Breather Dynamics of the SineGordon EquationBreather Dynamics of the SineGordon Equation

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COMMUN.THEOR.PHYS.592013664670VO1.59,NO.6,JUNE15,2013BREATHERDYNAMICSOFTHESINEGORDONEQUATIONSTEPHENJOHNSONANDANJANBISWAS,。,STEPHENJOHNSONANDANJANBISWASDEPARTMENTOFMATHEMATICALSCIENCES,DELAWARESTATEUNIVERSITYDOVER,DE199012277,USA2DEPARTMENTOFMATHEMATICS,FACULTYOFSCIENCE,KINGABDULAZIZUNIVERSITY,JEDDAH,SAUDIARABIARECEIVEDDECEMBER5,2012;REVISEDMANUSCRIPTRECEIVEDAPRIL22,2013ABSTRACTTHISPAPERSTUDIESTHEADIABATICDYNAMICSOFTHEBREATHERSOLITONOFTHESINEGORDONEQUATION.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’。’0’1V2/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,THECONSERVEDQUANTITIESARENOLONGERINTEGRALSOFC2’2’0IJ10QXQDX01D/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.THEVARIABL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breather dynamics of the sinegordon equation
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