Stability analysis from fourth order evolution equation for deep water capillary-gravity waves in the presence of air flowing over water

Fourth order nonlinear evolution equations, which are a good starting point for the study of nonlinear water waves as first pointed out by Dysthe (1979) and later elaborated by Janssen (1983), are derived for deep water capillary-gravity waves in the presence of air flowing over water. Stability analysis is then made for a uniform Stokes capillary gravity wave train. Graphs are plotted for the maximum growth rate of instability, the frequency at marginal stability and the frequency separation for fastest growing side-band component as a function of wave steepness. Significant deviations are noticed from the results obtained from the third-order evolution equation, which is the nonlinear Schrödinger equation.