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Semiclassical Limit of Focusing NLS for a Family of Square Barrier Initial Data
Author(s) -
Jenkins Robert,
McLaughlin Ken D. TR
Publication year - 2014
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21494
Subject(s) - semiclassical physics , pointwise , mathematics , limit (mathematics) , classification of discontinuities , nls , nonlinear system , mathematical analysis , mathematical physics , initial value problem , classical limit , square (algebra) , quantum mechanics , physics , quantum , biochemistry , chemistry , cytoplasm , nuclear localization sequence , geometry
The small dispersion limit of the focusing nonlinear Schrödinger equation (NLS) exhibits a rich structure of sharply separated regions exhibiting disparate rapid oscillations at microscopic scales. The non‐self‐adjoint scattering problem and ill‐posed limiting Whitham equations associated to focusing NLS make rigorous asymptotic results difficult. Previous studies have focused on special classes of analytic initial data for which the limiting elliptic Whitham equations are wellposed. In this paper we consider another exactly solvable family of initial data,the family of square barriers,ψ 0( x ) = q χ[− L , L ] for real amplitudes q . Using Riemann‐Hilbert techniques, we obtain rigorous pointwise asymptotics for the semiclassical limit of focusing NLS globally in space and up to an O(1) maximal time. In particular, we show that the discontinuities in our initial data regularize by the immediate generation of genus‐one oscillations emitted into the support of the initial data. To the best of our knowledge, this is the first case in which the genus structure of the semiclassical asymptotics for focusing NLS have been calculated for nonanalytic initial data. © 2013 Wiley Periodicals, Inc.