  We study the superfluid dynamics of the outer core of neutron stars by means
of a hydrodynamic model made of a neutronic superfluid and a protonic
superconductor, coupled by both the dynamic entrainment and the Skyrme SLy4
nucleon-nucleon interactions. The resulting nonlinear equations of motion are
probed in the search for dynamical instabilities triggered by the relative
motion of the superfluids that could be related to observed timing anomalies in
pulsars. Through linear analysis, the origin and expected growth of the
instabilities is explored for varying nuclear-matter density. Differently from
previous findings, the dispersion of linear excitations in our model shows
rotonic structures below the pair-breaking energy threshold, which lies at the
origin of the dynamical instabilities, and could eventually lead to emergent
vorticity along with modulations of the superfluid density.
