Integrable semi-discretization of the coupled nonlinear Schr¨odinger equations英文资料

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1、Integrablesemi-discretizationofthecouplednonlinearSchr¨odingerequationsTakayukiTsuchida†,HideakiUjinoandMikiWadatiDepartmentofPhysics,GraduateSchoolofScience,UniversityofTokyo,Hongo7–3–1,Bunkyo-ku,Tokyo113–0033,JapanReceived29April1998,infinalform25November1998Abstract.Asystemofsemi-discretecoup

2、lednonlinearSchr¨odingerequationsisstudied.Toshowthecompleteintegrabilityofthemodelwithmultiplecomponents,weextendthediscreteversionoftheinversescatteringmethodforthesingle-componentdiscretenonlinearSchr¨odingerequationproposedbyAblowitzandLadik.Bymeansoftheextension,theinitial-valueproblemofth

3、emodelissolved.Further,theintegralsofmotionandthesolitonsolutionsareconstructedwithintheframeworkoftheextensionoftheinversescatteringmethod.arXiv:solv-int/9903013v118Mar1999†E-mailaddress:tsuchida@monet.phys.s.u-tokyo.ac.jpIntegrablesemi-discretizationofthecouplednonlinearSchr¨odingerequations2

4、1.IntroductionTherehasbeenasurgeofinterestinthefamilyofnonlinearSchr¨odinger(NLS)equationsbecauseofitsmanyapplicationstovariouskindsofphysicalphenomena.Intheremarkablepapers[1,2]ZakharovandShabatsolvedtheNLSmodel,∂q∂2qi+−2qrq=0∂t∂x2(1.1)∂r∂2ri−+2rqr=0∂t∂x2withr=∓q∗bymeansoftheinversescatteringm

5、ethod(ISM).AfteranothersuccessoftheISMforthemodifiedKdVequation[3],Ablowitz,Kaup,NewellandSegur[4]unifiedaclassofsolitonequationsbyemployingvarioustimedependencesofthescatteringproblem.TheirformulationiscalledtheAKNSformulation.AnumberofauthorshavestudiedextensionsoftheAKNSformulationandpresented

6、manymodels,whichareintegrablebytheISM[5,6].Amongsuchmodels,asystemofcouplednonlinearSchr¨odinger(CNLS)equations∂q∂2qXmjji+−2qkrk·qj=0∂t∂x2k=1j=1,2,...,m(1.2)∂r∂2rXmjji−2+2rkqk·rj=0∂t∂xk=1isparticularlyremarkableindescribingdiversephysicalphenomena[7,8,9,10,11,12,13,14,15,16,17,18,19].Manakov[8]

7、consideredthetwo-componentCNLSequations((1.2)withm=2)withr=−q∗(j=1,2)asamodelforpropagationjjoftwopolarizedelectromagneticwavesandappliedtheISMtothemodelforthefirsttime.Interesthasrecentlyfocusedonthetwo-componentCNLSequationsinstu