Title

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

Authors

Authors

A. Tovbis;S. Venakides

Comments

Authors: contact us about adding a copy of your work at STARS@ucf.edu

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

Share

COinS