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lectures 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 (cid:154)(cid:130)5(cid:212)n(cid:137)˘ This page intentionally left blank 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 ·北京剧盹 Aωh(Jr E"it(J~ LV Ov<ya盯nikov \lail H. Ihrdgimo飞 Couocilor ofRussian Acadcrηy nfScicnce< Dcpartmcnt of Mnthcrnatics and 只~'Pnt'. Lavrcntyc\' lns{itu{c of Hydr 、d，叫到m，~< Blckingc InSlilutc ofTcchnol，唱、 Lavrentyev ?叫 15 371 79 Karlskron~. <:"'edcn Novosibirsk， 63∞ 90. Rω"" E-mail' nih(i毛hlh ~旷 F-m~il' ovs(a)hytlro.MC.nI , , 俨 2013 Highcr Edu陀alion f>r陀ss Lim ted Comp3ny. 4 Dewai Dajie. 1阳。120. R~;i;η~. l' R \'j,," 图书在版捕目 (C f P )数据微甘方程前性顶用ìt讲义== Lecturt's on the theory of grO\lp pr叩ertl<"::龟 o(di卧陀nti31 叫ll"UOn、英文 ! (俄罗斯) 跑走斯币尼科J王若北京高耳教育出版祉 2013.4 (非战忡物坪科学/罗明位 f 瑞典丁伊布柿韭'"失丰蝙) ISDN 9ï8 ï 04 036944 1 ①橄 U. ，:1)现 rn ①微分方程研究英文 '2'本群研究英立 [\' ~.v OlïS 字。 1:;2 5 小罔版术闯卡11'\ (":IP 数据枯宁( :!Ol飞)寄; 029566 号司E划l编幌子'.RF.浮，陪任编指手华英树面设计均TJ新 I<i式1号计字..灯白阿侄核对陈烟野哥哥{圣即制桥创出胶好?于 2写写数育出版社咨淘司且活 ~OO-810-0598 垃址北京市西城l主ll!外大街 4 哥同址 htrp:ltw飞.vw.hep，edu.rn ;Ij(\~偏码 100120 http;//飞飞飞V飞v.hcp.com.cn 日j 自由原4钊币犀间四月11苟明公司网上订购 http://w飞V咽..，I~ndr且C()，COl11 开 '" 787n,,"" \(192111'" \ 11 (, 11Itp://ww飞，v.landmço.com ffl 印 ?长 <).75 版次 20\3年 4 月革 l 版字数 2\0干宇印次 20\3埠 4 同事 1 府阳回| 晌书热钱。\0-585日111冉本?主如有缺页倒!lî、树荫锋精俗J♂锁 1窗咧日开耐图书销笛苦恼叮联系1湖法版权所有慢权必究 ?匆料号 3的料 ω 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