Symmetries of a class of nonlinear fourth order pa

arXiv:math/9901154v1[math.AP]1Jan1999JournalofNonlinearMathematicalPhysics1999,V.6,N1,66–98.ArticleSymmetriesofaClassofNonlinearFourthOrderPartialDifferentialEquationsPeterA.CLARKSONandThomasJ.PRIESTLEYInstituteofMathematicsandStatistics,UniversityofKentatCanterbury,Canterbury,CT27NF,UKReceivedSeptember01,1998AbstractInthispaperwestudysymmetryreductionsofaclassofnonlinearfourthorderpartialdifferentialequationsutt=κu+γu2
xx+uuxxxx+μuxxtt+αuxuxxx+βu2xx,(1)whereα,β,γ,κandμarearbitraryconstants.ThisequationmaybethoughtofasafourthorderanalogueofageneralizationoftheCamassa-Holmequation,aboutwhichtherehasbeenconsiderablerecentinterest.Furtherequation(1)isa“Boussinesq-type”equationwhicharisesasamodelofvibrationsofananharmonicmass-springchainandadmitsboth“compacton”andconventionalsolitons.Acatalogueofsym-metryreductionsforequation(1)isobtainedusingtheclassicalLiemethodandthenonclassicalmethodduetoBlumanandCole.Inparticularweobtainseveralreduc-tionsusingthenonclassicalmethodwhicharenotobtainablethroughtheclassicalmethod.1IntroductionInthispaperweareconcernedwithsymmetryreductionsofthenonlinearfourthorderpartialdifferentialequationgivenbyΔ≡utt−κu+γu2
xx−uuxxxx−μuxxtt−αuxuxxx−βu2xx=0,(1)whereα,β,γ,κandμarearbitraryconstants.ThisequationmaybethoughtofasanalternativetoageneralizedCamassa-Holmequation(cf.[24]andthereferencestherein)ut−ǫuxxt+2κux=uuxxx+αuux+βuxuxx.(2)ThisisanalogoustotheBoussinesqequation[9,10]utt=uxx+12u2
xx(3)Copyrightc1999byP.A.ClarksonandT.J.PriestleySymmetriesofaClassofNonlinearFourthOrderPartialDifferentialEquations67whichisasolitonequationsolvablebyinversescattering[1,13,14,30,71],beinganalternativetotheKorteweg-deVries(KdV)equationut=uxxx+6uux(4)anothersolitonequation,thefirsttobesolvedbyinversescattering[39].Twospecialcasesof(1)haveappearedrecentlyintheliteraturebothofwhichmodelthemotionofadensechain[62].Thefirstisobtainableviathetransformation(u,x,t)7→(2εα3u+εα2,x,t)withtheappropriatechangeofparameters,toyieldutt=α2u+α3u2
xx+εα2uxxxx+2εα3uuxxxx+2u2xx+3uxuxxx(5)withε0.ThisequationcanbethoughtofastheBoussinesqequation(3)appendedwithanonlineardispersion.Itadmitsbothconventionalsolitonsandcompactsolitonsoftencalled“compactons”.Compactonsaresolitarywaveswithacompactsupport(cf.[62,63,64,65]).Thecompactstructurestaketheformu(x,t)=3c2−2α22α3cos2n(12ε)−1/2(x−ct)o,if|x−ct|≤2π,0,if|x−ct|2π.(6)oru(x,t)=Acosn(3ε)−1/2hx−23α3
1/2tio,if|x−ct|≤2π,0,if|x−ct|2π.(7)Theseare“weak”solutionsastheydonotpossessthenecessarysmoothnessattheedges,howeverthiswouldappearnottoaffecttherobustnessofacompacton[62].Numericalexperimentsseemtoshowthatcompactonsinteractelastically,reemergingwithexactlythesamecoherentshape[65].See[48]forarecentstudyofnon-analyticsolutions,inparticularcompactonsolutions,ofnonlinearwaveequations.Thesecondequationisobtainedfromthescalingtransformation(u,x,t)7→2α3u/ε,√εx,t
,againwithappropriateparameterisation,utt=α2u+α3u2
xx+εuxxtt+2εα3uuxxxx+2u2xx+3uxuxxx(8)withε0.Thisequation,unlike(5)iswellposed.Italsoadmitsconventionalsolitonsandallowscompactonslikeu(x,t)=4c2−3α22α3cos2n(12ε)−1/2(x−ct)o,if|x−ct|≤2π,0,if|x−ct|2π,(9)oru(x,t)=Acosn(3ε)−1/2hx−32α2
1/2tio,if|x−ct|≤2π,0,if|x−ct|2π.(10)68P.A.ClarksonandT.J.PriestleyTheseagainareweaksolutions,andareverysimilartotheprevioussolutions:both(7)and(10)aresolutionswithavariablespeedlinkedtotheamplitudeofthewave,whereasboth(6)and(9)aresolutionswitharbitraryamplitudes,whilstthewavespeedisfixedbytheparametersoftheequation.TheFuchssteiner-Fokas-Camassa-Holm(FFCH)equationut−uxxt+2κux=uuxxx−3uux+2uxuxx,(11)firstaroseintheworkofFuchssteinerandFokas[34,36]usingabi-Hamiltonianapproach;weremarkthatitisonlyimplicitlywrittenin[36]–seeequations(26e)and(30)inthispaper–thoughisexplicitlywrittendownin[34].IthasrecentlybeenrederivedbyCamassaandHolm[11]fromphysicalconsiderationsasamodelfordispersiveshallowwaterwaves.Inthecaseκ=0,itadmitsanunusualsolitarywavesolutionu(x,t)=Aexp(−|x−ct|),whereAandcarearbitraryconstants,whichiscalleda“peakon”.ALax-pair[11]andbi-Hamiltonianstructure[36]havebeenfoundfortheFFCHequation(11)andsoitappearstobecompletelyintegrable.RecentlytheFFCHequation(11)hasattractedconsiderableattention.Inadditiontotheaforementioned,otherstudiesinclude[12,25,26,27,29,32,33,35,40,42,56,66].Symmetryreductionsandexactsolutionshaveseveraldifferentimportantapplicationsinthecontextofdifferentialequations.Sincesolutionsofpartialdifferentialequationsasymptoticallytendtosolutionsoflower-dimensionalequationsobtainedbysymmetryreduction,someofthesespecialsolutionswillillustrateimportantphysicalphenomena.Inparticular,exactsolutionsarisingfromsymmetrymethodscanoftenbeusedeffectivelytostudypropertiessuchasasymptoticsand“blow-up”(cf.[37,38]).Furthermore,explicitsolutions(suchasthosefoundbysymmetrymethods)canplayanimportantroleinthedesignandtestingofnumericalintegrators;thesesolutionsprovideanimportantpracticalcheckontheaccuracyandreliabilityofsuchintegrators(cf.[5,67]).TheclassicalmethodforfindingsymmetryreductionsofpartialdifferentialequationsistheLiegro