Numerical bifurcation and stability for the capillary–gravity Whitham equation

We adopt a robust numerical continuation scheme to examine the global bifurcation of periodic travelling 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. •Global bifurcation and orbital stability for the capillary–gravity Whitham equation•A robust numerical scheme for a rich variety of periodic travelling waves•A highly accurate numerical method for nonlinear stability and instability•Instability for higher modes and great wave heights for weak surface tension•Methodology useful for capillary–gravity and other nonlinear dispersive waves