Table Of Contentlectures on the Theory of
Group Properties of
Differential Equations
微分方程群性质理论讲义
Edited by N. H. Ibragimov
也 1lt奇'J:P.ι~l
HIGHEREOυCAτlONPRESS
•
NONLINEAR PHYSICAL SCIENCE
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NONLINEAR PHYSICAL SCIENCE
NonlinearPhysicalSciencefocusesonrecentadvancesoffundamentaltheoriesand
principles,analyticalandsymbolicapproaches,aswellascomputationaltechniques
innonlinearphysicalscienceandnonlinearmathematicswithengineeringapplica-
tions.
TopicsofinterestinNonlinearPhysicalScienceincludebutarenotlimitedto:
-Newfindingsanddiscoveriesinnonlinearphysicsandmathematics
-Nonlinearity,complexityandmathematicalstructuresinnonlinearphysics
-Nonlinearphenomenaandobservationsinnatureandengineering
-Computationalmethodsandtheoriesincomplexsystems
-Liegroupanalysis,newtheoriesandprinciplesinmathematicalmodeling
-Stability,bifurcation,chaosandfractalsinphysicalscienceandengineering
-Nonlinearchemicalandbiologicalphysics
-Discontinuity,synchronizationandnaturalcomplexityinthephysicalsciences
SERIES EDITORS
AlbertC.J.Luo NailH.Ibragimov
Department of Mechanical and Industrial DepartmentofMathematicsandScience
Engineering BlekingeInstituteofTechnology
SouthernIllinoisUniversityEdwardsville S-37179Karlskrona,Sweden
Edwardsville,IL62026-1805,USA Email:[email protected]
Email:[email protected]
INTERNATIONAL ADVISORY BOARD
PingAo,UniversityofWashington,USA;Email:[email protected]
JanAwrejcewicz,TheTechnicalUniversityofLodz,Poland;Email:[email protected]
EugeneBenilov,UniversityofLimerick,Ireland;Email;[email protected]
EshelBen-Jacob,TelAvivUniversity,Israel;Email:[email protected]
MauriceCourbage,Universite´Paris7,France;Email:[email protected]
MarianGidea,NortheasternIllinoisUniversity,USA;Email:[email protected]
JamesA.Glazier,IndianaUniversity,USA;Email:[email protected]
ShijunLiao,ShanghaiJiaotongUniversity,China;Email:[email protected]
JoseAntonioTenreiroMachado,ISEP-InstituteofEngineeringofPorto,Portugal;Email:[email protected]
NikolaiA.Magnitskii,RussianAcademyofSciences,Russia;Email:[email protected]
JosepJ.Masdemont,UniversitatPolitecnicadeCatalunya(UPC),Spain;Email:[email protected]
DmitryE.Pelinovsky,McMasterUniversity,Canada;Email:[email protected]
SergeyPrants,V.I.Il’ichevPacificOceanologicalInstituteoftheRussianAcademyofSciences.Russia;
Email:[email protected]
VictorI.Shrira,KeeleUniversity,UK;Email:[email protected]
JianQiaoSun,UniversityofCalifornia,USA;Email:[email protected]
Abdul-MajidWazwaz,SaintXavierUniversity,USA;Email:[email protected]
PeiYu,TheUniversityofWesternOntario,Canada;Email:[email protected]
L.y. Ovsyannikov
Lectures on the Theory of
Group Properties of
Differential Equations
微分方程群'性质理论讲义
Weifen Fangcheng Qunxingzhi Lilun Jiangyi
Edited by N.H. Ibragimov
Translated by E.D. Avdonina, N.H. Ibragimov
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Editor’s preface
WhenIstudiedatNovosibirskStateUniversity(Russian)Iwasluckytohavesuch
brilliantteachersinmathematicsas M.A.Lavrentyev,S.L.Sobolev,A.I.Mal’tsev,
Yu.G.Reshetnyakandothers.ButitwereL.V.Ovsyannikov’slecturesinOrdinary
differentialequations,Partialdifferentialequations,GasdynamicsandGroupprop-
erties ofdifferentialequationsthat wereofthemostbenefitforme.I attendedhis
course“Grouppropertiesofdifferentialequations”whenIwasathird-yearstudent.
His lectures provided a clear introduction to Lie group methods for determining
symmetries of differentialequations and a variety of their applications in gas dy-
namicsandothernonlinearmodelsaswellasOvsyannikov’sremarkablecontribu-
tiontothisclassicalsubject.Hislectureswerespectacularnotonlyduetothebril-
liantpresentationofthematerialbutalsoduetoabsolutelynewdiscoveriesforus.I
rememberoneofourmostemotionalstudent’srepeatedexclamationslike“Wonder-
ful,...Incredible!”everytimewhenOvsyannikovrevealedmostunusualproperties
ofsymmetriesorunexpectedmethods.
His lecture notes of this course were published in 1966 with the print of 300
copiesonly.SincethentheNoteswereneitherreprintednortranslatedintoEnglish,
thoughtheycontainthematerialthatisusefulforstudentsandteachersbutcannotbe
foundinmoderntexts.Forexample,theoryofpartiallyinvariantsolutionsdeveloped
byOvsyannikovandpresentedinx3.5,x3.6isusefulforinvestigatingmathematical
models described bysystems ofnonlineardifferentialequations.Itis importantto
makethisclassicaltextavailabletoeveryoneinterestedinmoderngroupanalysis.
In order to adapt the text for modern students I made several minor changes
intheEnglishtranslation.Inparticular,sectionshavebeendividedintosubsections
andfewmisprintshavebeencorrected.Partoftheproblemsformulatedin x3.7have
beencompletelyorpartiallysolvedsince1966.Butwedidnotmakeanycomments
onthismatterinthepresenttranslation.
January2013 NailH.Ibragimov
Preface
Thetheoryofdifferentialequationshastwoaspectsofinvestigation,namelylocal
andglobal,nomatterwhethertheequationsarisefromappliedproblemsofphysics
and mechanics or from abstract speculations (which is rather frequent in modern
mathematics).Thelocalaspectischaracterizedbydealingwiththeinnerstructure
of a familyofsolutionsandits investigationina neighborhoodof acertainpoint.
Theglobalapproachdealswithsolutionsdefinedinsomedomainandhavingagiven
behavioronitsboundary.
It would certainlybe erroneousto opposethese directionsto each other. How-
ever,it is nogoodtoignorethedifferencesinapproacheseither.Whiletheglobal
approachnecessitatesthefunctionalanalyticapparatus,thelocalviewpointallows
onetogetalongwithalgebraicmeansonly.Abrilliantexampleofaprofoundlo-
cal consideration is the famous Cauchy-Kovalevskaya theorem which is, in fact,
an algebraic statement. Moreover,it is an easy matter to notice that the theory of
boundary valueproblemsalso makes an essential applicationof various algebraic
propertiesofthewholefamilyofsolutions.Therefore,thelocalaspectofthealge-
braictheoryofdifferentialequationsisquitevital.
Thetheoryof grouppropertiesof differentialequationsdescriedin the present
lecturenotesisatypicalexampleofalocaltheory.Itisespeciallyvaluableininves-
tigatingnonlineardifferentialequations,foritsalgorithmsacthereasreliablyasfor
linearcases.
Inspiteofthefactthatthefundamentalsofthetheoryofgrouppropertieswere
elaboratedinworksoftheNorwegianmathematicianSophusLiemorethanahun-
dredyearsago,itsdevelopmentisdesirablenowadaysaswell.
Methodologicalpeculiarityofthepresenttextisthatitsfirstchapterusesonlythe
simplestalgebraicapparatusofone-parametergroups,whichisespeciallyadvisable
forresearchersengagedinappliedfields. This allowsoneto solvethe problemof
findingagroupadmittedbyagivensystemofdifferentialequationscompletely.The
secondchapteristailoredtoprovideadeeperinsightintothesubjectresultingfrom
solvingdeterminingequations.Thegroupstructureofthefamilyofsolutionsitself
is discussed in the third chapter, which also suggests some new elements for the
theory.Thelatterarerelatedtothenotionsofapartiallyinvariantmanifoldofthe
viii Preface
group,its defectof invariance,and the problemof reductionof partiallyinvariant
solutions.Thefinalsection x3.7suggestsaqualitativeformulationofseveralprob-
lemsdemonstratingpossibilitiesforfurtherdevelopmentofthetheorywithnoclaim
tobecomplete.
Thepresentlecturenotesarewrittenhotonthetracesofaspecialcoursegiven
bytheauthorinNovosibirskStateUniversityduringthe1965/1966academicyear.
Suchapromptdecisionwasmadetohavethelecturenotespublishedbythespring
examinations.Therefore,thelecturesmayappeartobe“raw”tomanyextent,and
theauthorisreadytobecompletelyresponsibleforthat.
Thequickreleaseofthelecturenoteswouldbeimpossiblewithoutthesupportof
administrationoftheuniversity.Amajortechnicalworkinpreparingthemanuscript
was done by the students V.G. Firsov,E.Z. Borovskaya,T.E. Kuzmina,N.I. Nau-
menko, M.L. Kochubievskaya and others. The author is sincerely grateful to all
thesepeople.
Novosibirsk,Russia,May1966 L.V.Ovsyannikov
Contents
Editor’spreface .................................................... v
Preface............................................................ vii
1 One-parametercontinuoustransformationgroupsadmittedby
differentialequations ........................................... 1
1.1 One-parametercontinuoustransformationgroup................. 1
1.1.1 Definition.......................................... 1
1.1.2 Canonicalparameter ................................. 3
1.1.3 Examples........................................... 4
1.1.4 Auxiliaryfunctionsofgroups.......................... 5
1.2 Infinitesimaloperatorofthegroup ............................ 7
1.2.1 Definitionandexamples .............................. 7
1.2.2 Transformationoffunctions ........................... 9
1.2.3 Changeofcoordinates ................................ 10
1.3 Invariantsandinvariantmanifolds............................. 12
1.3.1 Invariants........................................... 12
1.3.2 Invariantmanifolds................................... 14
1.3.3 Invarianceofregularlydefinedmanifolds ................ 15
1.4 Theoryofprolongation...................................... 17
1.4.1 Prolongationofthespace ............................. 17
1.4.2 Prolongedgroup..................................... 18
1.4.3 Firstprolongationofthegroupoperator ................. 19
1.4.4 Secondprolongationofthegroupoperator ............... 22
1.4.5 Propertiesofprolongationsofoperators ................. 24
1.5 Groupsadmittedbydifferentialequations....................... 28
1.5.1 Determiningequations................................ 28
1.5.2 First-orderordinarydifferentialequations................ 30
1.5.3 Second-orderordinarydifferentialequations ............. 32
1.5.4 Heatequation ....................................... 34
1.5.5 Gasdynamicequations................................ 42
