Semiclassical Limit of the Scattering Transform for the Focusing Nonlinear Schrodinger Equation

    Authors

    A. Tovbis;S. Venakides

    Comments

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    Abbreviated Journal Title

    Int. Math. Res. Notices

    Keywords

    SHABAT EIGENVALUE PROBLEM; ASYMPTOTICS; Mathematics

    Abstract

    The semiclassical limit of the focusing Nonlinear (cubic) Schrodinger Equation corresponds to the singularly perturbed Zakharov-Shabat (ZS) system that defines the direct and inverse scattering transforms (IST). In this paper, we derive explicit expressions for the leading-order terms of these transforms, which we call semiclassical limits of the direct and IST. Thus, we establish an explicit connection between the decaying initial data of the form q(x,0)=A(x)e(iS(x)) and the leading order term of its scattering data. This connection is expressed in terms of an integral transform that can be viewed as a complexified version of the Abel transform. Our technique is not based on the Wentzel-Kramers-Brillouin (WKB) analysis of the ZS system, but on the inversion of the modulation equations that solve the inverse scattering problem in the leading order. The results are illustrated by a number of examples.

    Journal Title

    International Mathematics Research Notices

    Issue/Number

    10

    Publication Date

    1-1-2012

    Document Type

    Article

    Language

    English

    First Page

    2212

    Last Page

    2271

    WOS Identifier

    WOS:000304047400003

    ISSN

    1073-7928

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