Symmetries and first integrals of ordinary differe

SymmetriesandrstintegralsofordinarydierenceequationsPEHydonDepartmentofMathematicsandStatisticsUniversityofSurreyGuildford,SurreyGU25XHEnglandAbstractThispaperdescribesanewsymmetry-basedapproachtothesolutionofordinarydierenceequations.Thisapproachmakesitpossibletodevisetechniquesforsolvingdierenceequations,byadaptingexistingdierentialequationtechniques.Inparticular,weobtainanewsystematicmethodofdeterminingone-parameterLiegroupsofsymmetriesinclosedform.Thismethodenablestheusertocalculatethegeneralsolutionofagivenordinarydierenceequationforwhichsucientlymanysuchsymmetriescanbeobtained.Severalexamplesareusedtoillustratethetechniqueforbothtransitiveandintransivesymmetrygroups.Itisshownthateverylinearsecond-orderordinarydierenceequationhasaLiealgebraofsymmetrygeneratorsthatisisomorphictosl(3).Thepaperconcludeswithanewsystematicmethodforconstructingrstintegralsdirectly,whichcanbeusedevenifnosymmetriesareknown.1.IntroductionOveracenturyago,SophusLieintroducedsymmetry-basedtechniquesforsolvingordinarydierentialequations(ODEs).Lie’sapproachenablestheusertodetermineLiegroupsofsymmet-riesofagivenODE.Ifasucientlylargesymmetrygroupcanbefound,itmaybeusedtosolvetheODE.ForanintroductiontosymmetrymethodsforODEs,seeOlver(1993),Bluman&Kumei(1989),Stephani(1989),orHydon(2000).Recently,Maeda(1987)showedthatautonomoussystemsofrst-orderordinarydierenceequations(OEs)canbesimpliedorsolvedusinganextensionofLie’smethod.MaedaalsoshowedthatthelinearizedsymmetryconditionforsuchOEsamountstoasetoffunctionalequations.Ingeneral,thesearehardtosolve,butMaedadescribedtwoexamplesforwhichaveryrestrictiveansatzyieldsLiesymmetries.Gaeta(1993)usedformalseriesexpansionstoderivesomesymmetriesofthosesystemsofOEsthatarediscretizationsofcontinuoussystems.GivenanODEwithknownLiepointsymmetries,onemayaskwhetheritispossibletodiscretizetheODEinawaythatpreservesatleastsomeofthesymmetries.Dorodnitsyn(1994)describeshowthiscanbeachieved,andlistssomeclassesofOEsthathaveagivenLiegroup.Maeda’sideashavebeenextendedtononautonomoussystemsandhigher-orderOEsbyQuispel&Sahadevan(1993)andLevietal.(1997).Thesepapersdescribedierentseries-basedmethodsforobtainingsomesolutionsofthelinearizedsymmetrycondition.Seriesexpansionscanbecalculatedifthesymmetryconditionhasaxedpoint,althoughitisusuallynotobvioushowtosumtheseriestoobtainsolutionsinclosedform.Unfortunately,thewell-knownmethodforcalculatinginvariantsrequiresthesymmetrygeneratortobeinclosedform.Thisisasubstantiallimitationontheusefulnessofseries-basedtechniques.Inthecurrentpaper,weintroduceasystematicmethodforobtainingLiesymmetries(inclosedform)ofagivenOE.Thenewmethodusesthelinearizedsymmetrycondition,which1isafunctionalequation,toaderiveanassociatedsystemoflinearpartialdierentialequations.ThissystemissimilartothesystemofdeterminingequationsforLiesymmetriesofagivenODE.Moreover,havingsetupthemathematicalframeworkforthenewmethod,wendthatitenablesustotransferallofthemainsymmetrymethodsforODEsacrosstoOEs;onlyminormodicationsareneeded.Thepaperdescribessomenontrivialapplicationsoftheunderlyingtransferprinciple(whichwillbediscussedelsewhere).Anco&Bluman(1998)havedescribedaconstructivemethodforobtainingrstintegralsofODEsdirectly,withoutusingLiesymmetries.Instead,themethodusestheadjointofthelinearizedsymmetrycondition.Inx5ofthecurrentpaper,weintroduceatechniqueforobtainingrstintegralsofOEsdirectly.UnlikethemethoddescribedbyAnco&Bluman,thistechniquedoesnotusetheadjointofthelinearizedsymmetrycondition.Nevertheless,ithasmanyfeaturesincommonwiththeODEmethod,anditiseasytouse.2.SymmetriesofordinarydierenceequationsInthefollowing,weconsiderNth-orderOEsoftheformun+N=!(n;un;un+1;:::;un+N1);(2:1)where!isagivensmoothfunction.Heretheindependentvariablenisaninteger.Someauthorsprefertousexnastheindependentvariable(particularlyiftheOEarisesasadiscretizationofanODE).Itdoesnotmatterwhichnotationisused,providedthatthereisabijectionthatmapsntoxn.(N.B.Themeshpoints,xn,neednotbeuniformlyspaced.)Forsimplicity,attentionisrestrictedtoregionsinwhich!un6=0.ArstintegraloftheOE(2.1)isanon-constantfunction,=(n;un;:::;un+N1);thatisconstantonsolutionsof(2.1).Inotherwords,anon-constantfunctionisarstintegralif(n+1;un+1;:::;un+N1;!(n;un;:::;un+N1))=(n;un;:::;un+N2;un+N1):(2:2)Thisconditionholdsasanidentityinthevariablesn;un;:::;un+N1.Tosimplifythenotation,weintroducetheshiftoperator(restrictedtosolutions):S:(n;un;:::;un+N2;un+N1)7!((n+1;un+1;:::;un+N1;!(n;un;:::;un+N1)):(2:3)Theactionofthisoperatoronanyfunctionisdenedbytheactiononthefunction’sarguments:S(F(n;un;:::;un+N1))=F(Sn;Sun;:::;Sun+N1)=F(n+1;un+1;:::;!):Therefore(2.2)amountstoS=:(2:4)TheOE(2.1)hasNfunctionallyindependentrstintegrals,1;:::;N,andthegeneralsolutionof(2.1)isi=ci;i=1;:::;N;(2:5)wherec1;:::;cNarearbitraryconstants.Here\functionallyindependentmeansthattheJacobiandoesnotvanish,thatis,@(1;:::;N)@(un;:::;un+N1)6=0:(2:6)2Thisconditionensuresthat(inprinciple,atleast)eachofun;:::;un+N1canbewrittenasafunctionofn;1;:::;N.Inparticular,ifun=F