A new application of Crapper's exact solution to waves in constant vorticity flows

In 1957 Crapper found an exact solution for capillary waves propagating at the surface of an irrotational flow of infinite depth. Here we provide conclusive analytical and numerical evidence that a Crapper wave makes the profile of a periodic traveling wave propagating, in the absence of the effects of gravity and surface tension, in a constant vorticity flow. This is achieved by constructing a Stokes expansion up to the third order in a small amplitude parameter and by numerically computing large amplitude waves.