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Similarity

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    Similarity

    Commun.Theor.Phys.(Beijing,China)54(2010)PP.603608

    ?ChinesePhysicalSocietyandIOPPublishingLtdVo1.54,No.4,October15,2010 SimilaritySolutionsforGeneralizedVariableCoefficientsZakharov——Kuznetsov

    EquationunderSomeIntegrabilityConditions

    M.H.M.MoussaandRehabM.ElShiekh

    DepartmentofMathematics,FacultyofEducation,AinShamsUniversity,Roxy,Hiliopolis,Cairo,Egypt

    (ReceivedJanuary14,2010)

    AbstractInthispaper.thesymmetrymethodhasbeencarriedovertothegenerMizedvariablecoecientsZakharov-

    Kuznetsovequation.Theinfinitesimalsymmetriesandtheoptimalsystemarededucedandfromthisoptimalsystem

    sevenbasicfieldsaredetermined.andforeveryvectorfieldj

    theoptimalsystemtheadmissibleformsofthecoemcients

    arefoundandthisalsoleadsustotransformthegiyenequationjntopartialdifferentialequationsjntwovariables.

    Afterusingsomereferencedtransformationsthementionedpartialdifferentialequationseventuallyreducet0ordinary

    differentialequations.Thesearchforsolutionstothoseequationshasyieldedmanyexactsolutionsinmostcases.

    PACSnumbers:02.30.Jr

    Keywords:symmetrymethod,thegeneralizedvariablecoefficientsZakharov

    Kuznetsovequation,exactso

    lutions

    1Introduction

    ThegeneralizedvariablecoefficientsZakharovKuz

netsov(GVZK)equationisgivenby

    ?+(t)"u+()u+()

    +()z+()uz=0,(1)

    where(),(t),p(t),(?),and-r(t)arearbitraryfunc

    tionsoft.Thisequationcontainsmanyimportantequa- tionsforexamplewhen-r(t)=0,thisequationbe

    comesthecombinedKdVBurgersequationwithvariable coefficients[JandwhenZ(t)=0,andp(t)=0Eq.(1) turnstoZKequationwithvariablecoemcients.whichde

    scribethenonlineardevelopmentofion-acousticwaves inamagnetizedplasmaundertherestrictionsofsmall waveamplitude,weakdispersion,andstrongmagnetic fields[.-7]alsowhenp(t)=0,and(t)=1Eq.(1) turnstothegeneralizedvariablecoecientsZakharov

    Kuznetsovequation.IS-9]Fortheconstantcoefficientsver

    sionofEq.f1)andspecialcasesseeRefs.11016I. 2SymmetryMethod

    WebrieflyoutlinedSteinberg'ssimilaritymethodof findingexplicitsolutionsofbothlinearandnon-linearpar

    tia1differentialequations.fir]Themethodbasedonfinding thesymmetriesofthedifierentialequationsisasfollows: SupposethatthedifferentialoperatorLcanbewritten intheform

    (u)=OPu

    (),

    whereu=u(t,x)andHmaydependont,x,u,andany derivativeofaslongthederivativeofudoesnotcontain morethan(P1),tderivatives.Considerthesymmetry operatorcalledinfintesimalsymmetry,whichbeingquasi

linearpartialdifferentialoperatoroffirstorder,hasthe

    form

    )?+

    i=1

    ),u).(3)

    DefinetheFr6chetderivativeof()by

    F(L,12,)=+.

    Withthesedefinitionsinthemindweneedtofollow thefollowingsteps:(i)ComputeF(L,u,);(ii)Corn

    puteF(L,u,());(iii)SubstituteH(u)for(Opu/Ot)in F(L,u,());(iv)Setthisexpressiontozeroandperform apolynomialexpansion;fv)Solvetheresultingpartialdif- ferentialequations.0ncethissystemofpartialdifferentia1 equationsissolvedforthecoefficientsofS(u),Eq.(2)can beusedtoobtainthefunctionalformofthesolutions. 3FundamentalEquation

    ThegeneralizedvariablecoefficientsZakharov-Kuz- netsovequationcanbeexpressedintheform L(u)=72?+(?)+@).+p(t)u

    where(?),

    tionsoft.

    +()+7()!,!,=o,(5)

    (?),p(t),(),and7)arearbitraryfunc

    4DeterminationofSymmetries

    InordertofindthesymmetriesofEq.(5),wesetthe followingsymmetryoperator

    S(u)=A(x,Y,t,u)ut+B(x,Y,t,u)u

    +C(x,Y,t,u)u+E(x,Y,t,u)

    CalculatingtheFr6chetderivativeF(L,,v)ofL(u)in thedirectionofv,givenbyEq.(2),andreplacingvby

S(u)inweget

    F(L,r",s()):St+(t)+uxS

    +Z(t)[u2Sx+2uuS]

    +p(t)Sx+(),+,y(t)!,.(7)

    SubstitutingthevaluesofdifferentderivativesofS(u)in FwiththeaidofMapleprogram,wegetapolynomial expansioninuz,ut,!,,uut,...,etc.Onmakinguseof Eq.(5)inthepolynomialexpressionforF,rearranging termsofvariouspowersofderivativesofuandequating M_H.M.MoussaandRehabME1ShiekhVo1.54

    themtozero,weget

    A=A(t),B=B(t,),C=c(y),

    "=u=0,+2Ey"=0,

    2pBApt+3ABAtP=0,

    Bt+aE7EupBzBz+2fluE+8Bx

    +auB(Aflhu(Aa)tu=0,

    3Bx(AA)t=0,27c(A@t+7B=0,

    ++flu+pG.++=0.(8)

    Onsolvingsystem(8),weseethattheinfinitesimalsA, B,C,andEsatisfyingtheaboveequationsare: A=1[(al+a2)+.4],

    dr(t)

    =Q(t),

    B=alx+a5,C=a3y+a6,E=a2u,(9)

    whereai,i=1,2.,6arearbitraryconstants.Thefunc

    tionsQ),@),p@),(?),and70)aregovernedbythe

    relations:

    (al+2a2)fl(Af1)t=0,3alA(AA)=0

    2alp(Ap)t=0,(al+2a3)7(AT)t=0.(10)

    ThesymmetriesunderwhichEq.(5)isinvariantcanbe

    spannedbythefollowingsixinfinitisimalgenerators =

    r(t)

    

    0

    +

    o

    ,:

    r(t)

    

    0+u0

    ,

    =

    o

    Oy,=

    1Ot,tJ0,0.(11)

    5ClassificationofGroupInvariantSolutions

    Ingeneral,toeach8-parametersubgroupHofthefull symmetrygroupGofasystemofdifferentialequations, therewil1correspondafamilyofgroupinvariantsolutions.

    Sincetherearealmostalwaysaninfinitenumberofsuch subgroups,itisnotusuallyfeasibletolistallpossible group-invariantsolutionstothesystem.Weneedaneffec

    tive,systematicmeansofclassifyingthesesolutions,lend

    ingtoan"optima1system"ofgroupinvariantsolutions

    fromwhicheveryothersuchsolutioncanbederived.Since elementsgEGnotinthesubgroupHwilltransforinan -invariantsolutiontosomeothergroupinvariantsolu H

    tions,onlythosesolutionsnot8orelatedneedtobelisted inouroptimalsystem.

LetGbeaLiegroup.Anoptimalsystemofs

    parametersubgroupsISalistofconjIugacyinequivalent s-paraznetersubgroupswiththeprobertythatanyother subgroupisconjugatetopreciselyonesubgroupinthelist. Theproblemoffindinganoptimalsystemofsubgroupsis equivalenttothatoffindinganoptimalsystemofsub

    algebras.Foronedimensionalsubalgebras.thisclassifi

    cationproblemisessentiallythesameastheproblemof classifyingtheorbitsoftheadjointrepresentation(Olver 1986),[181wheretheadjointactionisgivenbytheLieseries 2

    Ad(exp(eVi))Yj=YjE,]+[,[,]]...,(12)

    wherel,=V5isthecommutatorforthe

    Liealgebra,andisaparameter.Toobtaintheoptiaml systemofthevectorfields(11)weshouldfirstconstruct thecommutatorTable1asfollows

    Table1Thecommutatortableofthevectorfields(11) WiththehelpoftheLieseries(12)andthecommu- tatortable,theadjointtablefortheLiealgebra(11)can beeasilyconstructedasshowninTable2.

    Table2Theadjointtable

    Toobtaintheoptiamlsystem,wenowtakeageneral element

    V=al+a2+a3+a4V4+a5+a6V6,(13)

    andsubjectittovariousadjointtransformationstosire- plifyitasmuchaspossible;thuswehavededucedthe followingbasicfieldswhichformanoptimalsystemfor thegeneralizedvariablecoefficientsZakharov——Kuznetsov

    equation(i)+0+6;(ii)(a)c+,(ii)(b)

+c;(iii)+w;(iv)(a)+m+,(iv)

    (b)+mVa;(v)+nV4;(vi)(a)V4+kV5+,

    (vi)(b)V4+;(vii)V4+f;(viii)(a)+,

    (viii)(b)V6;(ix);(X),wherea,b,c,w,m,

    n,k,andfarearbitraryconstants.Becausethediscrete symmetry(x,Y,t,)(x,Y,t,)willmap(ii)(b),(iv)

    (b),(Vi)(b),(viii)(b)to(ii)(a),(iv)(a),(vi)(a),(viii) (a)respectively,alsothegenerators(viii)(a),(ix),and (x)givetrivialcasessincetheydonotdependont,and therefore,intheoptimalsystem,weconfineourselvesto sevengeneratorsonly~

    6SimilarityReductionsandReduced

    OrdinaryDifferentialEquations

    Inordertoobtaintheinvarianttransformationineach oftheabovecaseswewritethecharacteristicequationin theform

    dtdx

    (,Y,t,)B(x,Y,t,)=----------------—一C(x,Y,t,)

    '

    (4)

    Oncethisequationissolvedfortheabovesevencasesthe invariantvariablesandthecorrespondingreductionsto partialdifferentialequationsareobtainedandbyusing somereferencetransformationsthosepartialdifferential No.4SimilaritySolutionsforGeneralizedVariableCoefficientsZakharov-KuznetsovEqua

    tionunderSomeIntegrabilityConditions605

    equationswillbereducedtoordinarydifferentialequa-coefficientsofthegivenproblem.Our

    resultsaretabu-

    tionsunderintegrabilityconditionsbetweenthevariablellatedinthefollowingTables3-5?

    Table3Theinvariantvariablesandthecorrespondingformsofthecoefficientfunctions

Cabe

    Theinvaxiantvariables Formsoftoecientfunctions (i)xF/(+.)(t)yF/(+)(t),((,n)r./(+.)(t) (ii)lnr(t)zyF(t),(e,77)r(t)

    (iii),(e,)r(t)

    (iv)(1/m)r(t)Yexp(-(1/m)P(t)),(','7)

    (v)

    (vi)

    (vii)

    Xkr(t)

    z1r(t1

    Yexp((1/n)r(t)),((,)

    ,((,)

    ,((,)

    Z(t)=klF)I1./(+.)(?),

    p(t)=k2F)r(1--a)/(+.)(t), A(t)=k3F)r(2--a)/(+.)(t), -y(t):k4F(t)r(..)/(+.)(t)

    z(t)=k~C(t)r),

    p(t)=k6F(t)r),

    A(?)=kTF-1(t)r(t),

    (t)=ksF.(t)c).

    Z(t)=kgr(t)r(t),

    p(t)=kl0F-1(t)r(t), (t)=k11F-1(t)r),

    ,y(t)=k12F.(t)r(?).

    Z(t)=k13F),p(t)=klaF) (t):k15F),

    v(t)=k16exp((2/m)P(t))r(t).

Z(t)=klTF),p(t)=klsF,(t)

    (t)=k19F),

    (t)=k20exp((2/n)F(t))F(t).

    Z(t)=k21r),p(t)=k22F)

    (t)=k23F),7(t)=k24F)

    Z(t)=k25F),p(t)=k26F)

    A(t)=k27F,(?),7(t)=k2sF)

    Table4Thecorresp0ndingreducedpartialdifferentialequation 4,(ee+k2,([1/(i+n)]e,c一【b/O+n)]叼一【0/(1+0)]l+k3,('+,,(+kif,(=0 k7jc?c+k8l?qq+k5fl'+ff~}e+fk6je(+cqjq=0,

    ku|(e(+ki2leqq+k9f}(+}lel+klofee一?T1=0,

    ki5f''?+k16(qn+klaffe+}l'1/m)j'+k14f'e1/m)rlfn=0,m?0

    k19f(''+k20|'qq+klTfl<+ff~+kl8|0I/n)nfn=01n?0

    k23ft't+k24}'qq+k21ff'+lf'+k22l(e一一k}c=0,

    k27|ejrk28|enjrk25f|e|j'+k26lt|?=0

    Table5Somereferencedtransformationstoreducetheabovepartialdifferentialequationsto

    ordenarydifferentialequations

    M.H.M.MOUSSa,andRehabM.E1ShiekhVo1.54

    7DeterminationofExactSolutions

    Now,wehavegoingtoouroriginaltask,findingexact solutionsforthereducedODEs,whichbybacksubstitu- tiongivesnewexactsolutionsforthegeneralizedvariable coefficientsZakharovKuznetsovequationasfollows:

    CaseJT0determinethesolutionfortheCIDEcorre

    spendingtothiscase,weassumethatthissolutiontakes thefollowingform

    9:Ao++

    where0,A1,42,B1,andB2arearbitraryconstanttobe determined.SubstitutingfromEq.(15)intothereduced ODEgivenbycase(i)andcollectingthevariouspowersof

    0thenequatingthemtozero,wegetsystemofalgebraic equationsintheconstantsA0,AI,A2,B1,B2,a,c1,c2, 1,2,3,and4.Solvingthissystemwiththeaidof Mapleprogram,wegetthefollowingtwosetsofsolutions Thefirstset

    0:,A1=,B2=12(k3c+4c),

    k1=k2=Ao=A2=Bo=B1=0.

    ThecorrespondingexactsolutionforEq.(1)isgivenby (,,t)=(+C.2y],r(t)

    12[k3+4(c;/c)r(?)

    +(c2/c1)y]

    Tesecondset

    (16)

    .:1,A1:,B1:2k2c1,3:,

    c11

    kl=A0=A2=B0=B2=0.

    Correspondingthesecondsetwearrivedatthefollowing solutionforthegeneralizedvariablecoefficientsZakharov- Kuznetsovequation

    (,?)=(+c2)r()+2k2(17)

    Casei)ToobtainasolutionfortheODEcorrespond

    ingthiscase.weassumethatk7=k8andk5=k6=0,

    thenwegetthefollowingformforg

    g=LambertW(exp(0)),

    whereCoisanintegrationconstant.SothatEq.(1)has thefollowingsolution

    u(x,Y,t)=-F(t)

    xLambert(Coexp(1nr(t)x+)).(18)

    Case(uoTosolvetheODEcorrespondingtothiscase,

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