous multilinear quad-graph PΔEs
linearizablebyapointtransformation. 394
AppendixA. ConstructionoflatticeequationsandtheirLaxpair 397
AppendixB. Transformationgroupsforquadlatticeequations. 407
AppendixC. AlgebraicentropyofthenonautonomousBollequations 413
1. Algebraicentropytestfor𝐻4and𝐻6trapezoidalequations 413
2. AlgebraicentropyforthenonautonomousYdKNequationanditssubcases. 416
AppendixD. Translation from Russianof RIYamilov, Onthe classificationof
discreteequations,reference[841]. 421
1. Proofoftheconditions(D.2–D.4). 422
2. Nonlineardifferentialdifferenceequationssatisfyingconditions(D.2–D.4). 424
3. ListofnonlineardifferentialdifferenceequationsoftypeIsatisfyingconditions
(D.2,D.4). 425
AppendixE. Noquad-graphequationcanhaveageneralizedsymmetrygivenbythe
Narita-Itoh-Bogoyavlenskyequation 433
Bibliography 435
SubjectIndex 473