  A soliton ensemble is a particular kind of approximation of the solution of
an initial-value problem for an integrable equation by a reflectionless
potential that is well adapted to singular asymptotics like the
small-dispersion limit. We study soliton ensembles for the Benjamin-Ono
equation by using reasonable hypotheses to develop local approximations that
capture highly oscillatory features of the solution and hence provide more
information than weak convergence results that are easier to obtain. These
local approximations are deduced independently from empirically-observed but
unproven distributions of eigenvalues of two related matrices, one Hermitian
and another non-Hermitian. We perform careful numerical experiments to study
the asymptotic behavior of the eigenvalues of these matrices in the
small-dispersion limit, and formulate conjectures reflecting our observations.
Then we apply the conjectures to construct the local approximations of slowly
varying profiles and rapidly oscillating profiles as well. We show that the
latter profiles are consistent with the predictions of Whitham modulation
theory as originally developed for the Benjamin-Ono equation by Dobrokhotov and
Krichever.
