ConstructionofallsingletravelingwavesolutionstoDullin-Gottwald-Holmequation

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(Heilongjiang University of Finance and Economics, Harbin 150025, China)

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ConstructionofallsingletravelingwavesolutionstoDullin-Gottwald-Holmequation

HOUMan-dan

(HeilongjiangUniversityofFinanceandEconomics,Harbin150025,China)

Bythecompletediscriminationsystemforpolynomialmethod,thispaperconstructedallsingletravelingwavesolutionstotheDullin-Gottwald-Holmequation.

completediscriminationsystemforpolynomialmethod;Dullin-Gottwald-Holmequation;travelingwavesolution

Itiswellknownthatthenonlinearpartialdifferentialequations(NLPDEs)arewidelyusedtodescribecomplexphenomenainvariousfieldsofsciences,particularlyinphysics.ExacttravelingwavesolutionofNLPDEsisoneofimportantinvestigationsinmathematicsphysics.Therearesomemethodsforgettingtheexactsolutionsoftheseequations,suchasthetransformedrationalfunctionmethod[1],themultipleexp-functionmethod[2-3],theextendedtanh-functionmethod[4],thesine-cosinemethod[5],thecompletediscriminationsystemforpolynomialmethod[6-9],andsoon.

TheDullin-Gottwald-Holm(DGH)equationwaspresentedbyDullinetal.[10],whichis

ut+c0ux+3uux-α2(uxxt+uuxxx+2uxuxx)+γuxxx=0

(1)

Recently,manypapersweredevotedtothestudyoftheDGHequation[11-17].GuoandLiuhaveobtainedexpressionsofcuspwavein[11],peakedwavesolutionsin[12],thenewboundedwavesolutions,compacton-likewavesolutionsandkink-likewavesolutionswithα2>0inreference[13].Dullinetal.haveshowedtheexistenceofthreetypesofboundedwavesforthecaseα2>0inreference[14].Inreference[15],someexplicitandimplicitexpressionsoffourtypesofboundedwaveshavebeenobtainedwithα2<0byTangandZhang.Inreference[16],anumberofexactsolutionshavebeengivenbytheExp-functionmethod.Newexactperiodicwavesolutionshavebeenobtainedbysemi-inversemethodandintegralbifurcationmethodinreference[17].Inreference[18],theorbitalstabilityofthetrainofNsolitarywaveshasbeenstudiedwiththedispersiveparameterγ=-c0α2.

However,weaimtoobtainallsingletravelingwavesolutionstotheDGHequation(1).Insection2,underfunctiontransformandthetravelingwavetransformation,weturnedEq. (1)intoanintegralequation.Insection3,byusingLiu’scompletediscriminationsystemforpolynomialmethod[6-9],weobtainedtheclassificationofallsingletravelingwavesolutionsofEq. (1).AllthesolutionshavebeenverifiedthattheyaresolutionsofEq. (1).

1　The integral forms of Eq. (1)

Underthetravelingwavetransformation,(x,t)=u(ζ),ζ=x-λt,wehave

(c0-λ)u′+3uu′-α2(-λu?+uu?)+λu?=0

(2)

whereλ(≠0)isthewavespeedand“′”isthederivativewithrespecttoζ.

IntegratingEq. (2)once,wehave

(3)

whereg0isanintegralconstant.

MultiplyingEq.(3)on2u′andintegratingitonceleadsto

(α2u-α2λ-γ)u′2=u3+(c0-λ)u2+g0u+g1

(4)

whereg1isanintegralconstant.

TheelementaryintegralformofEq.(4)is

±(ζ-ζ0)=

(5)

whereζ0isanintegralconstant.

2 Construction of the solution u

Case1.α=0,γ≠0 .AccordingtoEq.(5),wehave

(6)

Takingthetransformation

(7)

wehave

f(w)=w3+b1w+b0

(8)

Denote

(9)

Δandb1makeupthecompletediscriminationsystemoff(w).

u=

(10)

andifa>w>b,then

u=

(11)

andifw>b>a,then

u=

(12)

Case2 Δ=0,b1=0,thenf(w)=w3,wehave

(13)

Case3 Δ>0,b1<0,thenf(w)=(w-a)(w-b)(w-c),wherea

u=

(14)

andifw>c,wehave

(15)

u=

(16)

Case5 α≠0

f(w)=w3+b1w+b0

(17)

Case6 Δ=0,b1=0,thenf(w)=w3.

Ifθ=1,whenw>max(0,σ)orw

(18)

(19)

Case8 θ=1.Whenw>max(b,σ)orw<σ,thesolutionofEq. (1)is:

ifamax(b,σ),then

(20)

andifmin(b,σ)

(21)

Case9 θ=-1.Whenmin(b,σ)

ifamax(b,σ),then

(22)

andifmin(b,σ)

(23)

Case10 Δ>0,b1<0,thenf(w)=(w-a)(w-b)(w-c),wherea>b>c.

(24)

Case11 Δ<0,thenf(w)=(w-a)(w2+μw+v) ,wherea,μvarerealandμ2-4v<0.

(25)

3　Conclusions

Inthispaper,byusingLiu’scompletediscriminationsystemforpolynomialmethod,weconstructedallsingletravelingwavesolutionstotheDGHequation.

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2015-09-13.

黑龍江省高等教育教學(xué)改革項(xiàng)目(JG2014010930)

侯嫚丹(1978-)，女，碩士，副教授，研究方向：數(shù)學(xué)模型的應(yīng)用.

O241

A

1672-0946(2016)04-0485-04

構(gòu)建Dullin-Gottwald-Holm方程全部單一行波解

侯 嫚 丹

(黑龍江財(cái)經(jīng)學(xué)院，哈爾濱 150025)

通過(guò)多項(xiàng)式完全判別系統(tǒng)的方法，為Dullin-Gottwald-Holm方程構(gòu)建了所有單一行波解.

多項(xiàng)式完全判別系統(tǒng)方法;Dullin-Gottwald-Holm方程; 行波解