  This paper studies the nonlinear stability of capillary-gravity waves
propagating along the interface dividing two immiscible fluid layers of finite
depth. The motion in both regions is governed by the incompressible and
irrotational Euler equations, with the density of each fluid being constant but
distinct. A diverse collection of small-amplitude solitary wave solutions for
this system have been constructed by several authors in the case of strong
surface tension (as measured by the Bond number) and slightly subcritical
Froude number. We prove that all of these waves are (conditionally) orbitally
stable in the natural energy space. Moreover, the trivial solution is shown to
be conditionally stable when the Bond and Froude numbers lie in a certain
unbounded parameter region. For the near critical surface tension regime, we
prove that one can infer conditional orbital stability or orbital instability
of small-amplitude traveling waves solutions to the full Euler system from
considerations of a dispersive PDE model equation.
  These results are obtained by reformulating the problem as an
infinite-dimensional Hamiltonian system, then applying a version of the
Grillakis--Shatah--Strauss method recently introduced by Varholm, Wahl\'en, and
Walsh. A key part of the analysis consists of computing the spectrum of the
linearized augmented Hamiltonian at a shear flow or small-amplitude wave. For
this, we generalize an idea used by Mielke to treat capillary-gravity water
waves beneath vacuum.
