Fifth order evolution equation of gravity-capillary waves

Abstract

We extend the evolution equation for weak nonlinear gravity–capillary waves by including fifth-order nonlinear terms. Stability properties of a uniform Stokes gravity–capillary wave train is studied using the evolution equation obtained here. The region of stability in the perturbed wave-number plane determined by the fifth-order evolution equation is compared with that determined by third- and fourth-order evolution equations. We find that if the wave number of longitudinal perturbations exceeds a certain critical value, a uniform gravity–capillary wave train becomes unstable. This critical value increases as the wave steepness increases. doi:10.1017/S144618111700027X