Semiclassical Limit of the Scattering Transform for the Focusing Nonlinear Schrodinger Equation
Abbreviated Journal Title
Int. Math. Res. Notices
SHABAT EIGENVALUE PROBLEM; ASYMPTOTICS; Mathematics
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.
International Mathematics Research Notices
"Semiclassical Limit of the Scattering Transform for the Focusing Nonlinear Schrodinger Equation" (2012). Faculty Bibliography 2010s. 3398.