Semiclassical limit of the scattering transform for the focusing nonlinear Schrödinger equation
The semiclassical limit of the focusing Nonlinear (cubic) Schr ̈ odinger 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. © 2011 The Author(s).
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