  The long-time asymptotic behavior of solutions to the focusing nonlinear
Schr\"odinger (NLS) equation on the line with symmetric, nonzero boundary
conditions at infinity is studied in the case of initial conditions that allow
for the presence of discrete spectrum. The results of the analysis provide the
first rigorous characterization of the nonlinear interactions between solitons
and the coherent oscillating structures produced by localized perturbations in
a modulationally unstable medium. The study makes crucial use of the inverse
scattering transform for the focusing NLS equation with nonzero boundary
conditions, as well as of the nonlinear steepest descent method of Deift and
Zhou for oscillatory Riemann-Hilbert problems. Previously, it was shown that in
the absence of discrete spectrum the $xt$-plane decomposes asymptotically in
time into two types of regions: a left far-field region and a right far-field
region, where to leading order the solution equals the condition at infinity up
to a phase shift, and a central region where the asymptotic behavior is
described by slowly modulated periodic oscillations. Here, it is shown that in
the presence of a conjugate pair of discrete eigenvalues in the spectrum a
similar coherent oscillatory structure emerges but, in addition, three
different interaction outcomes can arise depending on the precise location of
the eigenvalues: (i) soliton transmission, (ii) soliton trapping, and (iii) a
mixed regime in which the soliton transmission or trapping is accompanied by
the formation of an additional, nondispersive localized structure akin to a
soliton-generated wake. The soliton-induced position and phase shifts of the
oscillatory structure are computed, and the analytical results are validated by
a set of accurate numerical simulations.
