英文版b2t

Commun.
Theor.
Phys.
(Beijing,China)40(2003)pp.
137–142cInternationalAcademicPublishersVol.
40,No.
2,August15,2003SymbolicComputationandConstructionofSoliton-LikeSolutionstothe(2+1)-DimensionalBreakingSolitonEquationCHENYong,LIBiao,andZHANGHong-QingDepartmentofAppliedMathematics,DalianUniversityofTechnology,Dalian116024,China(ReceivedOctober14,2002;RevisedDecember17,2002)AbstractBasedonthecomputerizedsymbolicsystemMaple,anewgeneralizedexpansionmethodofRiccatiequationforconstructingnon-travellingwaveandcoecientfunctions'soliton-likesolutionsispresentedbyanewgeneralans¨atz.
Makinguseofthemethod,weconsiderthe(2+1)-dimensionalbreakingsolitonequation,ut+buxxy+4buvx+4buxv=0,uy=vx,andobtainrichnewfamiliesoftheexactsolutionsofthebreakingsolitonequation,includingthenon-travellingwaveandconstantfunctionsoliton-likesolutions,singularsoliton-likesolutions,andtriangularfunctionsolutions.
PACSnumbers:03.
40.
KfKeywords:generalizedexpansionmethodofRiccatiequation,symboliccomputation,breakingsolitonequa-tion,soliton-likesolutions,solitons1IntroductionInrecentyears,muchattentionhasbeenpaidonthestudyof(2+1)-dimensionalsolitonsystemsparticu-larlyaftertheadventofdromionswhichareexponen-tiallylocalizedinthestructuresdrivenbysomestraight-lineghostsolitonsunearthedbyBoitiandhisco-workes[1]andForkasandSantini.
[2]SomesignicantmodelssuchastheKadomtsev–Petviashvili(KP)equation,[3]Davey–Stewartson(DS)equation,[4,5]Nizhnik–Noviko–Vesselov(NNV)equation,[6,7]theasymmetricNNV(ANNV)equation,[6,8]andbreakingsolitonequation,[9,10]etc.
havealsobeenestablishedinnonlinearphysics.
Tondsomeexactphysicallysignicantcoherentsoli-tonsolutions(whicharelocalizedinalldirections)in(2+1)dimensionsismuchmoredicultthanin(1+1)dimensions.
Recently,twokindsof"variableseparating"procedureshavebeenestablishedforsometypesof(2+1)-dimensionalintegrablemodels(seeRefs.
[5],[7],[8],and[10]–[13]).
Thesecondtypeofvariableseparationmethodhadbeenestablishedforsometypesof(2+1)-dimensionalintegrablemodelliketheDSequation,[5]NNVandANNVequations,[7,8]andsomespecialtypesofexactsolutionsoftheseequationscanbeobtainedbyselectingthearbitraryfunctionsappropriately.
Inthispaper,wewouldpresentanewmethodnamedgeneralizedexpansionmethodofRiccatiequation,whichismainlystemmedfromthetanhmethod,[14,15]extendedtanh-functionmethod,[1621]modiedextendedtanh-functionmethod,[22]andgeneralizedhyperbolic-functionmethod,[23]anduseittoconsiderthesoliton-likesolutionsforthe(2+1)-dimensionalbreakingsolitonequation,[9,10]ut+buxxy+4buvx+4buxv=0,uy=vx.
(1)Equations(1)describesthe(2+1)-dimensionalinteractionofaRiemannwavepropagatingalongthey-axiswithalongwavealongthex-axis,anditseemstohavebeengenerated.
[24]Recently,byuseofavariableseparationap-proach,Ruan[10]studiedthecoherentstructuresofEq.
(1)andobtainedsomespecialtypesofthedriomionsolutions,lumps,ringsolutions,curvedsolitons,andbreathersbyselectingthearbitraryfunctionsappropriately.
Inthispaper,usingourmethod,weobtainrichnewfamiliesoftheexactsolutionsofthe(2+1)-dimensionalbreakingsolitonequation,includingthenon-travellingwaveandconstantfunctionsoliton-likesolutions,singu-larsoliton-likesolutions,triangularfunctionsolutions.
Theplanofthispaperisasfollows.
InSec.
2,wedescribebrieythegeneralizedexpansionmethodofRic-catiequation.
InSec.
3,weapplythemethodto(2+1)-dimensionalbreakingsolitonequationandbringoutrichsoliton-likesolutions.
Conclusionswillbepresented-nally.
2GeneralizedExpansionMethodofRiccatiEquationLetussimplydescribethegeneralizedexpansionme-thodofRiccatiequation.
Foragivensystemofnonlinearevolutionequations(NEES)inthreevariablesx,y,andt,E1(u,v,ut,vt,ux,vx,uy,vy,uxx,vxx,uxt,vxt,uxy,vxy,uyt,vyt,0,E2(u,v,ut,vt,ux,vx,uy,vy,uxx,vxx,uxt,vxt,uxy,vxy,uyt,vyt,0,(2)weseekthefollowingformalsolutionsofthegivensystembythenewmoregeneralans¨atz,u(x,y,t)=a0+mi=1[aiφi(ξ)+biφi1(ξ)R+φ2(ξ)+kiφi(ξ)],TheprojectsupportedbyNationalNaturalScienceFoundationofChinaunderGrantNo.
1007201andtheNationalKeyBasicResearchDevelopmentProjectProgramunderGrantNo.
G1998030600Email:chenyong@dlut.
edu.
cn万方数据138CHENYong,LIBiao,andZHANGHong-QingVol.
40v(x,y,t)=A0+nj=1[Ajφi(ξ)+Bjφj1(ξ)R+φ2(ξ)+Kjφj(ξ)],(3)wherem,nareintegerstobedeterminedbybalancingthehighestorderderivativetermswiththenonlineartermsinEq.
(2),Risarealconstant,whilea0=a0(x,y,t),A0=A0(x,y,t),ai=gi(x,y,t),bi=bi(x,y,t),ki=ki(x,y,t),Aj=Ai(x,y,t),Bj=Bj(x,y,t),Kj=Kj(x,y,t)(i=1,m;j=1,n),andξ=ξ(x,y,t)arealldierentiablefunctions,andφ(ξ)satisesdφ(ξ)dξ=R+φ2(ξ).
(4)Itiseasytoseethattheans¨atz(3)ismoregeneralthantheans¨atzinthegeneralizedhyperbolic-functionmethod,[23]tanhmethod,[14,15]extendedtanh-functionmethod,[1621]modiedextendedtanh-functionmethod.
[22]Firstly,com-paredwiththetanhmethod,extendedtanh-function,aswellasthemodiedextendedtanh-functionmethod,therestrictiononξ(x,y,t)asmerelyalinearfunctionx,y,tandtherestrictiononthecoecientsai,bi,ki,Aj,Bj,Kj(i=0,m;j=0,n)andξasconstantsareremoved.
Secondly,comparedwiththegeneralizedhyperbolic-functionmethod,wecannotonlyrecovertheexactsolutionsforagivenNEEswhicharethesuperposi-tionofdierentpowersofthesechξfunction,tanhξfunc-tionortheircombinations,butalsowecan,withnoextraeorts,ndothernewandmoregeneraltypesofsolutions,suchassingularsoliton-likesolutions,coth-typesolutions,triangularperiodic-likesolutions,tan-typesolutions,andtheseformalfunctions'combination,evenrationalsolu-tions,etal.
Moreimportantly,weaddtermskiφi(ξ)innewans¨atz(3),somoretypesofsolutionswouldbeexpectedforsomeequations.
Thereexistthefollowingstepstobeconsideredfur-ther:Step1Determiningthevaluesofmandnofsystem(3)byrespectivelybalancingthehighest-orderpartialderiva-tivetermsandthenonlineartermsinsystem(2).
Step2SubstitutingEqs.
(3)alongwithEq.
(4)intoEq.
(2),multiplyingthemostsimplifyingcommonde-nominatorintheobtainedsystem,settingthecoecientsofφr(ξ)(R+φ2(ξ))s(r=0,1,s=0,1).
(Notehereφr(ξ)denotesrpowerofφ(ξ)and(R+φ2(ξ))sdenotesspowerofR+φ2(ξ))tozero,weobtainasetofover-determinedpartialdierentialequationswithregardtodierentialfunctionsai,bi,ki,Aj,Bj,Kj(i=0,m;j=0,n)andξ.
Step3Solvingtheover-determinedpartialdierentialequationsbyuseofthePDEtoolspackageofMaple,wewouldendupwiththeexplicitexpressionsforai,bi,ki,Aj,Bj,Kj(i=0,m;j=0,n)andξ,orthecon-strainsamongthem.
Step4ItiswellknownthatthegeneralsolutionsofRiccatiequation(4)areφ(ξ)=√Rtanh(√Rξ),R0,√Rcot(√Rξ),R>0,1/ξ,R=0.
(5)ThusaccordingtoEqs.
(3),(5)andtheconclusionsinStep3,thesoliton-likesolutionsofEqs.
(2)canbeob-tained.
Forthegeneralizationoftheans¨atz,naturallymorecomplicatedcomputationisexpectedthaneverbefore.
EveniftheavailabilityofcomputersymbolicsystemslikeMapleorMathematicaallowsustoperformthecom-plicatedandtediousalgebraiccalculationanddierentialcalculationonacomputer,ingeneral,itisverydicult,sometimeimpossible,tosolvethesetofover-determinedpartialdierentialequationsinStep3.
Asthecalcula-tiongoeson,inordertodrasticallysimplifytheworkormaketheworkfeasible,weoftenchoosespecialfunctionformsforai,bi,ki,Aj,Bj,Kj(i=0,m;j=0,n)andξonatrial-and-errorbasis.
3The(2+1)-DimensionalBreakingSolitonEquationInthissection,byusethegeneralizedexpansionmethodofRiccatiequation,weconsiderthe(2+1)-dimensionalbreakingsolitonequation,i.
e.
,Eqs.
(1).
[9,10]Bybalancingthehighest-ordercontributionsfromboththelinearandnonlineartermsinEqs.
(1),weobtainm=2,n=2inEq.
(3).
Thereforeweassumethesolu-tionsofEqs.
(1)intheformu(x,y,t)=a0+a1φ(ξ)+a2φ(ξ)+b1R+φ2(ξ)+b2φ(ξ)R+φ2(ξ)+k1φ1(ξ)+k2φ2(ξ),(6)v(x,y,t)=A0+A1φ(ξ)+A2φ2(ξ)+B1R+φ2(ξ)+B2φ(ξ)R+φ2(ξ)+K1φ1(ξ)+K2φ2(ξ),(7)whereai=ai(y,t)(i=0,1,2),bi=bi(y,t)(i=1,2),ki=ki(y,t)(i=1,2),Ai=Ai(y,t)(i=0,1,2),Bi=Bi(y,t)(i=1,2),Ki=Ki(y,t)(i=1,2)andξ=xp+q(pisaconstantandq=q(y,t))arealldierentialfunctions,andφ(ξ)satisesEq.
(4).
SubstitutingEqs.
(6)and(7)alongwithEq.
(4)intoEq.
(1),multiplyingφ5(ξ)R+φ(ξ)2andφ3(ξ)intherstequationandthesecondequation,respectively,thensettingthecoecientsofφs(ξ)(R+φ(ξ)2)r/2(r=0,1;s=0,1,2,tozerointheobtainedsystemofpartialdierentialequation,wecandeducethefollowingsetofover-determinedpartialdierentiableequationswithrespecttotheunknowndierentiablefunctionsa0,a1,a2,b1,b2,k1,k2,A0,A1,A2,B1,B2,K1,K2,andq(Noteinthispaper,aiy=ai(y,t)/y,andsoon.
)4bpk1A04bpa0K14bpk2A1+4bpb1B2R2+4bpa2K1R+4bpb2B1R2+a0t+4bpa1RA0+4bpa0A1R万方数据No.
2SymbolicComputationandConstructionofSoliton-LikeSolutionstothe(2+1)-DimensionalBreakingSolitonEquation139+2bp2k2yqtk1+4bpk1A2R+2bp2qya1R22bp2qyk1R+2bp2a2yR2+qta1R4bpa1K2=0,(8)qya1pA1+a2y=0,(9)2pK2+k1y2qyk2=0,(10)pK1R+k2yqyk1R=0,(11)6bpR(pqyk1R2pk2yR+2K1k2+2k1K2)=0,(12)2pA2R+a1y+2qya2R=0,(13)12bpR(b1K2+B1k2)=0,(14)pB2R+b1y+qyb2R=0,(15)2pB2+2qyb2=0,(16)4bpb1A0R4bpk2B14bpb1K2+5bp2b2yR2+8bpb1A2R2+8bpa1B2R2+5bp2qyb1R2+8bpa2R2B1+qtb1R+8bpb2A1R2+b2tR+4bpb2K1R+4bpk1B2R+4bpa0B1R=0,(17)6bp(pb1qy+pb2y+2a1B2+2B1a2+2A2b1+2A1b2)=0,(18)12bpa1B1R+b1t+3bp2b1yR+12bpb2A2R2+33bp2qyb2R2+12bpb2A0R+12bpb1A1R+12bpa0B2R+12bpa2B2R2+3qtb2R=0,(19)4bpR2(B1k1+b1K1+K2b2+B2k2)=0,(20)R(5bp2qyb2R2+4bpa1B1R+qtb2R+4bpa0B2R+4bpb2A0R+bp2b1yR+4bpb1A1R4bpb2K2+b1t4bpk2B24bpb1K14bpk1B1)=0,(21)4bpa0B1+4bpb1A0+qtb1+20bpa1B2R+11bp2b2yR+b2t+20bpa2B1R+4bpb2K1+20bpb1A2R+4bpk1B2+20bpb2A1R+11bp2qyb1R=0,(22)8bpa1B1+2bp2b1y+8bpb2A0+28bpa2B2R+52bp2qyb2R+8bpb1A1+28bpb2A2R+2qtb2+8bpa0B2=0,(23)8bp(3b2qyp+2B2a2+2A2b2)=0,(24)4bpa1K2R8bp2qyk1R2qtk1R12bpk2K14bpa0K1R+8bp2k2yR+k2t12bpk1K24bpk2A1R4bpk1RA0=0,(25)8bpb1B1R+16bp2qya2R2+a1t+8bpa0A2R+8bpb2B2R2+8bpa2RA0+8bpa1A1R+2qta2R+2bp2a1yR=0,(26)4bpa2K1+diff(a2,t)+12bpa1A2R+4bpa1A0+16bpb1B2R+16bpb2B1R+8bp2qya1R+12bpa2A1R+8bp2a2yR+4bpk1A2+4bpa0A1+qta1=0,(27)8bpb1B1+24bpb2B2R+8bpa0A2+2qta2+16bpa2A2R+2bp2a1y+8bpa1A1+40bp2qya2R+8bpa2A0=0,(28)6bp(pa2y+pqya1+2B1b2+2B2b1+2A1a2+2a1A2)=0,(29)8bpa0K2R8bpk2RA040bp2qyk2R28bpk1K1R2qtk2R+2bp2k1yR216bpk2K2=0,(30)2bp2k1yR16bp2qyk2R8bpk2A08bpk1K12qtk28bpa0K2+k1t=0,(31)8bpR2(b1K2+B1k2)=0,(32)8bp(3qya2p+2a2A2+2B2b2)=0,(33)8bpk2R(3qyR2p+2K2)=0,(34)a0yqyk1+qya1RpA1R+pK1=0,(35)2pA2+2qya2=0,(36)2R(qyk2pK2)=0,(37)pB1+b2y+qyb1=0.
(38)UsingthepowerfulPDEtoolspackageofMaple,solvingthesetofpartialdierentialequations(8)(38),wecanobtainthefollowingresults.
(Noteintherestofthispaper,q(y,t)denotesarbitraryfunctionwithrespecttoy,t,andC1,C2arearbitraryconstants).
Case1a1=b1=k1=k2=b2=A1=B1=B2=K1=K2=0,a0=2Rp2,a2=32p2,A0=qt4bp,A2=32pqy,q=q(y,t).
(39)Case2a1=b1=b2=k1=k2=A1=B1=B2=K1=K2=0,万方数据140CHENYong,LIBiao,andZHANGHong-QingVol.
40a0=1110Rp2,a2=32p2,A0=18bp2qyR+5qt20bp,A2=32pqy,q=q(y,t).
(40)Case3a1=b1=b2=k1=k2=A1=B1=B2=K1=K2=0,a0=C1,a2=32p2,A0=8bp2qyR+qt+4bC1qy4bp,A2=32pqy,q=q(y,t).
(41)Case4a1=a2=b1=b2=k1=A1=A2=B1=B2=K1=0,a0=C1,k2=32p2R2,A0=8bp2qyR+qt+4bC1qy4bp,K2=32qyR2p,q=q(y,t).
(42)Case5a1=b1=k1=k2=A1=B1=K1=K2=0,a0=54Rp2,a2=34p2,b2=34p2,A0=qt4bp,A2=34pqy,B2=34pqy,q=q(y,t).
(43)Case6a1=b1=k1=k2=A1=B1=K1=K2=0,a0=54Rp2,a2=34p2,b2=34p2,A0=qt4bp,A2=34pqy,B2=34pqy,q=q(y,t).
(44)Case7a1=b1=k1=k2=A1=B1=K1=K2=0,a0=C2,a2=34p2,b2=34p2,A0=5bp2qyR+qt+4bC2qy4bp,A2=34pqy,B2=34pqy,q=q(y,t).
(45)Case8a1=b1=k1=k2=A1=B1=K1=K2=0,a0=C1,a2=34p2,b2=34p2,A0=5bp2qyR+qt+4bC1qy4bp,A2=34pqy,B2=34pqy,q=q(y,t).
(46)Case9a1=b1=b2=k1=A1=B1=B2=K1=0,a0=C1,a2=32p2,k2=32p2R2,A0=8bp2qyR+qt+4bC1qy4bp,A2=32pqy,K2=32qyR2p,q=q(y,t).
(47)FromEqs.
(5)(7)and(39)(47),wecanobtainthefollowingsolutionsforthe(2+1)-dimensionalbreakingsolitonequation.
Type1FromCase1,wecanobtainthefollowingsolutions:u11=2Rp2+32p2Rtanh2[√R(xp+q(y,t))],v11=qt4bp+32pqyRtanh2[√R(xp+q(y,t))],R0,(50)u14=2Rp232p2Rcot2[√R(xp+q(y,t))],v14=qt4bp32pqyRcot2[√R(xp+q(y,t))],R>0.
(51)Type2FromCase2,wecanobtainthefollowingsolutions:u21=1110Rp2+32p2Rtanh2[√R(xp+q(y,t))],v21=18bp2qyR+5qt20bp+32pqyRtanh2[√R(xp+q(y,t))],R0,(54)万方数据No.
2SymbolicComputationandConstructionofSoliton-LikeSolutionstothe(2+1)-DimensionalBreakingSolitonEquation141u24=1110Rp232p2Rcot2[√R(xp+q(y,t))],v24=18bp2qyR+5qt20bp32pqyRcot2[√R(xp+q(y,t))],R>0.
(55)Type3FromCases3and4,wecanobtainthefollowingsolutions:u21=C1+32p2Rtanh2[√R(xp+q(y,t))],v21=8bp2qyR+qt+4bC1qy4bp+32pqyRtanh2[√R(xp+q(y,t))],R0,(58)u24=C132p2Rcot2[√R(xp+q(y,t))],v24=8bp2qyR+qt+4bC1qy4bp32pqyRcot2[√R(xp+q(y,t))],R>0.
(59)Type4FromCases5and6,wecanobtainthefollowingsolutions:u41=54Rp2+34p2Rtanh2(√Rξ)±itanh(√Rξ)sech(√Rξ),v41=qt4bp+34pqyRtanh2(√Rξ)±itanh(√Rξ)sech(√Rξ),R0,(62)u44=54Rp234p2Rcot2(√Rξ)±cot(√Rξ)csc(√Rξ),v44=qt4bp34pqyRcot2(√Rξ)±cot(√Rξ)csc(√Rξ),R>0,(63)whereξ=xp+q(y,t).
Type5FromCases7and8,wecanobtainthefollowingsolutions:u51=C2+34p2Rtanh2(√Rξ)±itanh(√Rξ)sech(√Rξ),v51=5bp2qyR+qt+4bC2qy4bp+34pqyRtanh2(√Rξ)±itanh(√Rξ)sech(√Rξ),R0,(66)万方数据142CHENYong,LIBiao,andZHANGHong-QingVol.
40u54=C234p2Rcot2(√Rξ)±cot(√Rξ)csc(√Rξ),v54=5bp2qyR+qt+4bC2qy4bp34pqyRcot2(√Rξ)±cot(√Rξ)csc(√Rξ),R>0,(67)whereξ=xp+q(y,t).
WhensettingC2=C1intheabovesolutions,wecanobtainanothersetofsolutionsforEq.
(1).
Type6FromCase9,wecanobtainthefollowingsolutions:u61=C1+32p2tanh2(√Rξ)±coth2(√Rξ),v61=8bp2qyR+qt+4bC1qy4bp+32pqyRtanh2(√Rξ)±coth2(√Rξ),R<0,(68)u62=C132p2tan2(√Rξ)±cot2(√Rξ),v62=8bp2qyR+qt+4bC1qy4bp32pqyRtan2(√Rξ)±cot2(√Rξ),R<0,(69)whereξ=xp+q(y,t).
4ConclusionsInsummary,basedonthecomputerizedsymboliccomputation,byintroducinganewmoregeneralans¨atzthantheans¨atzintheextendedtanh-functionmethod,modiedextendedtanh-functionmethod,andgeneralizedhyperbolic-functionmethod,wehaveproposedageneralizedexpansionmethodofRiccatiequationforsearchingforexactsolutionsofNEEsandimplementedincomputerizedsymbolicsystemMaple.
MakinguseofourmethodandwiththeaidofMaple,westudythe(2+1)-dimensionalbreakingsolitonequationandobtainnewfamiliesoftheexactsolutions.
Inourobtainedexactsolutionstherestrictiononξ(x,y,t)asmerelyalinearfunctionx,y,tandtherestrictiononthecoecientsai,bi,ki,Aj,Bj,Kj(i=0,m;j=0,n)andξasconstantsareremoved,andwithnoextraeorts,thesingularsolitonicsolutionandtriangularfunctionsolutionscouldbeobtained.
Tomaketheworkfeasible,howtochoosetheformsforai,bi,ki,Aj,Bj,Kj(i=0,m;j=0,n)andξintheans¨atzwouldbethekeystepinthecomputationofourmethod.
Themethodproposedinthispaperforthe(2+1)-dimensionalbreakingsolitonequationmaybeextendedtondexactsoliton-likesolutionsofotherNEEs.
AcknowledgmentTheauthorswouldliketoexpresstheirsincerethankstotherefereesfortheirhelpfulsuggestion.
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万方数据SymbolicComputationandConstructionofSoliton-LikeSolutionstothe(2+1)-DimensionalBreakingSolitonEquation作者:CHENYong,LIBiao,ZHANGHong-Qing作者单位:DepartmentofAppliedMathematics,DalianUniversityofTechnology,Dalian116024,China刊名:理论物理通讯(英文版)英文刊名:COMMUNICATIONSINTHEORETICALPHYSICS年,卷(期):2003,40(8)被引用次数:17次参考文献(25条)1.
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李德生若干非线性演化方程精确求解法的研究[学位论文]博士2004本文链接:http://d.
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