  We present the results of a numerical experiment inspired by the
semiclassical (zero-dispersion) limit of the focusing nonlinear Schroedinger
(NLS) equation. In particular, we focus on the Gaussian semiclassical soliton
ensemble, a family of exact multisoliton solutions obtained by repeatedly
solving the initial-value problem for a particular sequence of initial data.
The sequence of data is generated by adding an asymptotically vanishing
sequence of perturbations to pure Gaussian initial data. These perturbations
are obtained by applying the inverse-scattering transform to formal WKB
approximations of eigenvalues of the associated spectral problem with a
Gaussian potential. Recent results [Lee, Lyng, & Vankova, Physica D 24
(2012):1767--1781] suggest that, remarkably, these perturbations---interlaced
as they are with the integrable structure of the equation---do not excite the
acute modulational instabilities that are known to be present in the
semiclassical regime. Here, we provide additional evidence to support the claim
that these WKB-induced perturbations indeed have a very special structure. In
particular, as a control experiment, we examine the evolution from a family of
initial data created by an asymptotically vanishing family of analytic
perturbations which are qualitatively indistinguishable from the WKB-induced
perturbations that generate the Gaussian semiclassical soliton ensemble. We
then compare this evolution to the (numerically computed) true evolution of the
Gaussian and also to the evolution of the corresponding members of the
semiclassical soliton ensemble. Our results both highlight the exceptional
nature of the WKB-induced perturbations used to generate the semiclassical
soliton ensemble and provide new insight into the sensitivity properties of the
semiclassical limit problem for the focusing NLS equation.
