  We adopt a robust numerical continuation scheme to examine the global
bifurcation of periodic traveling waves of the capillary-gravity Whitham
equation, which combines the dispersion in the linear theory of
capillary-gravity waves and a shallow water nonlinearity. We employ a highly
accurate numerical method for space discretization and time stepping, to
address orbital stability and instability for a rich variety of the solutions.
Our findings can help classify capillary-gravity waves and understand their
long-term dynamics.
