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Breather Dynamics of the SineGordon EquationBreather Dynamics of the SineGordon Equation -- 5 元

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Commun.Theor.Phys.592013664670Vo1.59,No.6,June15,2013BreatherDynamicsoftheSineGordonEquationStephenJohnsonandAnjanBiswas,。,StephenJohnsonandAnjanBiswasDepartmentofMathematicalSciences,DelawareStateUniversityDover,DE199012277,USA2DepartmentofMathematics,FacultyofScience,KingAbdulazizUniversity,Jeddah,SaudiArabiaReceivedDecember5,2012revisedmanuscriptreceivedApril22,2013AbstractThispaperstudiestheadiabaticdynamicsofthebreathersolitonofthesineGordonequation.Theintegralsofmotionarefoundandthenusedinsolitonperturbationtheorytoderivethedifferentialequationgoverningthesofitonvelocity.Timedependentfunctionsariseandtheirpropertiesarestudied.Thesefunctionsarefoundtobeboundedandperiodicandaffectthesolitonvelocity.Thesofitonvelocityisnumericallyplottedagainsttimefordifferentcombinationsofinitialvelocitiesandperturbationterms.PACSnumbers02.30.Ik,02.30.Jr,42.81.Dp,52.35.SbKeywordssolitons,phonons,conservationlaws,perturbations,numerics1IntroductionThesineGordonequationanditssolitonsolutionshavebeenwel1studiedinthepast.117】ThesineGordonequationisknowntoadmitthreedistincttypesofsolutionskinks.breathers.andphonons.【l一Thetopo1ogicalsolitons.knownaskinks.havebeenexhaustivelystudiedduetotheirimportanceinJosephson{urictiontheory.12,89,13Breathershavebeenfoundusingseveraldifierentmethodsasspecialcasesofcertainparameters.56,12,15】Theseworksshowtheimportanceofbreathersandthenecessityforthestudyofbreathers.Muchlessisknownaboutthebreatherthanisknownaboutthekink.andthatiswhybreathersarethetopicofstudyinthispaper.Breathersareunusua1solitonsandthereforethemostinteresting.TheYareunusualbecause,contrarytotheusualdefinitionofasoliton,theyoscillatewithinasmallregion.Thisoscillationnotonlydistinguishesbreathersfromothersolitons,butitalsocausessomepeculiarphenomenawhichwillbeseeninthispaper.ThispaperwillstudyhowbreathersolitonsareaffectedbytheperturbationtermsfoundinJosephson{unc.tions.ItwillbefoundthattheseDerturbationsadiabaticallychangethesolitonvelocity.Theperturbationtermswil1lcadtotimedependentfunctionswithspecialpropertiesthatwillbestudied.Thesefunctionsareimportantbecausetheydirectlyaffectthesolitonvelocity.Itwillbeshownthatthesefunctionsareallperiodicandbounded.Someofthesefunctionspossessfurtherpropertiesthatwil1bestudiedandappliedtothedynamicsofthesolitonvelocity.Finallynumericalsimulationswillshowhowthevelocityreactstocertainvaluesoftheperturbationterms.2IntegralsofMotionThesineGordonequationSGEisqttc2qxwsin口0.1E.mailbiswas.anjangmail.corn2013ChinesePhysicalSocietyandIOPPublishingLtdThebreathersolutiontotheSGE1isgiveninRef.1byqx,t4arctansech『LuLC1vtItwillbeassumedwithoutlossofgeneralitythatc0,w0.and札0becauseanegativesigninwillcance1insidethearctanterm.InanefforttoreducethecornplexityofEq.2,wewillintroducetwonewfunctionsofxandtdefinedby咖,£wlxvtt一v/cxWiththesenewfunctions,妒and妒,Eq.2becomes口4arctansechn砂.4TheSGE1admitsatleasttwointegralsofmotion,i.e.%吼tc啦%。w。qzsinq0,qtqttcgt啦zwqtsinq0.5Theseareexactdifierentials】_01互1qt2互1c22w21cosq0,01互1吼2三c1cosq_c2伽0.6IntegratingEq.6overallandtakingtheconstantsofintegrationtobezero,becausethesearesolitons,causestherighthandtermstovanishleavinguswith仁,http//.iop.org/EJ/journal/ctphttp//ctp.itp.ac.cn南叫No.6CommunicationsinTheoreticalPhysicsaOt122C2W21COSqd7P一仁ETherefore,theintegralsofmotion,whichareconstantintime,are1q去c2q1cosqd8wherePisthemomentumandEistheenergy.Wewillnowintroduceanewintegralnotationinordertobetterrepresenttheseintegrals.l,n2,n3,n4㈤SolvingtheintegralsinEq.8withqfoundinEq.4,themomentumandenergyaregivenby蟹,。。/c2。01v2/c2霹2,1。/c。1一/c。c。,。u。/c。,。1v2c。4uvl,。,,1。/c1一。/c。EventhougheachindividualintegralinEq.10istimedependent,themomentumandenergyarenot.Throughnumericalsimulation,themomentumandenergyarefoundtobeP16vwu2,.11a11bItisimmediatelyseenthattherelativisticrelationshippc2EvissatisfiedinEq.11.Thisisbecausethebreather,justlikethekink,actslikearelativisticparticle.Itsenvelope,definedbyqx,I4arctans础x一vt/。,一/c2j/undergoesLorentzlengthcontraction,anditsperiod.wu41一2/cundergoestimedilation,133PerturbationTermsTheSGEisagoodmodelforseveralapplications,butitisonlyanapproximation.Manysituationsrequireextracorrectionalterms.whichareusuallysmall_Forexample.the1ongJosephsonJunctionisnotexactwithoutaqtterm.TheotherapplicationsoftheSGE,suchasultrashortopticalpulsesandcrystals,alsorequiresmallperturbationterms.【l6JTheperturbedSGEisgivenbyqttc%zwsin口eR,14whereEistheperturbationparameterandRcontainstheperturbationterms.Inthepresenceofperturbationterms,theconservedquantitiesarenolongerintegralsofc220IJ10qxqdx01d/二dx16f/qxqtdx一。。motion.Instead,theyundergoanadiabaticdeformationofsolitonparameters,whichinthecaseoftheSGEisjustthevelocityv.Theadiabaticdeformationofthemomentumandenergyaregivenby芸e/二础面dEe/二删.ThetimederivativeofEqdPdt1lais16wu2c3、,干1一v2/c。/。dvdt1516Solvingfordv/dtleadstodvtc3x/1u2/c2一。/二础17TheperturbationtermsthatwillbestudiedareR/3qt一y%qt入gtt%t%z18TheperturbationtermsallcomefromdifferentphysicalphenomenainJosephsonjunctions.ThetermaccountsfordissipativelossesduetotheRCcircuit.ThevariableisalwaysnegativeandisanessentialpartofaJosephsonjunction.Thetermcomesfromtheinhomogeneityofthelocalinductance.The5termrepresentsthediffusion.Thetermaccountsforthecapacityinh0mogeneity.Theotermrepresentslossesduetoacurrentalongthebarrier.Whentheotermispresent,itmustbenegative,butitisnotessentialtotheJosephson.junction.Thetermisthehigherorderspatialdispersion.I,TheperturbedSGEisqttc2qwsinqEqt,yq口十Aqttaqt工,q∞.19PerformingtheintegralinEq.17foreachperturbationterminEq.181yields16vwu2蟹,。,。,。2v/c,。,,u/c。,1乱。/c21一t,。/c2,仁tdx/二瓦0l、118、J\JPECommunicationsinTheoreticalPhysicsVo1.59qxqttdx,oC2T。1,4,4,0r.u%。0,2,1,1一。2Ic2v212,。UV。0l,一2l,l2c2霹,4,3,1一2Zt2r。l。,22霹2,qxqxxtdx{U。2HIO,2,0,2u。21sI2,2,。U21一G一4G2,4,2,0_6V10I4I2l26C2』T42,4,4,06vGio,6,2,26vGi2I41212C2。/.2T0624象G4,。C2G4,2丢12c2./\i26,。2日一1f1211U4G一G一1414116G砖3一一621,362G。E3,一62霹3,1s2V2日G5,.Vu2。C___2T36,s,,0fqzq∞∞zd\whereG1V2/c2andH1一U2V。c.TheintegralsinEq.20canberewrittenas伽。,,1qxqttdx/qxqtdx~P,一o。仁一1I㈣,/%td,3,wherethefunctionsfl,,2,and,3thavespecialproperties.4PropertiesofthePerturbationTermsThefunctionsflt,,it,,2t,and,3twillbeshowntobeperiodicandboundedforconstantV.InsertingEq.13intoEq.3gives,t妒,t7r,22an1nsmgLIlerelationsin7r一sine,itiSclearthatgvTqzvT吾gtvT,£T一吼,t,%vT,T一%,,吼vT,T一gtt,t,232420vTqxxt吾,25,一啦zt,,25Thuseveryperturbationintegralsatisfies仁t吾。如吾dz/二vT吾口xrvT/q,z,tDq,tdx/qxx,tDqx,tdx,26whereDisoneofthederivativeoperatorsandX,vT/2.Therefore,functionsfx£,t,,2£,and,3tareallperiodicwithperiodT/2.Becauseeachofthesefunctionsiscontinuousontheclosedintervalf0,T/21,theyareboundedonthatinterva1.Sinceeachofthesefunctionsisperiodic.theyaxeboundedforalltimet∈R.Itisclearbythedefinitionofflthatitisalwayspositive.Because%,0isevenandqxtx,0andqttX,0areodd,thatiSqxX,0啦,0,qxt一X,0一qxtX,0,qtt一X,0一qttX,0,27andbecauseanevenfunctionmultipliedbyanoddfunctionisodd,itiseasytoseethat00and,200.Thus,flnT/20and,2nT/20,Vn∈z.Onefinalpropertyoffl£isthatitstimeaverageoverahalfperiodiszero.9fT/2fw2导/ftdt吾一flo。.28Itisalsotruethatthetimeaverageoverahalfperiodofhtiszero.\●/、/一2一2£t订一2一2一砂N0.6CommunicationsinTheoreticalPhysics667c。删2/一一qxqttdxdt2F/0T㈣dtdx1捌t2一,qx,。吼,。d一1Tqt2CXD,t一一。。,t】dvT,吼等,一%,Xt00一0d/_。。%,0一。。gz,0,0d0InEq.29,wemadeachangeofvariablevT/2Whenu0.qxx,t一%,t,qtt一,tqttX,t,qzzt一x,tqxxtX,tTherefore,if0,then,2tSubstitutingEqs.18anddvdtEc316wu20and,30,Vt∈.21intoEq.17yieldst一箬。。,t去fi,2t,3一e一墨Sinceand,2arebothzeroforeachhalfperiodandtheyarebothscaledbyafactorofE,theireffectsondr/dtwillbenegligibleandcanbeignored.TheunperturbedlongJosephsonjunctionhasaninnate卢termwhere0.Inthecasethatonlytheperturbationtermisnonzero,d/dtcanbesolvedexactlyintermsofaninitialvelocityv0v0.5NumericalSimulations32Figure1isofastationarybreatherwithparameterscw1andv0overonehalfperiod.ThesolidlinesareattT/aandt3T/4.whilethedottedlinesaJieatevenlyspacedpointsintimebetweenT/4and3T/4.3.141600,00003.14161O一50510Fig.1Breatherwithparametersc叫u1and0.OnefullperiodisT8.89.Thesolidlinesshowthesolitonenvelopewhilethedottedlinesshowthesolitonatequallyspacedtimes.2930316Fig.2Breatherwithparameterscwu1andv0.1.OnefullperiodisT8.93.80Fig.3Breatherwithparametersewu1andv0.5.OnefullperiodisT10.26.GiventheSGE11withspecifiedparameterscandw,thefreeparameterucorrespondstotheamplitudeofthesoliton.InFig.1.u1correspondstoasolitonofamplitude71.InthecoupledpendulamodeloftheSGE.theparameterscandwwilldependonthethependulaandtheircoupling.Raisingasectionofthependulaupandreleasingthemtoswingbackandforthcreatesabreathersoliton.Thatamplitudewilldetermineu.Theparameter钆dependsonlyontheinitialwavefrontortheinitial668CommunicationsinTheoreticalPhysicsVb1.59velocityofthesoliton.Alargevalueof钆correspondstoasmallamplitudeandwidebreather.Inthelimitasgrowswithoutbound,theamplitudewillgotozerobutthewidthwillalsogrowwithoutboundandtheperiodwillconvergeto2/cw.Asmallvalueofucorrespondstoalargeamplitudeandwidebreather.Asuapproacheszero,theamplitudeapproaches21rbutthewidthandperiodbothgrowlargerwithoutbound.Inthecoupledpendulamodel,initiallysettingthependulabeyonda27rrotationnolongercreatesalargerbreathersoliton.Sincea27rrotationcorrespondstothependulaintheirrestingposition、thisconfigurationi8stable.Anyfurtherrotationwouldcreateabreathersolitonwithinthe27rrotatedregion.Figures2and3showbreathersmovingoveratime1614121080.00000.404periodequaltotheirrespectiveperiodsofoscillation.Figure2isaslowbreatherthatresemblestheshapeofastationarybreather.Figure3isafastbreatherwhoseshapeisfardifferentfromthatofastationarybreather.Notonlyisthemotionapparentasthesolitonmoves5.13unitsinthepositivexdirection,buttheentireshapeofthebreatherjsdi饪erent.Figures4showtheplotsof11,f,,2,andf3overonefullperiodfortheparameterscwf1.andv0.5.Thesefiguresclearlyshowallofthepropertiesmentionedintheprevioussection.ThefunctionsareallboundedandperiodicwithperiodT/2.hisalwayspositive.Functionsfand}2haveatimeaverageofzeroand0f200.1.5,0.01.55.1302,10.26040.0000t0.000020245.130210.26045.130210.260400000t51302lO2604fFig.4Functions,1,,,2,and,3withparametersc1,w1,u1,andu0.5plottedagainsttimetFigures512showhowthevelocityevolvesfordiffer..entperturbationterms.Thesefiguresallusethesameparameterscw1andE0.01.buteachfigureshowstheeffectsofadifierentcombinationofperturbationterms.Figure5showsthevelocitiesoveralongtimefor一1.Inthiscase,oIdYtheperturbationtermisnonzero.Itisimmediatelyseenthatthevelocityiseitherpositiveandapproaches1orisnegativeandapproaches一1.Itisalsoclearthatthegraphhasamirrorsymmetryabouttheaxis.Figure6showsvelocitiesfor1andallotherperturbationtermszero.Sincetheperturbationtermismultipliedby11,whichisalwayspositive,the,perturbationtermwillconstantlypushthevelocityeitherupordowndependingonthesignof.Figure7showsthevelocitiesfor一1andeveryotherperturbationtermzero.Thisgraphisquiteinter.estingbecauseitisinstarkcontrasttothegraphsseenthusfar.Thekevdi髓renceportrayedinhowthetermaffectsthevelocityisthatthevelocitynolongerhasthememorylessproperty.Forexample,theeffectsofthetermareclearlymemorylessinthatgivenacertainvelocity,thetermwillactacertainwayregardlessofwhathashappenedupuntilthatpointinthegraph.Inotherwords,givenacertainvelocityv,weknowhowdv/dtwillreactwiththe/3term.Eventheperturbationtermistimedependent,butFig.6showsthatregardlessofthatfact.itwillactinacertainwaywithsmalldeviationgivenanyinitialvelocity.However.thisisnolongertrueforthe盯term.Itisplainlyseenthatgivenaninitialvelocity.say00.75,dv/dtinthiscaseisinitiallynegativebutthenaftersometime,thesignofdv/dtmayreverseandumayincreasebacktoitsstartingpointof0.75.andthenatthislaterpointintime.eventhoughu0.75onceagain.dr/dtisnowpositive.Thisshowsthattheoper.No.6CommunicationsinTheoreticalPhysics669turbationtermcannotbesimplydescribedbysayingonerangeofinitialvelocitieswillcertainlybehaveinonewaywhereanotherrangeofinitialvelocitieswillcertainlybehaveindifferentway.ItinitiallyappearsasthoughanypositiveinitialvelocitywilleventuallyleadtoVapproaching1andanynegativeinitialvelocitywilleventuallyleadtoapproaching一1.butthismaynotbethecase.TheonethingthatisclearaboutFig.7isthatthelinesabovetheaxisaxeexactlymirroredbelowthe一axis.0.80.40.40.8050i00150200250Fig.5Breathervelocitywithonlytheperturbationterm.Fig.6Breathervelocitywithonlythe7perturbationterm.Figure8impliesthatforthegivenvaluesofand7,thetermdominatesandforcesthevelocitytoapositivevalue,wheretheandtermsbothcontributeaposiriveaffecttowarddr/dt.Figure9showstheeffectsforadifferentvalueof.InFig.8,theperturbationtermsareiand,y1,butinFig.9,theperturbationtermsare一1and0.5.ItisseeninFig.9thatthereisanunstablevelocitywhereanythinggreaterwilleventuallyapproach1andanythinglesswilleventuallyapproach一1.O.8O.4O.40.801O02003004ffl5∞Fig.7Breathervelocitywithonlytheperturbationterm.0.80.4一O.4一O.8050100150200亡Fig.8Breathervelocityfor1and1080.40.80100200300400500Fig.9Breathervelocityfor1and一0.5Figure10showstheeffectsofthetermincombinationwiththeterm.Aswouldbeexpected.thetermiSmoredominant,andthegraphlooselyresemblesFig.5.Thegraphisexactlymirroredovertheaxisbe.causeboththe8andotermshavethesamesymmetryovertheaxis.Figure11showstheeffectsofiusttheandterms.Theterm,asbeforeinFig.8,drivesthevelocityincreasinglyhighertothelimitof1.Figure12combinesallthreeperturbationtermsthatcontributeaneteffect.ThisgraphisquitesimilartoFig.8withonly670CommunicationsinTheoreticalPhysicsVo1.59the8andterms.8.47O.0一.40.80150200Fig.10Breathervelocityfor口一1ando20.80.4t.OO.4一O8050100150Fig.11Breathervelocityfor吖1and一1Figures6through12allshowarippleeffectintheirgraphs.Thisisduetotheperiodicnatureofeachoftheperturbationfunctions,fl}21andf3.Sincethe8perturbationisnotassociatedwithaperiodicfunction,itdoesnotexhibitthisbehavior.Theandtermshavebeenomittedbecausetheycontributenoneteffecttothevelocity.Includingthemmighthaveonlyresultedinamorepronouncedrippleeffect.Fig.12Breathervelocityfor一1,1,and一1.6ConclusionThispaperfoundtheintegralsofmotionofbreathersolitonsofthesineGordonequation.Theintegralsofmo.tionarethemomentumPandtheenergyE.Theseconstantswereusedinsolitonperturbationtheorytofindtheadiabaticvariationofsolitonparameters,namelythevelocity.Whileusingsolitonperturbationtheory.sometimedependentfunctiohsarose.Theirpropertieswerefoundspecifically,theyareboundedandperiodic.Thefunctionthatisalwayspositive,fl,hasaconstanteffectonthesolitonvelocity,butthefunctionsthathaveazerotimeaverage{}andf21havenoneteffectonthevelocity.Thevelocitywasnumericallycalculatedfordifierentvaluesoftheperturbationterms.Insomecasesmakingaslightchangetotheinitialvelocityortheperturbationtermscausedthevelocitytobehavequitedifierentlyovertjme.References『91】TDu。idMPey.rard,Physicsof1.S。z。佗CambridgeUniversityPressNewYork2006【10】.1.L~【2A.L.Fabian,R.Kohl,andA.Biswas,Commun.Nonlin,earSci.Numer.Simu1.1420091227.13lK.ForinashandC.R.Willis,Phys.D149f2001195.,.、l1214】D.R.GulevichandF.V.Kusmartsev,Phys.Rev.B74『1312006214303.S.Johnson,F.Chen,andA.Biswas,App1.Math.Con1.put.21720116372.E.Mann,Theor.Math.Phys.107f19961775.N.RiaziandA.R.Gharaati.Internat.J.Theor.Phys.3719981081.W.Rui,B.He,andY.Long,Commun.Non1.Sci.Numer.Simu1.14f200911245.S.M.ShahruzandL.G.Krishna,Automat.37f20011】495.51J.LiandM.Li,Chaos,Solitons,andFractals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