ON THE SEMICLASSICAL LIMIT OF THE FOCUSING NONLINEAR SCHRODINGER-EQUATION

We present numerical experiments that provide new strong evidence of t he existence of the semiclassical limit for the focusing nonlinear Sch rodinger equation in one space dimension. Our experiments also address the spatiotemporal structure of the limit. Like in the defocusing cas e, the semiclassical limit appears to be characterized by sharply deli mited regions of space-time containing multiphase wave microstructure. Unlike in the defocusing case, the macroscopic dynamics seem to be go verned by elliptic partial differential equations. These equations can be integrated for analytic initial data, and in this connection, we i nterpret the caustics separating the regions of smoothly modulated mic rostructure as the boundaries of domains of analyticity of the solutio ns of the macroscopic model. For more general initial data in common f unction spaces, the initial value problem is ill-posed. Thus the semic lassical limit of a sequence of well-posed initial value problems is a n ill-posed initial value problem. (C) 1998 Published by Elsevier Scie nce B.V.