  We study the orbital stability of smooth solitary wave solutions of the
Novikov equation, which is a Camassa-Holm type equation with cubic
nonlinearities. These solitary waves are shown to exist as a one-parameter
family (up to spatial translations) parameterized by their asymptotic endstate,
and are encoded as critical points of a particular action functional. As an
important step in our analysis we must study the spectrum the Hessian of this
action functional, which turns out to be a nonlocal integro-differential
operator acting on $L^2(\mathbb{R})$. We provide a combination of analytical
and numerical evidence that the necessary spectral hypotheses always holds for
the Novikov equation. Together with a detailed study of the associated
Vakhitov-Kolokolov condition, our analysis indicates that all smooth solitary
wave solutions of the Novikov equation are nonlinearly orbitally stable.
