Existence of Wilton Ripples for Water Waves with Constant Vorticity and Capillary Effects

Abstract

In this paper we study the water wave problem with capillary effects and constant vorticity when stagnation points are not excluded. When the constant vorticity is close to certain critical values we show that there exist Wilton ripples solutions of the water wave problem with two crests and two troughs per minimal period. They form smooth secondary bifurcation curves that emerge from primary bifurcation branches that contain a laminar flow solution and consist of symmetric waves of half of the period of the Wilton ripples, at some nonlaminar solution. We also prove that any Wilton ripple contains an internal critical layer provided its minimal period is sufficiently small.