  We consider several different bidirectional Whitham equations that have
recently appeared in the literature. Each of these models combine the full
two-way dispersion relation from the incompressible Euler equations with a
canonical shallow water nonlinearity, providing nonlocal model equations that
may be expected to exhibit some of the interesting high-frequency phenomena
present in the Euler equations that standard "long-wave" theories fail to
capture. Of particular interest here is the existence and stability of periodic
traveling wave solutions in such models. Using numerical bifurcation techniques
we construct global bifurcation diagrams for each system and compare the global
structure of branches, together with the possibility of bifurcation branches
terminating in a "highest" singular (peaked/cusped) wave. We also numerically
approximate the stability spectrum along these bifurcation branches and compare
the stability predictions of these models. Our results confirm a number of
analytical results concerning the stability of asymptotically small waves in
these models and provide new insights into the existence and stability of large
amplitude waves.
