Symmetries of a class of Nonlinear Third Order Par

arXiv:solv-int/9609004v113Sep1996SymmetriesofaclassofNonlinearThirdOrderPartialDifferentialEquationsP.A.Clarkson,E.L.Mansfield&T.J.PriestleyInstituteofMathematicsandStatisticsUniversityofKentatCanterburyCanterbury,CT27NF,[e-mail]@[e-mail]tudysymmetryreductionsofaclassofnonlinearthirdorderpartialdifferentialequationsut−ǫuxxt+2κux=uuxxx+αuux+βuxuxx,(1)whereǫ,κ,αandβarearbitraryconstants.Threespecialcasesofequa-tion(1)haveappearedintheliterature,uptosomerescalings.Ineachcasetheequationhasadmittedunusualtravellingwavesolutions:theFornberg-Whithamequation,fortheparametersǫ=1,α=−1,β=3andκ=12,admitsawaveofgreatestheight,asapeakedlimitingformofthetravellingwavesolution;theRosenau-Hymanequation,forthepa-rametersǫ=0,α=1,β=3andκ=0,admitsa“compacton”solitarywavesolution;andtheFuchssteiner-Fokas-Camassa-Holmequation,fortheparametersǫ=1,α=−3andβ=2,hasa“peakon”solitarywavesolution.Acatalogueofsymmetryreductionsforequation(1)isobtainedusingtheclassicalLiemethodandthenonclassicalmethodduetoBlumanandCole.1IntroductionInthispaperweareconcernedwithsymmetryreductionsofthenonlinearthirdorderpartialdifferentialequationgivenbyΔ≡ut−ǫuxxt+2κux−uuxxx−αuux−βuxuxx=0,(1.1)whereǫ,κ,αandβarearbitraryconstants.Threespecialcasesof(1.1)haveappearedrecentlyintheliterature.Uptosomerescalings,theseare:(i),the1Fornberg-Whithamequation[28,58,59],fortheparametersǫ=1,α=−1,β=3andκ=12,(ii),theRosenau-Hymanequation[54]fortheparametersǫ=0,α=1,β=3andκ=0,and(iii),theFuchssteiner-Fokas-Camassa-Holmequation[9,10,25,27]fortheparametersǫ=1,α=−1andβ=2.TheFornberg-Whitham(FW)equationut−uxxt+ux=uuxxx−uux+3uxuxx(1.2)wasusedtolookatqualitativebehavioursofwave-breaking[58].Itadmitsawaveofgreatestheight,asapeakedlimitingformofthetravellingwavesolution[28],u(x,t)=Aexp−12|x−43t|
,whereAisanarbitraryconstant.TheRosenau-Hyman(RH)equationut=uuxxx+uux+3uxuxx.(1.3)modelstheeffectofnonlineardispersionintheformationofpatternsinliquiddrops[54].Italsohasanunusualsolitarywavesolution,knownasa“com-pacton”,u(x,t)=(−83ccos2{14(x−ct)},if|x−ct|≤2π,0,if|x−ct|2π.Thesewavesinteractproducingarippleoflowamplitudecompacton-anticompactonpairs.TheFuchssteiner-Fokas-Camassa-Holm(FFCH)equationut−uxxt+2κux=uuxxx−3uux+2uxuxx,(1.4)firstaroseintheworkofFuchssteinerandFokas[25,27]usingabi-Hamiltonianapproach;weremarkthatitisonlyimplicitlywrittenin[27]—seeequations(26e)and(30)inthispaper—thoughisexplicitlywrittendownin[25].IthasrecentlybeenrederivedbyCamassaandHolm[9]fromphysicalconsiderationsasamodelfordispersiveshallowwaterwaves.Inthecaseκ=0,itadmitsanunusualsolitarywavesolutionu(x,t)=Aexp(−|x−ct|),whereAandcarearbitraryconstants,whichiscalleda“peakon”.ALax-pair[9]andbi-Hamiltonianstructure[27]havebeenfoundfortheFFCHequation(1.4)andsoitappearstobecompletelyintegrable.RecentlytheFFCHequation(1.4)hasattractedconsiderableattention.Inadditiontotheaforementioned,otherstudiesinclude[10,21,22,23,24,26,32,41,46].TheFFCHequation(1.4)maybethoughtofasanintegrablemodificationoftheregularizedlongwave(RLW)equation[7,47]uxxt+uux−ut−ux=0,(1.5)2sometimesknownastheBenjamin-Bona-Mahoneyequation.However,incon-trastto(1.4),theRLWequation(1.5)isthoughtnottobesolvablebyinversescattering(cf.,[42]);itssolitarywavesolutionsinteractinelastically(cf.,[37])andonlyhasfinitelymanylocalconservationlaws[45].HoweverphysicallyithasmoredesirablepropertiesthanthecelebratedKorteweg-deVries(KdV)equationut+uxxx+6uux=0,(1.6)whichwasthefirstequationtobesolvedbyinversescattering[31].WeremarkthattwootherintegrablevariantsoftheRLWequation(1.5)areuxxt+2uut−ux∂−1xut−ut−ux=0,(1.7)where∂−1xf
(x)=R∞xf(y)dy,whichwasintroducedbyAblowitz,Kaup,NewellandSegur[2],anduxxt+uut−ux∂−1xut−ut−ux=0,(1.8)whichwasdiscussedbyHirotaandSatsuma[34].Wealsonotethat(1.4),withκ=12,(1.5),(1.7)and(1.8)allhavethesamelineardispersionrelationω(k)=−k/(1+k2)forthecomplexexponentialu(x,t)∼exp{i[kx+ω(k)t]}.Recently,GilsonandPickering[32]haveshownthatnoequationintheentireclassofequations(1.1)willsatisfythenecessaryconditionsofeitherthePainlev´ePDEtestduetoWeiss,TaborandCarnevale[57]orthePainlev´eODEtestduetoAblowitz,RamaniandSegur[3,4]tobesolvablebyinversescattering.However,theintegrableFFCHequation(1.4)doespossessthe“weakPainlev´e”property(cf.,[49,50]),asdoestheFWequation(1.2).Allthesespecialtravellingwavesolutionsareessentiallyexponentialsolu-tions,orsumsofexponentialsolutions,andthuswouldsuggestsomesortoflinearityinthedifferentialequation.ThisisdiscussedbyGilsonandPickering[32],whoshowthat(1.1),withα6=0andβ(1+β)6=0,canbewrittenas(βux+u∂x+ǫ∂t)(uxx−μ2u−2κ/β)=0,(1.9)where∂x≡∂/∂x,∂t≡∂/∂tandμ2=−α/(1+β),providedthatǫα+β+1=0,whichincludestheFFCHequation(1.4).Forthetravellingwavereduction,u=w(z),z=x−ct,theresultingordinarydifferentialequationis(2κ−c)w′+ǫcw′′′−ww′′′−αww′−βw′w′′=0,(1.10)where′≡d/dz,whichalsomaybefactorisedasβw′+(w−ǫc)ddz(w′′−μ2w+γ)=0,(1.11)providedthatμ2=−α1+β,β(1+β)γ−2κ(1+β)+c(1+β+αǫ)=0.3Thisincludesallthreespecialcases(1.2)–(1.4);sinceβ(1+β)isstrictlynon-zerointhesethreecasesthenasuitableγcanalwaysbefoun