Theexp(−ϕ(ξ))-expansionMethodappliedtoNonlinearEvolutionEquations

Mei-meiZhao∗†,Chao-LiSchoolofMathematicsandStatistics,LanzhouUniversityLanzhou,Gansu730000,P.R.ofChina

AbstractByusingexp(−ϕ(ξ))-expansionmethod,wehaveobtainedmoretravellingwavesolu-tionstothemKdVequation,theDrinefel’d-Sokolov-Wilsonequations,theVariantBoussinesqequationsandtheCoupledSchr¨odinger-KdVsystem.Theproposedmethodalsocanbeusedformanyothernonlinearevolutionequations.

Keywordsexp(−ϕ(ξ))-expansionmethod,Homogeneousbalance,Travellingwavesolu-tions,Solitarywavesolutions,MKdVequation,Drinefel’d-Sokolov-Wilsonequations,VariantBoussinesqequations,CoupledSchr¨odinger-KdVsystem.

1IntroductionItiswellknownthatnonlinearevolutionequationsareinvolvedinmanyfieldsfromphysicstobiology,chemistry,mechanics,etc.Asmathematicalmodelsofthephenomena,theinves-tigationofexactsolutionstononlinearevolutionequationsrevealstobeveryimportantfortheunderstandingofthesephysicalproblems.Understandingthisimportance,duringthepastfourdecadesorso,manymathematiciansandphysicistshavebeingpaidspecialattentiontothedevelopmentofsophisticatedmethodsforconstructingexactsolutionstononlinearevo-lutionequations.Thus,anumberofpowerfulmethodshasbeenpresentedsuchastheinversescatteringtransform[1],theB¨acklundandtheDarbouxtransform[2-5],theHirota[6],thetrun-catedpainleveexpansion[7],thetanh-founctionexpansionanditsvariousextension[8-10],theJacobiellipticfunctionexpansion[11,12],theF-expansion[13-16],thesub-ODEmethod[17-20],thehomogeneousbalancemethod[21-23],thesine-cosinemethod[24,25],therankanal-ysismethod[26],theansatzmethod[27-29],theexp-functionexpansionmethod[30],Algebro-geometricconstructionsmethod[31]andsoon.Inthepresentpaper,weshallproposedanewmethodwhichiscalledexp(−ϕ(ξ))-expansionmethodtoseektravellingwavesolutionsofnonliearevolutionequations.the

∗CorrespondingAuthor.

†E-mailaddress:yunyun1886358@163.com(M.Zhao).

1

http://www.paper.edu.cn mainidearsoftheproposedmethodarethatthetravellingwavesolutionsofanonliearevo-lutionequationcanbeexpressedbyapolynomialinexp(−ϕ(ξ)),whereϕ(ξ)satisfiesODE(seeEq.(5)insection2),ξ=x−Vt,thedegreeofthepolynomialcanbedeterminedbycon-sideringthehomogeneousbalancebetweenthehighestorderderivativesandnonlineartermsappearinginagivennonliearevolutionequation,andthecoefficientsofthepolynomialfromtheprocessofusingtheproposedmethod.Itwillbeseenthatmoretravellingwavesolutionsofmanynonlinearevolutionequationscanbeobtainedbyusingtheexp(−ϕ(ξ))-expansionmethod.Thepaperisorganizedasfollows:Insection2,wewillrecalltheexp(−ϕ(ξ))-expansionmethod.Insection3,wewillillustratethemethodindetailwiththemKdVequation,theDrinefel’d-Sokolov-Wilsonequations,thevariantBoussinesqequationsandtheCoupledSchr¨odinger-KdVsystem.Insection7,thefeaturesoftheexp(−ϕ(ξ))-expansionmethodwillbebrieflysummarized.

2Descriptionoftheexp(−ϕ(ξ))-expansionmethodInthefollowing,wewilloutlinethemainstepsofexp(−ϕ(ξ))-expansionmethod.Consideranonlinearequation,sayintwoindependentvariablexandt,isgivenby

P(u,ut,ux,utt,uxt,uxx,...)=0,(1)whereu=u(x,t)isanunknownfunction,Pisapolynomialinu=u(x,t)anditsvariouspartialderivatives,inwhichthehighestorderderivativesandnonlineartermsareinvolved.Step1Combiningtheindependentvariablexandtintoonevariableξ=x−Vt,wesupposethat

u(x,t)=u(ξ),ξ=x−Vt,(2)thetravellingwavevariable(2)permitsusreducingEq.(1)toanODEforu=u(ξ)P(u,−Vu󰀂,u󰀂,V2u󰀂󰀂,−Vu󰀂󰀂,u󰀂󰀂,...),(3)Step2SupposethatthesolutionofODE(3)canbeexpressedbyapolynomialinexp(−ϕ(ξ))asfollows

u(ξ)=αm(exp(−ϕ(ξ)))m+...,(4)whereϕ(ξ)satisfiestheODEintheformϕ󰀂(ξ)=exp(−ϕ(ξ))+µexp(ϕ(ξ))+λ,(5)thesolutionsofODE(5)areWhenλ2−4µ>0,µ=0,

ϕ(ξ)=ln(−󰀁λ2−4µtanh(√λ2−4µ2(ξ+C1))−λ2µ),(6)2

http://www.paper.edu.cn Whenλ2−4µ>0,µ=0,ϕ(ξ)=−ln(λexp(λ(ξ+C1))−1),(7)Whenλ2−4µ=0,µ=0,λ=0,ϕ(ξ)=ln(−2(λ(ξ+C1)+2)λ2(ξ+C1)),(8)Whenλ2−4µ=0,µ=0,λ=0,ϕ(ξ)=ln(ξ+C1),(9)Whenλ2−4µ<0,

ϕ(ξ)=ln(󰀁4µ−λ2tan(√4µ−λ22(ξ+C1))−λ2µ),(10)αm,...,V,λandµareconstantstobedeterminedlater,αm=0,theunwrittenpartin(4)isalsoapolynomialinexp(−ϕ(ξ)),butthedegreeofwhichisgenerallyequaltoorlessthanm−1,thepositiveintegermcanbedeterminedbyconsideringthehomogeneousbalancebetweenthehighestorderderivativesandnonlieartermsappearinginODE(3).Step3Bysubstituting(4)intoEq.(3)andusingtheODE(5),collectingalltermswiththesameorderofexp(−ϕ(ξ))together,thelefthandsideofEq.(3)isconvertedintoanotherpolynomialinexp(−ϕ(ξ)).Equatingeachcoefficientofthispolynomialtozero,yieldsasetofalgebraicequationsforαm,...,V,λandµ.Step4Assumingthattheconstantsαm,...,V,λ,andµcanbeobtainedbysolvingthealgebraicequationsinstep3,sincethegeneralsolutionsofODE(5)havebeenwellknownforus,thensubstitutingαm,...,V,andthegeneralsolutionsofEq.(5)into(4).wehavemoretravellingwavesolutionsofnonliearevolutionequation(1).Inthesubsequentsectionswewillillustratetheproposedmethodindetailwithvariousnonliearevolutionequationsinmathematicalphysics.