On the semiclassical limit of the focusing nonlinear Schrödinger equation

We present numerical experiments that provide new strong evidence of the existence of the semiclassical limit for the focusing nonlinear Schrodinger equation in one space dimension. Our experiments also address the spatiotemporal structure of the limit. Like in the defocusing case, the semiclassical limit appears to be characterized by sharply delimited regions of space-time containing multiphase wave microstructure. Unlike in the defocusing case, the macroscopic dynamics seem to be governed by elliptic partial differential equations. These equations can be integrated for analytic initial data, and in this connection, we interpret the caustics separating the regions of smoothly modulated microstructure as the boundaries of domains of analyticity of the solutions of the macroscopic model. For more general initial data in common function spaces, the initial value problem is ill-posed. Thus the semiclassical limit of a sequence of well-posed initial value problems is an ill-posed initial value problem.