Travelling wave solutions of the generalized Benja

arXiv:0707.0760v2[nlin.SI]17Sep2007TravellingwavesolutionsofthegeneralizedBenjamin-Bona-MahonyequationP.G.Est´evezDepartamentodeF´ısicaFundamental,´AreadeF´ısicaTe´orica,UniversidaddeSalamanca,37008Salamanca,SpainS¸.Kuru∗,J.NegroandL.M.NietoDepartamentodeF´ısicaTe´orica,At´omicay´Optica,UniversidaddeValladolid,47071Valladolid,SpainAbstractAclassofparticulartravellingwavesolutionsofthegeneralizedBenjamin-Bona-Mahonyequationisstudiedsystematicallyusingthefactorizationtechnique.Then,thegeneraltravellingwavesolutionsofBenjamin-Bona-Mahonyequation,andofitsmodifiedversion,arealsorecovered.Pacs:05.45.Yv,52.35.Mw,52.35.Sb,02.30.Jr1IntroductionThegeneralizedBBM(Benjamin-Bona-Mahony)equationhasahigherordernonlinearityoftheformut+ux+aunux+uxxt=0,n≥1,(1)whereaisconstant.Thecasen=1correspondstotheBBMequationut+ux+auux+uxxt=0,(2)whichwasfirstproposedin1972byBenjaminetal[1].ThisequationisanalternativetotheKorteweg-deVries(KdV)equation,anddescribestheunidi-rectionalpropagationofsmall-amplitudelongwavesonthesurfaceofwaterin∗Onleaveofabsencefrom:DepartmentofPhysics,FacultyofScience,AnkaraUniversity06100Ankara,Turkey.Emailaddress:[e-mail](J.Negro).PreprintsubmittedtoChaos,Solitons&Fractals1February2008achannel.TheBBMequationisnotonlyconvenientforshallowwaterwavesbutalsoforhydromagneticwaves,acousticwaves,andthereforeithasmoreadvantagescomparedwiththeKdVequation.Whenn=2,Eq.(1)iscalledthemodifiedBBMequationut+ux+au2ux+uxxt=0.(3)Whenlookingfortravellingwavesolutions,theBBMandmodifiedBBMequationscanbereducedtoordinarydifferentialequationsthatpossessthePainlev´epropertyandwhichareintegrableintermsofellipticfunctions[2,3].ThegeneralizedBBMequationisalsointegrableintermsofellipticfunctions,providedthatsomerestrictionsontheparametersareimposed.RecentlymanymethodshavebeenpresentedtoobtainthetravellingwavesolutionsofthegeneralizedBBMequation:thetanh-sechandthesine-cosinemethods[4,5],anapproachbasedonbalancingprincipletoobtainsomeexplicitsolutionsintermsofellipticfunction[6],andanextendedalgebraicmethodwithsymboliccomputation[7].Ouraimistoinvestigatesystematicallythetravellingwavesolutionsoftheseequations,applyingthefactorizationtechnique[8,9].Thus,wewillgetallthepreviouslyknownsolutionsandsomenewones,supplyinganewgeneralapproach.AssumingthatthegeneralizedBBMequationhasanexactsolutionintheformofatravellingwave,thenitwillreducetoathirdorderordinarydifferentialequation(ODE).ThisequationcanbeintegratedtriviallytoasecondorderODE,whichcanbefactorizedintwoways:thefirstonebymeansofdifferentialoperatorsandthesecondonebyusingafirstintegral(thatcanalsobefactorizedintermsoffirstintegrals).ThesefactorizationsgiverisetothesamefirstorderODEthatprovidesthetravellingwavesolutionsofthenonlinearequation.ThisfirstorderODEforn=1andn=2isintegrable,butforothervaluesofn,wecanalsofindsomeparticularsolutionsbyimposingsomerestrictionsontheparameters.Thepaperisorganizedasfollows.Insection2,weintroducethefactorizationtechniquefornonlinearequations,andweshowhowtoapplyittofindthetravellingwavesolutionsofthegeneralizedBBMequation.Insection3,weconsidersomespecialcasestogetparticularsolutionsofgeneralizedBBM.Insection4,weobtainthesolutionsoftheBBMandmodifiedBBM.Finally,section5endsthepaperwithsomeconclusions.IntheAppendix,wegivealsosomeusefulinformationabouttheellipticfunctionsthatareusedintheprevioussections.22TravellingwavesofthegeneralizedBenjamin-Bona-Mahonyequa-tion2.1TravellingwavesolutionsLetusassumethatEq.(1)hasanexactsolutionintheformofatravellingwaveu(x,t)=φ(ξ),ξ=hx−ωt,(4)wherehandωarerealconstants.Ifwesubstitute(4)inEq.(1),weget−h2ωφξξξ+(h−ω)φξ+haφnφξ=0.(5)Afterintegratingwithrespecttoξ,wehaveφξξ−h−ωh2ωφ−a(n+1)hωφn+1=−R,(6)whereRisanintegrationconstant.Letusintroducethefollowinglineartrans-formationofthedependentandindependentvariablesξ=hθ,φ(ξ)=c(n+1)a!1/nW(θ)(7)whereθ=x−ctandc=ω/h.Inthisway(6)becomesthenonlinearsecondorderODEd2Wdθ2−Wn+1−kW=D,(8)wherethenewconstantsarek=1−cc,D=−Rh2ac(n+1)!1/n.(9)Therefore,ifweareinterestedinfindingthetravellingwavesolutionsof(1),wehavetosolvetheODE(8).2.2FactorizationofsomespecialtypeofnonlinearsecondorderODEInthissectionwewillintroduceafactorizationtechniqueappliedtononlinearsecondorderODEofthespecialformd2Wdθ2−βdWdθ+F(W)=0,(10)3whereF(W)isanarbitraryfunctionofWandβisconstant.Thisequationcanbefactorizedasddθ−f2(W,θ)#ddθ−f1(W,θ)#W(θ)=0(11)beingf1andf2twounknownfunctionsthatmaydependexplicitlyonWandθ.Inordertofindf1andf2,weexpand(11)d2Wdθ2−f1+f2+∂f1∂WW!dWdθ+f1f2W−W∂f1∂θ=0,(12)andthencomparingwith(10),weobtainthefollowingconsistencyconditionsf1f2=FW+∂f1∂θ,(13)f2+∂(Wf1)∂W=β.(14)Ifwefindasolutionforthisfactorizationproblem,itwillallowustowriteacompatiblefirstorderODEddθ−f1(W,θ)#W(θ)=0(15)thatprovidesa(particular)solution[8,9]tothenonlinearODE(10).Intheapplicationsofthispaperf1andf2willdependonlyonW.2.3FactorizationofthegeneralizedBBMequationWhenweapplythefactorizationtechniquedescribedabovetoEq.(8),thenwehaveβ=0,F(W)=−Wn+1−kW−D,andtheconsistencyconditionsgivenby(13)and(14)taketheformf1f2=−Wn−k−DW−1,(16)f2=−f1−W∂f1∂W.(17)Substituting(17)in