On the Semiclassical Limit for the Focusing Nonlinear Schrödinger Equation: Sensitivity to Analytic Properties of the Initial Data
S. R. Clarke and P. D. Miller
Department of Mathematics and Statistics
Monash University
Clayton, VIC 3168 Australia
Abstract:
We present the results of a large number of careful numerical experiments carried out to investigate the way that the solution of the integrable focusing nonlinear Schrödinger equation with fixed initial data, when taken to be close to the semiclassical limit, depends on the analyticity properties of the data. In particular, we study a family of initial data that have complex singularities an adjustable distance from the real axis. We also make use of a simple relation that provides the exact solution, at the center of the wave field, of the elliptic quasilinear system that appears formally as a leading-order model for the semiclassical dynamics. Among other things, we conclude that the semiclassical limit cannot be expected to be continuous with respect to the initial data, even for real analytic data, if there are certain complex singularities present. We argue that in order to have well-posedness of the semiclassical limit, the correct setting is a physically relevant space of functions with compactly supported Fourier transforms.