  High-multiplicity Kepler systems (referred to as Kepler multis) are often
tightly packed and may be on the verge of instability. Many systems of this
type could have experienced past instabilities, where the compact orbits and
often low densities make physical collisions likely outcomes. We use numerical
simulations to study the dynamical instabilities and planet-planet interactions
in a synthetically generated sample of closely-packed, high-multiplicity
systems. We focus specifically on systems resembling Kepler-11, a Kepler multi
with six planets, and run a suite of dynamical integrations, sampling the
initial orbital parameters around the nominal values reported in Lissauer et
al. (2011a), finding that most of the realizations are unstable, resulting in
orbit crossings and, eventually, collisions and mergers. We study in detail the
dependence of stability on the orbital parameters of the planets and
planet-pair characteristics to identify possible precursors to instability,
compare the systems that emerge from dynamical instabilities to the observed
Kepler sample (after applying observational corrections), and propose possible
observable signatures of these instabilities. We examine the characteristics of
each planet-planet collision, categorizing collisions by the degree of contact
and collision energy, and find that grazing collisions are more common than
direct impacts. Since the structure of many planets found in Kepler multis is
such that the mass is dominated by a rocky core, but the volume is dominated by
a low-density gaseous envelope, the sticky-sphere approximation may not be
valid, and we present hydrodynamic calculations of planet-planet collisions
clearly deviating from this approximation. Finally, we rerun a subset of our
dynamical calculations using instead a modified prescription to handle
collisions, finding, in general, higher multiplicity remnant systems.
