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1、THEN-SOLITONOFTHEFOCUSINGNONLINEARSCHRODINGEREQUATION¨FORNLARGEGREGORYLYNGANDPETERD.MILLERAbstract.WepresentadetailedanalysisofthesolutionofthefocusingnonlinearSchr¨odingerequationwithinitialconditionψ(x,0)=Nsech(x)inthelimitN→∞.Webeginbypresentingnewandmorea
2、ccuratenumericalreconstructionsoftheN-solitonbyinversescattering(numericallinearalgebra)forN=5,10,20,and40.Wethenrecasttheinverse-scatteringproblemasaRiemann-Hilbertproblemandprovidearigorousasymptoticanalysisofthisprobleminthelarge-Nlimit.Forthose(x,t)wherer
3、esultshavebeenobtainedbyotherauthors,weimprovetheerrorestimatesfromO(N−1/3)toO(N−1).WealsoanalyzetheFourierpowerspectruminthisregimeandrelatetheresultstotheopticalphenomenonofsupercontinuumgeneration.WethenstudytheN-solitonforvaluesof(x,t)whereanalysishasnotb
4、eencarriedoutbefore,andweuncovernewphenomena.Themaindiscoveryofthispaperisthemathematicalmechanismforasecondarycaustic(phasetransition),whichturnsouttodifferfromthemechanismthatgeneratestheprimarycaustic.Themechanismforthegenerationofthesecondarycausticdepends
5、essentiallyonthediscretenatureofthespectrumfortheN-soliton,andmoresignificantly,cannotberecoveredfromananalysisofanostensiblysimilarRiemann-Hilbertproblemintheconditionsofwhichacertainformalcontinuumlimitistakenonanadhocbasis.1.IntroductionItiswellknown(see,e.
6、g.,[13])thatthesolutionoftheinitial-valueproblemforthefocusingnonlinearSchr¨odingerequation12(1)iψt+ψxx+
7、ψ
8、ψ=02withinitialdata(2)ψ(x,0)=Nsech(x)isaspecialsolutioncalledtheN-soliton.Thissolutionisrapidlydecreasingin
9、x
10、andperiodicintwithperiodindependentofN.Tof
11、urtherdescribethissolution,werecallthat(1)isexactlysolvableviatheinverse-scatteringframeworkintroducedbyZakharovandShabatin[16].Therearethreestepsintheprocedure:(i)forwardscattering—theinitialdatageneratesthescatteringdatawhichconsistsofeigenvalues,proportion
12、alityconstants,andareflectioncoefficient;arXiv:nlin/0508007v1[nlin.SI]1Aug2005(ii)timeevolution—thescatteringdatahaveasimpleevolutionintime;(iii)inversescattering—thesolutionofthepartialdiffe
