  We study the existence, regularity, and symmetry of periodic traveling
solutions to a class of Gardner-Ostrovsky type equations, including the
classical Gardner-Ostrovsky equation, the (modified) Ostrovsky, and the reduced
(modified) Ostrovsky equation. The modified Ostrovsky equation is also known as
the short pulse equation. The Gardner-Ostrovsky equation is a model for
internal ocean waves of large amplitude. We prove the existence of nontrivial,
periodic traveling wave solutions using local bifurcation theory, where the
wave speed serves as the bifurcation parameter. Moreover, we give a regularity
analysis for periodic traveling solutions in the presence as well as absence of
Boussinesq dispersion. We see that the presence of Boussinesq dispersion
implies smoothness of periodic traveling wave solutions, while its absence may
lead to singularities in the form of peaks or cusps. Eventually, we study the
symmetry of periodic traveling solutions by the method of moving planes. A
novel feature of the symmetry results in the absence of Boussinesq dispersion
is that we do not need to impose a traditional monotonicity condition or a
recently developed reflection criterion on the wave profiles to prove the
statement on the symmetry of periodic traveling waves.
