Periodic Capillary-Gravity Water Waves of Small Amplitude

In this paper, we investigate two-dimensional capillary-gravity water waves of small amplitude, which propagate over a flat bed. We prove the existence of a local curve of solutions by using the Crandall–Rabinowitz local bifurcation theory, and show the uniqueness for the capillary-gravity water waves. Furthermore, we recover the dispersion relation for the constant vorticity setting. Moreover, we present a formal stability result for the bifurcation of the laminar solution. In addition, we prove the analyticity of the free surface.