On the Construction of Traveling Water Waves with Constant Vorticity and Infinite Boundary

Abstract

The issue of whether there is a closed orbit in the water waves in an infinite boundary condition is an outstanding open problem. In this work, we first discuss the various developments on the structure of water waves in the context of finite bottom conditions. We then focus on the behavior of water for the kinematic boundary for the infinite depth. We present some findings to address this issue by creating a water wave profile for the zero and constant vorticity conditions through the application of the Crandall–Rabinowitz theorem.