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

dc.contributor.authorLyng, Gregory D.en_US
dc.contributor.authorMiller, Peter D.en_US
dc.date.accessioned2007-09-20T18:24:36Z
dc.date.available2008-09-08T14:25:13Zen_US
dc.date.issued2007-07en_US
dc.identifier.citationLyng, Gregory D.; Miller, Peter D. (2007)."The N -soliton of the focusing nonlinear SchrÖdinger equation for N large." Communications on Pure and Applied Mathematics 60(7): 951-1026. <http://hdl.handle.net/2027.42/55992>en_US
dc.identifier.issn0010-3640en_US
dc.identifier.issn1097-0312en_US
dc.identifier.urihttps://hdl.handle.net/2027.42/55992
dc.description.abstractWe present a detailed analysis of the solution of the focusing nonlinear SchrÖdinger equation with initial condition ψ( x , 0) = N sech( x ) in the limit N → ∞. We begin by presenting new and more accurate numerical reconstructions of the N -soliton by inverse scattering (numerical linear algebra) for N = 5, 10, 20, and 40. We then recast the inverse-scattering problem as a Riemann-Hilbert problem and provide a rigorous asymptotic analysis of this problem in the large- N limit. For those ( x, t ) where results have been obtained by other authors, we improve the error estimates from O ( N −1/3 ) to O ( N −1 ). We also analyze the Fourier power spectrum in this regime and relate the results to the optical phenomenon of supercontinuum generation. We then study the N -soliton for values of ( x, t ) where analysis has not been carried out before, and we uncover new phenomena. The main discovery of this paper is the mathematical mechanism for a secondary caustic (phase transition), which turns out to differ from the mechanism that generates the primary caustic. The mechanism for the generation of the secondary caustic depends essentially on the discrete nature of the spectrum of the N -soliton. Moreover, these results evidently cannot be recovered from an analysis of an ostensibly similar “condensed-pole” Riemann-Hilbert problem. © 2006 Wiley Periodicals, Inc.en_US
dc.format.extent11033204 bytes
dc.format.extent3118 bytes
dc.format.mimetypeapplication/pdf
dc.format.mimetypetext/plain
dc.publisherWiley Subscription Services, Inc., A Wiley Companyen_US
dc.subject.otherMathematics and Statisticsen_US
dc.titleThe N -soliton of the focusing nonlinear SchrÖdinger equation for N largeen_US
dc.typeArticleen_US
dc.rights.robotsIndexNoFollowen_US
dc.subject.hlbsecondlevelMathematicsen_US
dc.subject.hlbtoplevelScienceen_US
dc.description.peerreviewedPeer Revieweden_US
dc.contributor.affiliationumDepartment of Mathematics, University of Michigan, Ann Arbor, MI 48109 ; Department of Mathematics, University of Wyoming, Laramie, WY 82071-3036en_US
dc.contributor.affiliationumDepartment of Mathematics, University of Michigan, Ann Arbor, MI 48109en_US
dc.description.bitstreamurlhttp://deepblue.lib.umich.edu/bitstream/2027.42/55992/1/20162_ftp.pdfen_US
dc.identifier.doihttp://dx.doi.org/10.1002/cpa.20162en_US
dc.identifier.sourceCommunications on Pure and Applied Mathematicsen_US
dc.owningcollnameInterdisciplinary and Peer-Reviewed


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