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    Commun.Theor.Phys.(Beijing,China)46(2006)pp.10911100

    c?InternationalAcademicPublishersVol.46,No.6,December15,2006 ConditionalStabilityofSolitary-WaveSolutionsforGeneralizedCompoundKdV EquationandGeneralizedCompoundKdV-BurgersEquation?

    ZHANGWei-Guo,1

    DONGChun-Yan,1

    andFANEn-Gui2

    1

    CollegeofScience,UniversityofShanghaiforScienceandTechnology,Shanghai200093,China

    2

    InstituteofMathematics,FudanUniversity,Shanghai200433,China (ReceivedNovember21,2005;RevisedMay9,2006)

    AbstractInthispaper,wediscussconditionalstabilityofsolitary-wavesolutionsinthesenseofLiapunovforthe

    generalizedcompoundKdVequationandthegeneralizedcompoundKdV-Burgersequations.Linearstabilityofthe

    exactsolitary-wavesolutionsisprovedfortheabovetwotypesofequationswhenthesmalldisturbanceoftravelling

    waveformsatis?essomespecialconditions.

    PACSnumbers:52.35.Mw,52.35.Sb

    Keywords:generalizedcompoundKdVequation,generalizedcompoundKdV-Burgersequation

    1Introduction

    ThegeneralizedcompoundKdVequation

    ut+aup

ux+bu2p

    ux+δuxxx=0,

    a,b,δ,p=const.,p>0(1)

    andthegeneralizedcompoundKdV-Burgersequation ut+aup

    ux+bu2p

    ux+ruxx+δuxxx=0,

    a,b,r,δ,p=const.,δ,p>0,r<0(2)

    ariseinavarietyofphysicalmodelsandtheyareim- portantmodelequationsinthenonlinear?eld.Stability oftheirsolutionshasextensiveapplicationinquantum ?eldtheory,physics,andsolid-statephysics.Thesolitary- wavesolutionswerediscussedforthetwoequations(see Refs.[1][7]).InRefs.[3]and[4],DeyandCo?eyin-

    vestigatedkinksolitary-wavesolutionsofEq.(1)incases p=1andp=2bymeansofdirectintegralmethod

    andprogressionmethod.InRef.[5],theauthorsstudied bell-pro?lesolitary-wavesolutionsofEq.(1)inthecase p=1andobtainedkink-pro?lesolitary-wavesolutions ofEq.(2).Recently,theexactsolutionsofEqs.(1)and (2)havebeenobtainedforanyrealnumberp>0(see Ref.[7]).Thestabilityofsolutionsisimportantintheory andapplicationofthedi?erentialequations.Therefore,it isvaluabletostudystabilityofsolitary-wavesolutions.In Refs.[8][14],theauthorsdiscussedstabilityofsolitary- wavesolutionsofEq.(1)inthecaseb=0,a=1and

    drewconclusionsthatwhenp<4,thesolutionsaresta- ble,whilewhenp>4,theyareunstable.InRef.[15],the authorsinvestigatedtheoscillatoryinstabilityoftravelling wavesforEq.(2)incaseb=0,a=1,andtheyobtained

    thelinearinstabilitytakesplacewhen(i)for?xedvelocity ofwavep>4,and|r|wassu?cientlysmall,and(ii)for ?xed|r|,p>4,andvelocityofwavevwassu?ciently large,and(iii)for?xed|r|,?xedvelocityofwavev,and pwassu?cientlylarge.

    Inthispaper,wewillinvestigateconditionalstability ofsolitary-wavesolutionsforEqs.(1)and(2).Thatisto say,wewantto?ndtheconditionswhichmakesolitary- wavesolutionsstable,andtherangeofthevaluepwhich makessolitary-wavesolutionsofEqs.(1)and(2)unsta- ble.Weassumethatthesmalldisturbanceofthetrav- ellingwaveformsatis?essomespecialconditions.Then, thegeneralformulaofthesmalldisturbanceisobtained. Usingtheseresults,wewillprovethatconditionalstabil- ityforbell-pro?lesolutionsofEq.(1)andforkink-pro?le solutionsofEq.(2),andobtaintheirstabilityconditions. Theseconditionsarerelationsbetweenthesystemparam- eterandinitialcondition,soitindicatesthatstabilityof solitary-wavesolutionsofEqs.(1)and(2)dependonsys- temparameterandinitialcondition.

    2ConditionalStabilityofBell-Pro?leSolitary-WaveSolutionsforEq.(1)

    BythemethodpresentedinRef.[7],wecanobtainthatthegeneralizedcompoundKdVequatio

    n(1)hasbell-pro?le

    solitary-wavesolutions,

    u(ξ)=

    hAeα(ξ+ξ0)

    (1+eα(ξ+ξ0)

    )2

    +Beα(ξ+ξ0)

    i1/p

=

    hAsech2α

    2

    (ξ+ξ0)

    4+Bsech2(α/2)(ξ+ξ0)

    i1/p

    ,(3a)

    whereξ=x?vt,ξ0isaconstant,andvdenotesvelocityofwave,and

    α=?p

    r

    v

    δ

    ,A=?2|v|(p+1)(p+2)

    s

    2p+1

    a2(2p+1)+bv(p+1)(p+2)2 ,

    B=?2?

    2a|v|

    v

    s

    2p+1

    a2

    (2p+1)+bv(p+1)(p+2)2

    .(3b)

    ?TheprojectsupportedbyNationalNaturalScienceFoundationofChinaunderGrantNo.103

    71023andShanghaiLeadingAcademic DisciplineProjectunderGrantNo.T0502)

    1092ZHANGWei-Guo,DONGChun-Yan,andFANEn-GuiVol.46

    whenδv>0andpmakesq1/p meaningfulforanynegativenumberq.Letu0denotetheexactbell-pro?lesolitary-wave

    solution(3),andu1beasmalldisturbanceofthetravellingwaveformforthesolutionu0.Assu

    methatξ0isstarting

    timeofthesmalldisturbanceand|u0(ξ0)|?|u1(ξ0)|. Letu=u0+u1.SubstitutinguintoEq.(1)andneglectingsmallquantitiesofhighorders,wehav

    e

    δu0ξξξ?vu0ξ+aup

    0

    u0ξ+bu2p

    0

    u0ξ=0,(4)

    δu1ξξξ?vu1ξ+apu

    p?1

    0u1u0ξ+au

    p

    0u1ξ+bu

    2p

    0u1ξ+2bpu

    2p?1

    0u1u0ξ=0.(5)

    IntegratingEq.(5)fromξ0toξimpliesthat δu1ξξ?vu1+aup

    0

    u1+bu2p

    0

    u1=E(ξ0),(6)

    whereE(ξ0)=δu1ξξ(ξ0)?vu1(ξ0)+au p

    0

(ξ0)u1(ξ0)+bu

    2p

    0

    (ξ0)u1(ξ0).SinceE(ξ0)dependsonlyontheinitialvalue,we

    cantake

    E(ξ0)=0.(7)

    Thus,itfollowsfromEq.(6)that

    δu1ξξ?vu1+au

    p

    0

    u1+bu

    2p

    0

    u1=0.(8)

    ComparingEq.(4)withEq.(8),asolutionofEq.(8)isobtained,

    u11=u0ξ=

    ?4α

    p

    hAsech2

    (α(ξ+ξ0)/2

    4+Bsech2α(ξ+ξ0)/2

    i1/p

    ×

    tanh[α(ξ+ξ0)/2]

    4+Bsech2[α(ξ+ξ0)/2]

    .(9)

    Usingthetheoremofordinarydi?erentialequation(seeRef.[16]),anotherspecialsolutionof

    Eq.(8)linearlyindepen- dentofu11isobtained, u12=u11

Zξ

    S

    u

    ?2

    11exp

    ?Zξ

    A

    0dξ

    ?

    dξ=u11

    Zξ

    S

    u

    ?2

    11dξ,(10)

    Here,wehavetotakethefollowing.

    (i)Whenξ0?=0,thenS=0inEq.(10)andξ0>0,forξ?[0,+?);andξ0<0,forξ?(?

    ?,0]inEq.(10).

    (ii)Whenξ0=0,thenS=1,forξ?[0,+?);S=?1,forξ?(??,0]inEq.(10).

    Thus,theexistenceofthesolution(10)isensured.

    Therefore,generalsolutionofEq.(8)(namelyformofthedisturbance)intheconditionsofE(

    ξ0)=0canbewritten

    as

    u1=D1u11+D2u12=D1u0ξ+D2u0ξ

    Zξ

    S

    u?2

    0ξ

    dξ,(11)

whereD1andD2aredeterminedbytheinitialdisturbance.Especially,whenξ

    0=x0?vt0?=0,wehave D1=

    u1(ξ0)u12ξ(ξ0)?u12(ξ0)u1ξ(ξ0) u11(ξ0)u12ξ(ξ0)?u12(ξ0)u11ξ(ξ0) ,D2=

    u1(ξ0)u11ξ(ξ0)?u11(ξ0)u1ξ(ξ0) u11(ξ0)u12ξ(ξ0)?u12(ξ0)u11ξ(ξ0) ,

    whichareobtainedbysolvingthefollowingequationinCramerprinciple,

    u1(ξ0)=D1u11(ξ0)+D2u12(ξ0),u1ξ(ξ0)=D1u11ξ(ξ0)+D2u12ξ(ξ0).

    Now,letu0(ξ)bedenotedas u0(ξ)=

    ?K1

    K2

    ?1/p

    ,(12)

    where

    K1=?|v|(p+1)(p+2) s

    2p+1

    a2

    (2p+1)+bv(p+1)(p+2)2 sech2

    h

    p

    2

    r

    v

δ

    (ξ+ξ0) i

    ,

    K2=2+ ?

    ?1?a

    |v|

    v

    s

    2p+1 a2(2p+1)+bv(p+1)(p+2)2

    ?

    sech 2

    hp

    2

    r

    v

    δ

    (ξ+ξ0) i

    .

    Itiseasytogetthat

    lim

    t??

    u11=lim

    |ξ|??

    |u0|=?lim

    |ξ|?? 2

    r

    v

    δ

    ?

    K1

    K2

    ?1/p 1

    K2

    th

    h

    p

    2

    r

    v

    δ

    (ξ+ξ0)

    i

    =0, lim t??

    u0ξξ=lim

    |ξ|?? u0ξξ=?2

    v

    δ

    plim

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