The N-soliton of the focusing nonlinear Schroedinger equation for N large

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1、THEN-SOLITONOFTHEFOCUSINGNONLINEARSCHRODINGEREQUATION¨FORNLARGEGREGORYLYNGANDPETERD.MILLERAbstract.WepresentadetailedanalysisofthesolutionofthefocusingnonlinearSchr¨odingerequationwithinitialconditionψ(x,0)=Nsech(x)inthelimitN→∞.Webeginbypresentingnewandmoreaccuratenumericalreconstruc

2、tionsoftheN-solitonbyinversescattering(numericallinearalgebra)forN=5,10,20,and40.Wethenrecasttheinverse-scatteringproblemasaRiemann-Hilbertproblemandprovidearigorousasymptoticanalysisofthisprobleminthelarge-Nlimit.Forthose(x,t)whereresultshavebeenobtainedbyotherauthors,weimprovetheerr

3、orestimatesfromO(N−1/3)toO(N−1).WealsoanalyzetheFourierpowerspectruminthisregimeandrelatetheresultstotheopticalphenomenonofsupercontinuumgeneration.WethenstudytheN-solitonforvaluesof(x,t)whereanalysishasnotbeencarriedoutbefore,andweuncovernewphenomena.Themaindiscoveryofthispaperisthem

4、athematicalmechanismforasecondarycaustic(phasetransition),whichturnsouttodifferfromthemechanismthatgeneratestheprimarycaustic.ThemechanismforthegenerationofthesecondarycausticdependsessentiallyonthediscretenatureofthespectrumfortheN-soliton,andmoresignificantly,cannotberecoveredfromanan

5、alysisofanostensiblysimilarRiemann-Hilbertproblemintheconditionsofwhichacertainformalcontinuumlimitistakenonanadhocbasis.1.IntroductionItiswellknown(see,e.g.,[13])thatthesolutionoftheinitial-valueproblemforthefocusingnonlinearSchr¨odingerequation12(1)iψt+ψxx+

6、ψ

7、ψ=02withinitialdata(2)ψ

8、(x,0)=Nsech(x)isaspecialsolutioncalledtheN-soliton.Thissolutionisrapidlydecreasingin

9、x

10、andperiodicintwithperiodindependentofN.Tofurtherdescribethissolution,werecallthat(1)isexactlysolvableviatheinverse-scatteringframeworkintroducedbyZakharovandShabatin[16].Therearethreestepsintheproce

11、dure:(i)forwardscattering—theinitialdatageneratesthescatteringdatawhichconsistsofeigenvalues,proportionalityconstants,andareflectioncoefficient;arXiv:nlin/0508007v1[nlin.SI]1Aug2005(ii)timeevolution—thescatteringdatahaveasimpleevolutionintime;(iii)inversescattering—thesolutionofthepartia

12、ldiffe